VP-ellipsis and DRT

Helle Frisak Sem

12th of August, 1994

Contents

1 Discussion of data 6 1.1 Ellipsis and pronouns . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Ellipsis and control . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3 Reflexives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Cascaded ellipsis . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5 Nested ellipsis . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Short sketch of DRT. 29

3 To find one’s way through a DRS. 34 3.1 Some preliminary concepts. . . . . . . . . . . . . . . . . . . . 35 3.2 Linear construction systems. . . . . . . . . . . . . . . . . . . . 38 3.3 Co-syntactic construction systems. . . . . . . . . . . . . . . . 40 3.4 Linear^{+} and co-syntactic rules in [KR93]. . . . . . . . . . . . 43 3.5 Narrow construction systems. . . . . . . . . . . . . . . . . . . 47 3.6 Restricted construction systems. . . . . . . . . . . . . . . . . 55 3.7 Narrow^{+} and restricted^{+} rules in [KR93]. . . . . . . . . . . . 59

4 Extending DRT with ellipsis. 64 4.1 Identification of an elliptic clause. . . . . . . . . . . . . . . . . 64 4.2 The identification of source and parallel elements. . . . . . . . 66 4.3 The dominance relation. . . . . . . . . . . . . . . . . . . . . . 68 4.4 Construction rule for elliptic copy. . . . . . . . . . . . . . . . 71

5 Final remarks. 80

Introduction

In this paper, a framework for analysing ellipsis in DRT is developed. Section 1, contains a discussion of data with VP ellipsis, with special focus on the pattern of readings and the “missing reading” problem. The discussion is held on a pretheoretic level, in order to try and extract from the data some basic requirements that a framework for interpreting ellipsis must fulfil, while avoiding that a theoretical framework impose features or explanations that do not arise from the data themselves, but are artefacts of the theoretical framework. The pattern of missing readings for psychological predicates are found to correlate with other phenomena, like backwards anaphora and scope properties.

The rest of the presentation builds on Part I and Chapter 4 in Part II of [KR93]. Acquaintance with DRT and [KR93] is presupposed. A short sketch of DRT together with a small example is provided in Section 2.

Section 3 develops a theory for detectable DRS construction systems. The aim is to be able to retrieve material from the DRSs without having extra knowledge about the construction history, and from a given condition, corresponding to a subexpression of the discourse that the DRS represents an analysis of, to find all the conditions and referents, and the structural relations that hold between them, just by examining the DRS. DRS construction systems where there is an algorithm for this are called detectable. A set of requirements with respect to the formulation of the DRS construction rules are developed and discussed, and the rules of DRT as formulated in Part I and Chapter 4 in Part II of [KR93] are compared with the standards developed in Section 3. Some minor adjustments of some of the construction rules in [KR93] are suggested in order to obtain a DRS construction system that conforms with the requirements for detectability.

An extension of the fragment of Part I and Chapter 4 in Part II of [KR93] to cover VP-ellipsis, stripping, gapping, pseudogapping, and NP-ellipsis is suggested in Section 4. Finally, the proposed extension is compared with the requirements that were found in Section 1.

The suggestion for ellipsis in DRT still needs development, both with respect to syntactic parallelism between source and elliptic clause, extending the DRT fragment to cover phenomena like control, and tense, and to develop a mechanism for a better algorithm for the order of reduction for reducible conditions in a DRS, in order to cover the scope properties, backwards anaphora and the pattern of missing readings in connection with psychological predicates. A thorough discussion of data for tense and quantified NPs in VP-ellipsis, and other kinds of ellipsis than VP-ellipsis, and how the connected problems can be solved in connection with the proposed construction rule for ellipsis, is needed in connection with the interpretation of ellipsis in DRT.

1 Discussion of data

Ellipsis is a phenomenon that tells you to get information from preceding text. A simple example is the following.

(1) John runs, and Bill does too

The word does on the intended reading does not have its full verb meaning, like in Bill does his homework. The meaning intended is the one that we can paraphrase by John runs, and Bill runs too or John runs, and Bill does run too. We will say that Bill does too is an elliptic clause, and that John runs is its source^{1}. Elliptic constructions may be used for reasons of style or efficiency, and we will see below that the use of ellipsis in connection with other anaphoric expressions may have interesting effects. Even in a simple case like (2), we can see that the use of ellipsis is important also for semantic reasons.

(2) John washed his car, and Bill did too

There are three possible readings of (2), if we count the possible readings where his is taken to refer to some person different from John or Bill as one reading:

(3) (a) Both John and Bill washed their own car (b) Both John and Bill washed John’s car

(c) Both John and Bill washed some third persons car

Without the use of ellipsis, we could use John washed his car, and Bill washed his car (too) to cover all the possible readings of (2). But in addition to the readings mentioned in (3), there would be other readings, since there is no need to read the two occurrences of his in the same way. The first occurrence could refer to some third person, whereas the second occurrence could refer to John or Bill, or the first occurrence could refer to John, whereas the second occurrence could refer to some third person.

One conclusion we can draw already at this point, is that making a copy of the antecedent expression, washed his car, and analysing the sentences separately, will give us too many readings. The antecedent material must be copied on some level when at least parts of the analysis have been done. A minimal requirement to account for the readings of (2) is that it is known whether his refers to John or not:

1Following the terminology of [DSP91].

• Source internal coreference^{2} and non-coreference for the pronouns in the source is given information for elliptic copy.

We will call those elements in the source or elliptic clause that correspond to elements in the elliptic or source clause, respectively, for parallel elements, so that in (2), John is a parallel element of the source, Bill is a parallel element of the elliptic clause, and John and Bill are said to be parallel to each other. When a pronoun in the source clause corefers with a parallel element, the elliptic clause may have several readings. In (3) we see that there are two readings of the elliptic clause in (2) when his and John corefer, one reading where the hidden his of the elliptic clause also corefers with John, the strict copy reading, and one reading where it corefers with Bill, the sloppy copy reading.^{3}

1.1 Ellipsis and pronouns

Is the knowledge of which pronoun occurrences that corefer with which parallel elements in the source for an elliptic clause all we need in order to know what readings the elliptic clause may have? If it is, then all possible combinations of strict copy and sloppy copy readings of the pronouns involved should be possible for the elliptic clause. One example that supports this view is the following^{4}^{5} :

(4) Sandra_{i} sent her_{i} resume to her_{i} agent and Anita did too

The elliptic clause in (4) has 4 readings, which are exactly the four possibilities we have when each of the pronouns of the source may be copied strictly or sloppily for the reading of the elliptic clause:

(5) (a) Anita sent Anita’s resume to Anita’s agent

(b) Anita sent Anita’s resume to Sandra’s agent (c) Anita sent Sandra’s resume to Anita’s agent

2By ‘corefer’ is not meant ‘refers to the same entity’, but ‘must have the same reference whenever they refer’.

3Terminology from [Ros67].

4Example (4) is from [GP90, page 156].

5In the discussion below, the pronouns in the source material, are indexed. This is not to indicate that sentences come with indices, only that this information matters to the analysis of the elliptic clause, as noted in the requirement on page 1. In the discussion of the examples below, we are interested in the readings possible for the elliptic clause when the source has the reading indicated by the indices.

(d) Anita sent Sandra’s resume to Sandra’s agent

Other examples, however, falsifies the assumption that coreference with the parallel elements is all we need^{6}.

(6) Bill_{i} believed that he_{i} loved his_{i} wife, and Harry did too

The elliptic clause of (6) has only 3 readings. The “missing” reading is that:

(7) Harry believed that Bill loved Harry’s wife

One candidate for an explanation of why this reading is missing, whereas no reading is missing for (4), is found appealing to syntax. In (4), none of the occurrences of her c-commands^{7} the other, whereas in (6) he c-commands his. This explanation is supported by looking at what happens when we switch the arguments of the embedded sentence in (6):

(8) Bill_{i} believed that his_{i} wife loved him_{i}, and Harry did too

In (8), him does not c-command his, nor vice versa, and, as expected,

(8) has four readings. That c-command is sufficient to explain the pattern of missing readings is falsified by the following example:

(9) Edith_{i} said that her_{i} husband had upset her_{i}, and Martha did too

We would expect (9) to have four readings, but it has only three. The “missing” reading is that Martha said that Martha’s husband had upset Edith.

6A variant of (6), using so do instead of VP ellipsis is discussed in [OD74].

7A node A c-commands a node B iff A does not dominate B and the first branching node dominating A also dominates B. I am here referring to syntactic analyses that analyse sentences by something like the rule S → NP VP, so that a subject will c-command whatever is in the VP, and where the N is the head of the NP. If the sentence is analysed by something like S → X NP or S → X \overline{S}, where NP or \overline{S} is the last argument of the main verb, the subject will of course not c-command what is in the X or \overline{S}. A DP analysis, in which the determiner is the head of the ‘noun phrase’ rather that the N, destroys the c-command explanation for the pattern of readings for (6) and (8). On the other hand, this kind of analysis would make the examples (9) and (10) support the ccommand explanation. I can see no independent reason why an NP analysis should be used for examples like (6) and (8) whereas a DP analysis should be used for (9) and (10), however. I will thus continue the discussion from the point of view of NP analyses. A similar discussion would be necessary from the point of view of DP analyses.

If we try to switch the arguments in the embedded sentence of (9), like we did for (6), we get an intriguing result!

(10) Edith_{i} said that she_{i} had upset her_{i} husband, and Martha did too

The elliptic clause in (10) has all the four combinatorially possible readings. C-command cannot possibly account for this. In (10), where she c-commands her, there is no effect of the c-command, all four combinations of sloppy and strict copy are possible. In (9) on the other hand, where the second occurrence of her does not c-command the first, one combination is missing.

We may hold on to some aspects of the c-command explanation: On some level, there is a hierarchy between the arguments of a relation; an argument dominates what is embedded^{8} in arguments subordinate to it. We have seen that this dominance pattern do not coincide with the c-command pattern in all cases^{9}. We will use this dominance pattern to explain the pattern of possible readings in case of ellipsis in the following way. If a pronoun that corefers with a parallel element is in a dominant position to some other pronoun coreferent with the same parallel element, then a strict copy reading of the dominant pronoun enforces a strict copy reading of the subordinate. If love is the verb, then it is important for the number of possible readings whether a pronoun as its first argument is copied strictly or not, whereas in the case of upset, it is important whether a pronoun as its second argument is copied strictly or not.

What happens if there are more levels of embedding? The more complex a sentence gets, the more difficult it becomes to judge its readings, but let us try and have a look at the following:

8Reflexives will be discussed in Section 1.3.

9We have only studied dominance patterns for two-argument verbs. Consider the following sentences with the three-argument verb send.

(i) Sandra said that John had sent her her letters (ii) *Sandra said that John had sent her letters her (iii) Sandra said that John had sent her to her agent (iv) Sandra said that John had sent her letters to her I believe that a continuation with and Anita did too will have four readings in cases (i)

and (iv) and three readings in case (iii), indicating that neither the direct nor the indirect object of send in case (i) dominates the other, whereas in the case of the send … to construction, the second argument dominates the prepositional argument.

Edith_{i} said that her_{i} finding her_{i} husband nude had upset her_{i}, and Martha did too (11)

I believe that if the third occurrence of her is copied strictly, that is, if the one that gets upset in the reading of the elliptic clause of (11) is Edith, then the only possible reading is that Martha said that Edith’s finding Edith’s husband nude had upset Edith. If Martha is the one that gets upset, it seems that if Edith does the finding, she must find her own husband, whereas if Martha does the finding, she might find either her own husband, or Edith’s. The “missing” readings are:

(12) (a) Martha’s finding Martha’s husband nude upsets Edith (b) Martha’s finding Edith’s husband nude upsets Edith

(c) Edith’s finding Martha’s husband nude upsets Edith

(d) Edith’s finding Martha’s husband nude upsets Martha

This means that we have support for the theory that the strict or sloppy options for elliptic copy is sensitive to a hierarchy where the relations involved decides which argument dominates which other arguments.

What is it that makes love and find different from upset with respect to which of their arguments dominate the other semantically? In the case of love, only the first argument need be aware of what happens, the second argument need not even be animate, as in, e.g., John loves green marmalade. In the case of upset, the second argument is the one that must be aware of what happens, whereas the first argument need not be animate, as, e.g., in Green marmalade upsets John.

This suggests that the semantic roles of the arguments of the verbs, or thematic roles^{10} (θ-roles) have effects on the anaphoric relations. Psychological predicates, with one experiencer argument and one stimulus argument are of particular interest to us. These predicates allow in many languages for a double lexicalisation, one where the experiencer is realised as the subject, and one where the stimulus is realised as the subject^{11} , i.e., like/please, fear/frighten, regard…as XP/strike…as XP, be concerned/concern. The pattern of readings of elliptic clauses with psychological predicates in the source are interesting:

10A presentation of the linguistic discussion about thematic roles will not be given here.

See [Dow87, Dow89], [Hel88], [Jac72, Jac87]. See also [Eng90].

11See [Dow87] or the presentation in [Eng90] for an explanation from the point of view of thematic roles, or rather, Agent Proto Roles and Patient Proto Roles for how these verbs can have a double lexicalisation and at the same time be subjected to a thematic ordering constraint with respect to which argument is realised as the subject and which argument is realised as the object.

(13) (a) Bill_{i} says that he_{i} likes his_{i} wife, and Harry does too (b) Bill_{i} says that his_{i} wife pleases him_{i}, and Harry does too

(c) Missing: Harry says that Bill likes/is pleased by Harry’s wife

(14) (a) Edith_{i} says that she_{i} fears her_{i} husband, and Martha did too (b) Edith_{i} says that her_{i} husband frightens her_{i}, and Martha did too (c) Missing: Martha says that Edith fears/is frightened by Martha’s

husband

(15) (a) Bill_{i} says that he_{i} regards his_{i} wife as pompous, and Harry does

too

(b) Bill_{i} says that his_{i} wife strikes him_{i} as pompous, and Harry does too

(c) Missing: Harry says that Bill regards as pompous/is struck by the pompousness of Harry’s wife

In (13)-(15), the (a) and (b) sentences contain embedded sentences with verbs that mean approximately the same, apart from that what is surfaced as the first argument in the (a) examples is surfaced as the second argument in the (b) examples; the one who likes is the one that gets pleased, the one that fears is the one that gets frightened and the one that regards is the one that gets struck, all being the roles that has the perspective in the respective sentences. The experiencer role in the embedded sentence in the source clause in each of (13.a,b)-(15.a,b) is a pronoun coreferring with the parallel element, and the other role in the embedded sentence contains a pronoun that also corefers with the parallel element. The elliptic clause in (13)-(15) all have only three readings, (a) and (b) in each case lack the reading indicated by (c). It is semantically approximately the same reading for the two cases of each pair, but since the syntax is different between the (a) and (b) cases, it is the strict-sloppy reading that is missing forthe (a) cases, and the sloppy-strict reading that is missing for the (b) cases. Switching the arguments in the same way as we did from (6) to (8) and from (9) to (10) will in each of the examples (13)-(15) lead to four readings. No matter which argument is realised as the subject, it is for these verbs the experiencer role that is the dominant with respect to the anaphoric connection that is important for strict and sloppy copy readings.

That the semantic roles of the arguments of verbs have effects on the anaphoric connections of their arguments have been noted elsewhere, see [Eng90] and the references therein.

(16) (a) Every father_{i} loves his_{i} children.

(b)*Its_{i} father loves every child_{i}.

(c)*Every child_{i} concerns its_{i} father.

(d) His_{i} children concern every father_{i}.

In (16.a,b) we see the distribution of a universal quantifier like every and a pronoun bound by it in the case of the predicate love. If every NP ccommands the pronoun in case the predicate is love, then the pronoun becomes bound by the quantifier, if not, then the binding fails. In (16.c,d)^{12}, where the predicate in question is the psychological concern, the situation is reversed. The c-commanding quantifier fails to bind the pronoun in (16.c), whereas the c-commanding pronoun is bound by the c-commanded quantifier in (16.d).

Engdahl finds the same effect with respect to reflexives^{13}:

(17) (a)*Sam concerns himself^{14}

(b) Sam is concerned about himself

Engdahl argues against an explanation based on a thematic hierarchy for why some anaphoric relations are impossible even though they seem fine from a point of view of syntactic structure. She gives two reasons, the first one being that even though it may serve to explain why

(18) *Sam was shaved by himself

is out, it cannot be extended to bound pronouns in general. We have seen above, and Engdahl shows it in more detail, that psychological predicates have similar effects for reflexives and anaphora bound by quantifers. An example corresponding to (18) with a pronoun bound by a quantifier is fine:

12From [Eng90, page 386].

13From [Eng90, page 386].

14Only out on the reading that can be paraphrased as “something about Sam causes him to be concerned”. OK on other readings (“Sam concerns himself with …”).

(19) Every man was shaved by his brother

The other reason is that an example like (17.a) shows that we would have to postulate two hierarchies, one for subjectivisation and one for reflexivisation. “But if we have to appeal to two distinct hierarchies, the explanative potential of the thematic hierarchy approach is considerably diminished.”^{15} Engdahl’s own proposal is to look at another factor, namely the perspective of the situation described by the sentence. She writes^{16}:

“When interpreting a sentence where one of the arguments is an experiencer, it is natural to take that participant’s perspective. However, if that argument is a bound anaphor, we cannot establish the perspective until we have resolved what the antecedent for the bound anaphor is. Hence the sensation of indirectness that seems to crop up when we are faced with such sentences.”

Behind this sensation of indirectness lies the idea that a bound anaphor must get its reference through the reference of its antecedent in order to covary with it. Let us assume that in general, not only in case of quantifiers and bound anaphora, a subordinate element coreferent with a dominant element is not only coreferent with it, but gets its reference through the reference of the dominant element. The dominant element must then be resolved before the subordinate, in order to establish the coreference for the subordinate element. This explains the distribution of anaphora for Engdahl, since she claims that an argument having the perspective cannot get its analysis through another argument of the predicate of which it is itself an argument, without loosing the perspective for that predicate. It also will explain the distribution of strict and sloppy copy readings for us, under the same assumption of dominance for the argument having the perspective.

Our explanation goes like this: Elements that are recognised as coreferent with parallel elements get a sloppy copy reading in case of ellipsis. All other elements get a strict copy reading. Elements need not be recognised as coreferent with parallel elements even if they are. But if an element gets its reference through the reference of some other element, the former element cannot independently of the latter element be recognised as coreferent with a parallel element.^{17}

15[Eng90, page 388].

16[Eng90, page 388].

In [Eng90, pages 388ff], she points out that some psychological predicates may have two slightly different meanings, one stative, which we can paraphrase as “something about the 1st argument …” and one active, which we can paraphrase as “something the 1st argument does …”. The active meaning can in these cases be enforced by adding deliberately to the first argument, or putting it in the progressive form. If we have the active meaning, the first argument is understood to control the action, and, e.g., backwards anaphora is no longer possible^{18}. The internal perspective is in this case on the first argument, and not on the second. Engdahl argues that “the internal perspective cannot be linked automatically to a particular argument position, e.g., the subject, or to a particular argument role, e.g., the experiencer, but that the internal perspective gets established in a fashion depending on aspectual properties and properties like agentivity and controllability”^{19}.

What happens if we add deliberately to examples like (9) and (10)?

(20) (a) Edith_{i} said that her_{i} husband deliberately had upset her_{i}, and Martha did too

(b) Edith_{i} said that she_{i} deliberately had upset her_{i} husband, and Martha did too

17This recognition of coreference with parallel elements can also be used as an intuitive explanation of what goes on in the analysis in [OD73, OD74], where a strict copy arises from evaluating the value of an embedded function before making the elliptic copy, and thus loosing track of possible elements that may be coreferent with the parallel element inside this embedded function. Thus all elements embedded in that function is given a strict reading, too.

18Engdahl uses, e.g., amuse as an example:

(i) Her_{i} children amuse every mother_{i} (ii) *Her_{i} children are amusing every mother_{i} (iii) *Her_{i} children deliberately amuse every mother_{i} Both the progressive form and the addition of deliberately enforces us to read amuse as having an active reading, where the subject is in control of the action. As we see, this has consequences for the anaphoric connections. In (i), the perspective is on the second argument, and the backwards anaphoric connection between her and every mother is not only OK, but the only one acceptable unless we want the active reading of amuse. In (ii)

and (iii), the perspective is on the first argument, and a backwards anaphoric connection is not possible.

19[Eng90, page 390].

In (20), (a) has four readings and (b) has three, lacking the reading that Martha said that Edith deliberately had upset Martha’s husband. This is in accordance with what we would predict, since when we add deliberately, we get the active reading of had upset, where the subject is in control of the action, and the internal perspective is then on the subject.

We conclude this section by summing up:

• The syntactic hierarchy (c-command) is not appropriate to account for the missing readings of elliptic clauses with pronouns in their source.

• On some level, there is a (partial) dominance order between the arguments of each predicate. This dominance order coincides with the dominance order given by the c-command hierarchy unless this conflicts with the demand that an argument that has the internal perspective dominates other arguments.

• The semantics of a predicate, and aspectual properties and properties like agentivity and controllability decide which of its arguments have the internal perspective over elements in other arguments of this predicate.

• If an element is subordinate to a coreferent element according to the dominance order, then the subordinate element gets its reference through the element that dominates it.

• Elements that are recognised as coreferent with a parallel element gets a sloppy copy reading, other elements get a strict copy reading.

• Elements need not be recognised as coreferent with parallel elements even if they are.

• An element cannot be recognised as coreferent with a parallel element unless all elements that it gets its reference through are.

1.2 Ellipsis and control

We have seen that pronouns coreferring with parallel elements in a source for an elliptic clause may lead to ambiguity. We now consider control, where non-overt arguments of embedded clauses may corefer with parallel elements:

(21) John wants to go skiing and Bill does too

In (21), the subject of skiing corefers with John. The sentence John wants to go skiing means that John wants for John to go skiing^{20}. There is no ambiguity with respect to what the elliptic clause in (21) means, it can only mean that Bill wants for Bill to go skiing. The controlled argument does not seem to have the option of being copied in an independent way. This is confirmed by the more complex example (22)^{21}.

Edith_{i} said that finding her_{i} husband nude had upset her_{i}, and Martha did too (22)

Due to the fact that it is the second argument that has the perspective in finding her husband nude had upset her, and this second occurrence of her corefers with the subject of the source clause, I believe that if Edith is the controller of finding in (22), this can only be through the second occurrence of her. That is, the subject of finding iscontrolled by the second occurrence of her, which again is coreferent with Edith. It seems very difficult to imagine the first occurrence of her as the controller^{22}. With the second occurrence of her as the controller of the subject of finding, (22) has 3 readings.

(23) (a) Martha said that Martha’s finding Martha’s husband nude had

upset Martha

(b) Martha said that Martha’s finding Edith’s husband nude had upset Martha

20We will say that John in this case controls, or is the controller of, the subject of skiing. The non-overt subject of skiing is referred to as the controlled argument.

21This example is originally from [Sag76]. It is also discussed in [FM92, pages 22-23], where the combination of control and c-command is used to explain the missing reading of (22). If the second occurrence of her is taken to control the subject of finding, this works, since the subject of finding c-commands the first occurrence of her. We have already seen, however, that the corresponding reading is lacking for (9), and that c-command cannot explain this. That the corresponding reading is missing even for (24) further supports that the combination of control and c-command is not the correct explanation for the missing reading of (22)

22If we are in a setting where we are talking about adultery and the sad experience of discovering that your spouse is cheating on you, the following may be an example where the first occurrence of her controls the subject of finding.

(i) Edith_{i} said that finding her_{i} husband in bed with a stranger had upset her_{i}, and Martha did too

If the first occurrence of her controls the subject of finding, we would predict three readings, namely the readings corresponding to readings (23.a,c) and a reading corresponding to (23.b), except that the one that did the finding was Edith, not Martha.

(c) Martha said that Edith’s finding Edith’s husband nude had upset Edith

This is exactly the same pattern of readings as for (9), with the addition that the one that did the finding was the same as the one that got upset. If the finder is explicitly mentioned, as in (24), we have a similar pattern of readings. The elliptic clause in (24), lacks the reading where Martha said that Sandra’s finding Martha’s husband nude had upset Edith.

Edith_{i} said that Sandra’s finding her_{i} husband nude had upset her_{i}, and Martha did too (24)

Summing up the discussion in this section, we conclude:

• A controlled argument in the source of an elliptic clause does not have an independent status with respect to ellipsis. The controlled argument must have the same reading as its controller also in the elliptic clause.

1.3 Reflexives

If both the dominant and a subordinate argument of a relation are pronouns, and these corefer, then one of them must be a reflexive. There is a discussion^{23} with respect to whether reflexives must be given a sloppy copy reading in case of ellipsis, or whether they can also have a strict copy reading in examples like

(25) John can’t save himself, but his brother can

If the elliptic clause in (25) can have a strict copy reading as well as a sloppy copy one, then we must conclude that the reflexive himself can be copied as if it were a non-reflexive pronoun coreferent with the parallel element. It has to be reflexive in the source in order to be coreferent with John, but the coreference with the subject of save is not a property that is preserved by ellipsis.

Both control and reflexives are sometimes claimed to only have sloppy readings in case of ellipsis, they are ‘obligatory sloppy’ cases. Obligatory sloppiness in some cases also seems to be well in accordance with the view that pronoun occurrences are encoded for whether they can be copied strictly or sloppily in case of ellipsis. We saw above in connection with control that ‘obligatory sloppy’ is not a very good term, and that it should better be replaced by, e.g., ‘obligatory coreferent with its antecedent’. This move is motivated by, e.g., (26) and examples with control, like (22).

23See [Sag76], [Wil77], [Kit91], [DSP91], [Hes92].

(26) Bill_{i} believed that he_{i} saved himself_{i}, and Harry did too

If himself is obligatory coreferent with some element in the reading of the elliptic clause in (26), it is with the copy of he and not with Harry. In case we have this obligatory coreference, we would predict only two readings of the elliptic clause in (26), namely the readings where Harry believed that someone saved himself, where someone is either Bill or Harry. These are clearly possible readings of (26). If (26) can also have the reading that Harry believed that Harry saved Bill, we have support for the hypothesis that reflexives do not have obligatory coreference with their antecedent in the reading of elliptic clauses. The fourth possibility, that Harry believed that Bill saved Harry is clearly not a reading of (26). This is in accordance with our predictions regarding the dominance effect on strict and sloppy readings, since the subject of save dominates the object.

We find support for the hypothesis that the copy of reflexives do not have to corefer with the copy of their antecedent in the readings of elliptic clauses in the following example^{24}:

(27) John tried to kill himself before Bill did

The possible readings of (27)^{25} seem to be both that Bill tried to kill Bill and that Bill tried to kill John. We have earlier argued that a controlled argument does not have independent status with respect to ellipsis, and that its copy in the reading of the elliptic clause is obligatory coreferent with the copy of its controller. Thus, Bill is correctly predicted to be the only one that can try to kill someone in the readings of the elliptic clause of (27). A possible option for an independent strict or sloppy copy reading both in (25) and (27), as well as in (26), points at the reflexiveness of reflexives like himself is not represented on the level where ellipsis takes place, at least for those speakers who accept these readings.

24(27) is discussed in [DSP91, page 443-444], where they argue against the analysis of control in [Chi83, Chi84]. Their arguments presuppose that the reflexiveness is coded in the relevant representation, so that John tried to kill himself must be represented as try(John,λx.kill(x,x) and not try(John,λx.kill(x,John).

25We consider only the case where the source for Bill did is the full John tried to kill himself, and not some source lacking tried to.

1.4 Cascaded ellipsis

If the source of an elliptic clause is ambiguous, then the elliptic clause is too. Ellipsis have been used to test whether an alleged ambiguity is a real ambiguity, and not vagueness or lack of specification, see [ZS74]. If one type of reading is possible in the source, and another type of reading is possible in the elliptic clause, the source clause cannot be ambiguous, but must contain a vague expression. If only the same type of readings are possible in both source and elliptic clause, then the expression in question is ambiguous.

An important question in connection with the analysis of ellipsis is whether and to what extent, the elliptic copying mechanism is flexible, and may induce ambiguities, or whether the elliptic copying mechanism is exact, so that any ambiguity in the elliptic clause must come from an ambiguity in the source clause.

To see if the source and elliptic clause have only the same types of readings, or if mixed readings are allowed, it must of course be possible to establish by independent means which types of readings are involved. Thus, in a case like (2)^{26} , it is not sufficient to establish that the elliptic clause is ambiguous to claim ambiguity for the source clause:

(2) John washed his car, and Bill did too

To be able to claim ambiguity in the source, we must be able to decide whether we have the possibility of two different types of readings for the source and elliptic clauses or not. However, in any case John washes his own car, he also washes John’s car, and in any case he washes John’s car, he also washes his own car. For (2) there is no way we can decide whether there are combinations of the two different types of readings or not, unless we assume that the source is ambiguous in the first place, and that ellipsis is sensitive to this ambiguity, making the argument circular. To claim that the elliptic clause is clearly ambiguous is not sufficient to demonstrate ambiguity in the source, since we do not know independently whether ambiguity may arise from something in the elliptic copying process.

26We disregard the possibility that his corefer with someone else than John. That the source clause of (2) isambiguous between a reading where his corefers with John and readings where it doesn’t, is beyond doubt. This can also be tested by means of ellipsis, since there are no crossed readings, that is, readings where his corefers with John in the source and someone apart from John or Bill in the elliptic clause, or where his corefers with someone different from John in the source, but has changed to someone else in the elliptic clause.

Cascaded ellipsis, that is elliptic clauses in a row, have been used both to claim that the elliptic copying process is flexible and that it is exact^{27}. Note that we must be very careful when studying cascaded ellipsis examples in this way, and take into account not only the ellipsis copying mechanism, but also the choice of source and parallel elements. Only expressions in the source of an elliptic clause coreferring with parallel elements in that source have the option of sloppy copy readings for the elliptic clause, in addition to strict copy readings.

The following example comes from [OD73], as an English paraphrase of an example from [Sch73].

John_{i} realises that he_{i} is a fool, but Bill_{j} doesn’t, even though his_{j} wife does (28)

Dahl claims that (28) has 3 readings, namely that:

(29) (a) Bill doesn’t realise that Bill is a fool, even though Bill’s wife_{i} realises that she_{i} is a fool

(b) Bill doesn’t realise that Bill is a fool, even though Bill’s wife realises that Bill is a fool

(c) Bill doesn’t realise that John is a fool, even though Bill’s wife realises that John is a fool

We agree with Dahl in that (29.b) is a possible reading of (28), but let us first assume that this reading does not exist before we discuss the case that it exists.

In [Sag76], Sag claims that (28) lacks the reading (29.b). In case it does, we have support for the view that pronouns are encoded for whether they must be copied strictly or sloppily on the level where elliptic copying takes place. This support is not dependent on whether John realises that he is a fool or the first elliptic clause is chosen as the source for the second elliptic clause:

If John realises that he is a fool is the source for both elliptic clauses, then the lack of crossed readings between the first and second elliptic clauses support the hypothesis that crossed readings between an elliptic clause and its source is not possible, implying that the source is ambiguous.

27See discussions in [OD73], [DSP91, pages 423ff] and [Sem93], in favour of flexible copy analyses and [ZS74, page 22], [Sag76, pages 135ff], [GP90, pages124-125] in favour of exact copy analyses.

If the first elliptic clause is the source for the second elliptic clause, then the lack of crossed readings between these show on the one hand that the first elliptic clause is ambiguous between the strict reading and the sloppy reading, which we already knew. On the other hand, if the first elliptic clause on a sloppy reading leads only to a sloppy reading for the second elliptic clause, this points at the hidden pronoun in the first elliptic clause is encoded for a sloppy reading on the representation level where elliptic copy takes place. It is in accordance with this to assume that the pronoun in John realises that he is a fool is also encoded for a sloppy or strict copy option.

Thus, if (28) lacks reading (b), it is fair to claim that one finds support for the hypothesis that the ambiguity resides in the source.

That a fourth reading does not exist, namely that he is copied strictly for the reading of the first elliptic clause, but has a sloppy reading for the second elliptic clause, can only be used as an argument in favour of the ambiguity residing in the source if John realises that he is a fool is taken to be the source for both elliptic clauses. If the first elliptic clause, or some level in the analysis of it, is taken to be the source for the second elliptic clause, then on the reading that Bill doesn’t realise that John is a fool, there is no element coreferring with the parallel element Bill for which there is a sloppy copy option. We will argue below that the first elliptic clause is not only possible as a source for the second elliptic clause in cases like (28), but also the most likely source unless specific instructions are given to take the original source.

Let us now discuss the case that mixed readings like (29.b) is a possible reading of cases of cascaded ellipsis like (28). Even when accepting (29.b) as a possible reading of (28), we do not have a clear cut counterexample to the claim that the source must be ambiguous, since there may be some special mechanism that allows for the his in his wife to be the one that is himself a fool^{28}. With such a special mechanism, (29.b) is not seen as a sloppy-strict reading, but as a sloppy-sloppy reading where the hidden his in the second elliptic clause is coreferent with an element embedded in the parallel element his wife, instead of with the parallel element itself. In order to test if a sloppy-strict reading is possible in cases of cascaded ellipsis, we should therefore find an example where there are no embedded elements in the parallel element of the second elliptic clause, and see if we still find the reading corresponding to (29.b) possible:

28See, e.g., [FM92, pages 28-29], where the reading of Max thinks he is strong, Oscar does, too, but his father doesn’t that Oscar thinks that Oscar is strong but Oscar’s father doesn’t think that Oscar is strong is explained by such a mechanism.

John_{i} didn’t wash his_{i} car, but Bill did, even though Harry already had (30)

We think that (30) has three readings, exhibiting the same pattern as for (28). In particular, it has the reading that Bill washed Bill’s car even though Harry already had washed Bill’s car. It is hard to see how this reading could be made possible by the change of polarity and auxiliary verb, if the ambiguity with respect to the strict and sloppy copy is already encoded in John didn’t wash his car. The use of change of polarity, change of tense or auxiliary, and contrasting conjunctions like but and even though may serve to make this reading more plausible than otherwise, however.

Another test to whether a special mechanism picking up elements embedded in the parallel element of the elliptic clause enabling a seemingly strict copy reading in the second elliptic clause when it is in fact a sloppy copy one might have been the following example:

(31) John_{i} washed his_{i} car, while Bill didn’t, but John’s brother did

If such a mechanism is accessible, one would expect to find the reading of (31) where Bill didn’t wash Bill’s car and John’s brother washed John’s car, at least if John washed his car is the source for the second elliptic clause. We take it that (31) has 3 readings, one consistently strict, one consistently sloppy, and one where Bill didn’t wash Bill’s car but John’s brother washed Bill’s car, following the same pattern as (28) and (30).

The pattern of possible readings for (28), (30), and (31) suggests that in each of these cases, the first elliptic clause, and not its source, is the source for the second elliptic clause. It is therefore interesting to try and find out what pattern of readings we obtain if we try forcing the source of the first elliptic clause to be the source of the second elliptic clause too:

John_{i} washed his_{i} car, and Bill did too, even though in John’s case, Harry already had (32)

If the ambiguity arises in the copying mechanism, and even though in John’s case can be read as an instruction to use John washed his car as the source for the second elliptic clause, our predictions are that (32) has three readings, namely that:

(33) (a) Bill washed Bill’s car, even though Harry washed Harry’s car (b) Bill washed Bill’s car, even though Harry washed John’s car (c) Bill washed John’s car, even though Harry washed John’s car

Only the readings (33.b,c) seem to come easy here. This is not surprising, since even though in John’s case and already points at Harry’s washing has some important connection to John’s washing, like if, normally, John does not wash his car when Harry has done what he is claimed to have done. Harry’s having washed Harry’s car doesn’t seem to be any sensible reason for John not to wash his, though. In a context where it is a well known fact that John is determined never to wash his car if Harry has just washed his, this reading seems more likely, however.

John_{i} washed his_{i} car, and Bill did too, even though in the first case, Harry already had (34)

In example (34), the phrase in the first case seems to be an instruction on a level more close to the surface structure than in John’s case, which seems to be on a more semantic level. Also for (34), the readings corresponding to (33.b,c) are the readings that come easily. Under the same context assumptions as for (32), I think that a reading corresponding to (33.a) is equally acceptable for (34) as for (32).

The above examples suggest that the ellipsis copying mechanism is flexible with respect to strict and sloppy copy readings of elements coreferring with parallel elements in the source. If each pronoun coreferent with a parallel element was encoded for whether it could induce a strict or sloppy copy reading in case of ellipsis, and that this encoding was preserved by the elliptic copy mechanism, we couldn’t explain the pattern of possible readings for (30)-(32) and (34). If the pronouns are being encoded for strict versus sloppy copy options we would get the same number of readings independently of whether an earlier elliptic clause or its source is chosen as the source for a later elliptic clause in cascaded ellipsis. But in (30)-(34) we found that the choice of source did make a difference with respect to what readings were obtained. This points at the pronouns not being encoded for whether they can induce a strict or sloppy copy reading in case of ellipsis, and that a flexible copy analysis will explain the facts better than an exact copy analysis. If an elliptic clause can be used as the source for another elliptic clause, we must also conclude that the elliptic copying process cannot take place on a very surface close level. In order to serve as a possible source for a later elliptic clause, the analysis, including elliptic copying and the decision of coreference for pronouns, must have taken place.

The following example is put forward in [GP90, page 124] in defence of exact copy analysis.

(35) John_{i} revised his_{i} paper before Bill did, but after the teacher did

There are only two possible readings of (35), the consistently sloppy one and the consistently strict one. We have argued at length above to show that we do need flexible copy, and the existence of (35) cannot take away the readings possible for other sentences. But how can we explain that there are only two readings for (35)? If we accept the readings corresponding to (29.b) for the cases of cascaded ellipsis above, we cannot explain the lack of a corresponding reading for (35) by appealing to pronouns encoded for strict versus sloppy readings or using an exact copy analysis. But there are structural differences between the cascaded ellipsis examples above and (35). In (35), we do not have three conjoined sentences on the same level, but one main sentence with two subordinate, and conjoined, sentences. This may induce a stronger demand for parallelism than were necessary for the cases above.

An example like (35) is not in itself enough evidence to refute flexible copy analyses. A flexible copy analysis is not prevented from interacting with other principles. On the other hand, just one solid example of cascaded ellipsis where one clause is read sloppily and the other strictly, is enough to falsify the exact copy analyses. On the basis of the above discussion, we conclude:

• Exact copy analysis for ellipsis cannot account for the facts

1.5 Nested ellipsis

The example (36) is discussed both in [GP90, page 122ff] and in [DSP91, page 432ff].

(36) John_{i} revised his_{i} paper before the teacher did and Bill did too.

In (36) there is a source for an elliptic clause, John revised his paper before the teacher did which itself contains an elliptic clause and its source. I choose to call this nested ellipsis. The outermost source clause is clearly ambiguous, it is ambiguous between a reading where both the teacher and John revised John’s paper, and a reading where both the teacher and John revised their own paper. This ambiguity must also show up for the elliptic clause that has John revised his paper before the teacher did as its source; there is no reading in which the teacher revised John’s paper in John’s case, but his own paper in Bill’s case, or in which the teacher revised his own paper in John’s case, but Bill’s or John’s paper in Bill’s case. All combinations respecting this restriction is listed below^{29}:

(37) (a) John revised John’s paper before the teacher revised John’s paper, and Bill revised John’s paper before the teacher revised John’s paper.

(b) John revised John’s paper before the teacher revised the teacher’s paper, and Bill revised Bill’s paper before the teacher revised the teacher’s paper.

(c) John revised John’s paper before the teacher revised John’s paper, and Bill revised Bill’s paper before the teacher revised Bill’s paper.

(d) John revised John’s paper before the teacher revised the teacher’s paper, and Bill revised John’s paper before the teacher revised the teacher’s paper.

(e) John revised John’s paper before the teacher revised John’s paper, and Bill revised Bill’s paper before the teacher revised John’s paper.

(f) John revised John’s paper before the teacher revised John’s paper, and Bill revised John’s paper before the teacher revised Bill’s paper.

Readings (37.a,b) are the consistently strict and the consistently sloppy readings. There is no disagreement as to whether (36) has these readings. Reading (37.c) is perhaps the preferred reading of (36). This may be so because teachers are known to revise the papers of their pupils, and John and Bill may quite likely be pupils. Gawron and Peters recognise only these three readings for (36). If we swap the teacher and Bill in (36), the reading corresponding to (37.d) seems more plausible^{30}.

29We are disregarding that the teacher may be read relationally, so that there may be one or two teachers involved here. We are also disregarding the possibility that John revised his paper may be the source for Bill did.

30The readings where the subject and hidden pronoun in the before clause in the source are coreferent with their counterparts in the elliptic clause may coincide with the reading

Reading (37.e) is difficult to obtain. [DSP91] argues in favour of it, referring to the following^{31}:

Dewey announced his victory after the newspapers did, but so did Truman.

(38)

The historical facts are that Dewey announced Dewey’s victory after the newspapers announced Dewey’s victory, and Truman announced Truman’s victory after the newspapers announced Dewey’s victory.

The reading (37.f) is not a possible reading of (36).

If each of the readings corresponding to (37.a-e) can be demonstrated to exist for sentences which are structurally similar to (36), we must be able to explain all these readings. The source in (36) has only one pronoun, and it is very difficult to explain five different readings of (36) if this pronoun is encoded for strict or sloppy in the first place, and this encoding is inherited by its elliptic copies^{32}.

obtained if the before-clause is read as a reference to a point in time, and do not take part in the ellipsis of the last elliptic clause other than as a reference to this specific point in time. We can take away this possibility by explicitly making it necessary to recalculate a fresh point in time in the interpretation of Bill did too, as in viz., Today, John_{i} revised his_{i} paper before the teacher did and on Monday Bill did too, ensuring a properly nested ellipsis. We can find the same pattern of readings with respect to strict and sloppy possibilities in this case.

31We can obtain this reading for (36) also. Assume that the teacher always starts revising the pupils papers by revising John’s paper. Furthermore, suppose that John and Bill has as a specific goal to revise their own papers before the teacher starts revising papers.

32[FM92, footnote 28] argues that only the readings corresponding to (37.a-c) are possible readings for a similar example, but that (37.d,e) are marginally acceptable for some speakers. In the framework of [FM92], pronoun occurrences are marked for independent reference or dependent reference; pronouns have α and β occurrences. The reference of α occurrences are fixed, whereas β occurrences may change reference, regulated by what is called dependencies. If a β occurrence change reference, the dependencies that hold in the elliptic clauses must be so-called i-copies of dependencies in the source. If one pronoun occurrence c-commands another, it will show up in all dependencies which cover an expression containing both occurrences. And if two such pronouns are coreferent with a parallel element in the source, any dependency for the elliptic clause between the c-commanded pronoun and the ellipsis parallel element must also contain the c-commanding pronoun, and since both of these are coreferent with the parallel element in the source, the possibility of a strict copy for the c-commanding pronoun together with a sloppy copy of the c-commanded pronoun is ruled out. Reading (37.c), in which there are both strict and sloppy copies, all copies are explained as dependent by means of the subordinate status of the before clause, so that a dependency may hold between the subject of the superordinate clause, John in the source and Bill in the outermost elliptic clause, and the hidden

To get a better overview of the possible readings of (36), it is useful to look at the two different possible readings of its large source clause separately:

(39) John_{i} revised his_{i} paper before the teacher did

We mentioned above that (39) is ambiguous between a strict copy reading where the teacher revised John’s paper and a sloppy copy reading where the teacher revised the teacher’s paper, and that this ambiguity must show up in the possible readings of Bill did too in (36). If (39) has the sloppy copy reading, there is only one element in it that corefers with the parallel element John. If the ambiguity may arise in the ellipsis copying process, and only elements that corefer with the parallel element(s) may have sloppy readings (as well as strict ones), we would expect two possible readings for Bill did too in this case, one with a sloppy copy reading and one with a strict copy reading of his. Among the combinations in (37), only two have a sloppy copy reading of the teacher did for the large source, namely (37.b,d). Reading (37.b) has a sloppy reading of the overt his for the outermost ellipsis, and reading (37.d) has a strict reading of the overt his for the outermost ellipsis.

If (39) has the strict reading of the teacher did, we expect the four readings (37.a,c,e,f) since there are two elements that corefer with the parallel element John; the overt his and the one hidden by the elliptic clause the teacher did. We have just argued that only three of these are possible readings, however. But we have seen other sentences where two elements in the source corefer with the parallel element, and not all of these have four possible readings for the elliptic clause. Also (6) and (9) had only three readings. We do not have the same case here, however. If we look at a case where there is a pronoun coreferring with the parallel element embedded in a before clause, we see that generally, we expect four readings:

pronoun in the respective subordinate before clauses. Readings (37.d,e) are explained by a discourse subordination analysis also between John revised his paper before the teacher did and Bill did too for the speakers who accept these readings, so that the latter is subordinate to the former. Under this assumption, all the pronoun copies are understood as dependent, even the ones that corefer with John. The main problem with such an optional discourse subordination effect as an explanation, is that the use of it must be constrained in order not to open for a lot of unwanted readings in other cases. In [FM92, page 32f], the reason why Every sailor said he was crazy, and Bill did, too only has a sloppy reading, whereas Every sailor said he was crazy before Bill did has both a strict and a sloppy reading, is explained by the subordinate status of the before-clause versus the co-ordinate status of the and-clause. If discourse subordination may be possible for the and-clause in Every sailor said he was crazy, and Bill did, too, a strict reading should be possible also in this case.

John_{i} revised his_{i} paper before the dentist called on him_{i}, and Bill did too (40)

But in (39), there is no overt pronoun coreferring with John. The hidden his in the teacher did that corefers with John is an elliptic copy of the overt his, and corefers with John only in virtue of being the strict copy of the overt his.

It is quite reasonable to claim that a strict copy element gets its reference through the reference of its original. But then the pattern of possible readings for (36), (37.a-e), falls out automatically! We argued in Section 1.1 that the reason that some elements may have a sloppy copy option is that they are recognised as coreferent with a parallel element. This recognition is not obligatory; an element may be coreferent with a parallel element without this being recognised. Then the element is given a strict copy reading, in the same way as elements that are not coreferent with a parallel element. The special pattern of “missing readings” is explained as a consequence of the relative independence of some elements: Elements that get their reference through the reference of some other element cannot be recognised as coreferent with a parallel element independently of the elements that they get their reference through. Only when the elements that the element in question gets its reference through is recognised as coreferent with a parallel element, the element in question may be recognised as coreferent with the parallel element too, and thus be given a sloppy copy reading in addition to the strict copy reading, which is always possible. We can sum this up as:

• A strict copy element gets its reference through its original.

2 Short sketch of DRT.

Knowledge of DRT as it is described in [KR93] will be presupposed in the following presentation, since we cannot give a full presentation here. A short sketch of the central features of DRS construction may be useful, however.

DRT consists of a GPSG^{33} style grammar. A general DRS-Construction algorithm takes a discourse consisting of a sequence of sentences with syntactic structure and an empty DRS, adding one sentence at the time to the DRS and by repeatedly applying construction rules to the reducible conditions in the DRS, we obtain a DRS without reducible conditions. Then the next sentence is added to the DRS, analysed by means of the construction rules, and so on and so forth. At last, DRT contains a semantics for the DRSs, with models, and truth and verification definitions. We will restrict ourselves for the time being to how to construct the correct DRSs for elliptic clauses in a fragment without tense.

The construction rules consists of the following:

(i) Specification of the triggering configuration for the construction rule. A triggering configuration is a syntactic tree (but for the exception that some of the leaves may be syntactic variables), that must match the top part of a condition, or a subtree of a condition, or in some special cases a DRS part^{34}, that must be found in the DRS under construction, in order for the construction rule in question to apply. A condition that is simple^{35} and contains a triggering configuration is reducible, otherwise it is irreducible^{36}.

(ii) Instructions to introduce fresh referents.

33See [GKPS85]. Kamp and Reyle write “Our syntax follows in outline the Generalized Phrase structure Grammar (GPSG) of Gazdar and his associates (1985)”, [KR93, page 24].

34By DRS part we mean a set of referents and conditions other than the set of one simple condition. In [KR93], this occurs only in the rules Abstraction, Summation and CR.DA (Rule for distribution over a set obtained by abstraction). [KR93, pages 341, 344 and 389].)

35A simple or atomic condition is a condition with no DRSs in it. Conditions with DRSs are called complex. See [KR93, page 110].

36See the definition of reducible on, e.g., [KR93, page 87]. We add the restriction that the conditions are simple here because we view rules with triggering configurations specifying DRS parts as construction rules. If only rules with triggering configurations that specify simple conditions are construction rules, we get exactly the same set of reducible conditions by the definition in [KR93, page 87] as by the definition given here.

(iii) Instructions to choose referents from the DRS obtained so far.

(iv) Instructions to introduce new conditions. These may be simple, that is syntactic-like^{37} trees or first-order like expressions, like α(x) or xRy, where α is a relation and x, y discourse referents, and R a relation from a specified set (=, <, …), or complex, that is, involving DRSs.

(v) Instructions to delete the reducible condition.^{38}

(vi) Bookkeeping instructions.^{39}

Let us have a look at a very simple example, the DRS construction for John loves his wife. We will assume that this sentence will be given the syntactic structure given in Figure 1.^{40} When analysing a sentence under a given syntactic analysis, we will start by adding its syntactic analysis tree into a DRS, either a fresh one, if the discourse we are analysing starts with the sentence in question, or the DRS obtained from the analysis of the preceding discourse.

The syntactic structure for John loves his wife given in Figure 1 contains triggering configurations, and is thus a reducible condition when in a DRS. The triggering configurations that can be found as parts of the tree in Figure 1 are given in Figure 2.

When faced with more than one triggering configuration, we must decide which one of them shall be attended to first. In a case like this, where both triggering configurations are found in the same condition, we must take care

37By syntactic-like I mean that the conditions are like syntactic trees as defined by the grammar, except that some subtrees may be replaced by discourse referents.

38Some of the instructions may be on the form :‘Replace the reducible condition in question with …’, where … are new conditions, I consider those instructions to be a combination of a (set of) introduction-of-conditions instruction(s) together with a deletion instruction. Implicit in this interpretation of replacement instructions is that since there can be at most one deletion instruction per construction rule, there can only be at most one replacement instruction per construction rule.

39By thisI refer to the instructions regarding the reference point, Rpt and the temporal perspective point, TPpt. See [KR93, Chapter 5].

40The syntactic structure given in Figure 1 is only the mere skeleton of the full syntactic structure. To each of the nodes there will also be a set of features and their values, like, e.g., Gen = male on the NP node above John. We will as a rule not write these features in our syntactic structures here, but we will feel free to use them when necessary. The gender information is obvious for a human reader, and when the construction rules indicate that such gender information should be added to the DRS, we take the liberty to do so without having marked that this information is attached to a node in the condition that is reduced.

Figure 1: Syntactic structure for John loves his wife.

[figure: loves; NP; V; VP; VP′; NP; PN; John; S; N; Det; ’s; NP; PRO; he; wife]

Figure 2: Triggering configurations in Figure 1.

[figure: (i); VP′; NP; PN; α; S; (ii); NP; V′; VP; N; Det; ’s; NP]

of the one with the highest top node first^{41} ^{42}. In this case, it is important that the triggering configuration with the highest top node is reduced first, or at least that it is reduced before the result of the reduction of the other triggering configuration is reduced, since otherwise the referent for John will not be present in the DRS when needed as an antecedent for his.

The construction rules for both of the triggering configurations found in Figure 2 will tell us to introduce a fresh discourse referent, replace the subtree of the condition to reduce corresponding to the triggering configuration with the (topmost) NP as a root with that referent, and add a new condition on the form γ(x), where x is the fresh referent and γ is α in case (i) and the subtree with the topmost NP node as its root in case (ii). By γ(x) in the case that γ is a tree, we mean a tree just like γ except that the label of its root node, X, is replaced by X(x). The result is shown in Figure 3^{43}. We will also write [γ](x) and [γ(x)] for γ(x).

41See [KR93, page 87].

42By always applying a relevant DRS construction rule to the triggering configuration with the highest top node, the c-command hierarchy for sentence-internal anaphora will be respected, and backwards anaphora like the one in (16) will not be possible, since no referent for the antecedent will be present when analysing the anaphoric expression before its antecedent.

Figure 3: DRS for John loves his wife after reduction according to the triggering configurations in Figure 2.

x y

John(x)

Gen(x) = male Gen(y) = fem

[figure: VP′; x; VP; V; y; S; loves]

[figure: NP(y); N; Det; ’s; NP; PRO; he; wife]

In this DRS we have three irreducible conditions and one reducible. The condition with NP(x) as its top node contains the two triggering configurations given in Figure 4. Also in this case the order of the reduction is given by where in the reducible condition the triggering configurations are found^{44}. Applying the corresponding construction rules,^{45} we get the completed^{46} DRS given in Figure 5.

43The construction rules used are CR.PN on page 121 and CR.α’s β on page 243 in [KR93].

44The relative order of reduction between the two triggering configurations here does not have any effect on the immediate result, but, as mentioned above, it is important that the DRS contains a discourse referent for John before analysing his wife in order for his to corefer with John.

45CR.α’s on page 245 and CR.LIN on page 124 in [KR93] (Final version of CR.LIN is given on page 338, and is identical to the version on page 124, except for allowing for plurals.) Actually, there seems to be a rule missing for this case, since CR.LIN demands that its triggering configuration is the full condition to reduce, and in addition, in the reducible condition in the DRS in Figure 3, the N does not have a referent argument and the triggering configuration of CR.LIN ((ii) in Figure 4), cannot be found even as a

Figure 4: Triggering configurations in Figure 3.

(i) [figure: NP(u); N; Det; ’s; NP]  (ii) {N(v) \atop α}

One thing is of particular interest to us here; when a referent is chosen to represent a subtree of a reducible condition, the conditions resulting from the part of the tree substituted by that referent in the condition that was reduced, are marked with the referent in question at their top nodes. This

subtree of the reducible condition. We will therefore assume that there is a construction rule with the triggering configuration

[figure: NP(y); N; Det]

which in the condition to be reduced replaces the subtree with the indicated N as its root node with the referent y, and adds the condition just like the N subtree, except that its root node is labelled N(y), which in our case will be a condition corresponding to the triggering configuration (ii) in Figure 4. If the system in part I of [KR93] is extended with this rule, we will be able to analyse constructions like the reducible condition in Figure 3. There is a small price to pay, however: our system will then have reducible conditions with more than one triggering configuration neither of which is higher than the other. The only way to get around this would be to make combination rules for each case where an N-branch can occur as one branch and there can be reducible material in another branch. This is not a desirable solution when the only reason for doing so is not to have two triggering configurations without one of them being higher than the other, as it may lead to unnecessary complicated rules since two disjoint subtrees of the triggering configuration must be reduced in the same rule. Kamp and Reyle write that “there is no fundamental reason we can see why a DRS-condition could not in principle contain two triggering configurations neither of which is higher than the other” and “We hypothesize that, were further extensions ever to give rise to DRS-conditions containing incomparable triggering configurations, the correct stipulation concerning the process of such a condition should be: the condition may be reduced either via the one of these configurations or via the other.” ([KR93, page 88]). That seems to work fine for this case. There are no side effects to the application of the construction rule corresponding to the triggering configuration (i) in Figure 4, nor to the construction rule described here, that will affect the application of the other if applied before it.

46Completed DRSs are “DRSs to which no further construction rules can be applied” ([KR93, page 118]).

Figure 5: Completed DRS for John loves his wife.

x y z

John(x)

Gen(x) = male Gen(y) = fem z = x Gen(z) = male wife(y)

[figure: VP′; x; VP; V; y; S; loves]

[figure: NP(y); y; Det; ’s; z]

means that we can read off from the DRS what conditions are the analyses of material embedded in an argument of a given relation. This structure of embeddings together with the knowledge of which arguments are dominant to other arguments for the relation in a given condition is what we need to get the correct pattern of readings in all the cases we have considered, except for the cases of nested ellipsis.

We will now give the relevant notions a precise definition, and state some restrictions that DRS construction rules must follow in order to serve our purpose.

3 To find one’s way through a DRS.

One problem that meets us repeatedly in analysing ellipsis on the level of DRS, is the problem of isolating analysis material originating from complex expressions of the discourse. We must identify the source and its parallel elements, and we must isolate the material that shall undergo elliptic copy. If doing this on the level of a DRS, we do not have direct access to the complex expressions involved, but must make do with a loosely structured set of conditions and referents: the DRS. In the following, we will assume that if a source clause is to be identified in a DRS, there must be one condition, C, that represents the full source clause, so that all the other source material, be it source parallel elements or the material that must be copied for the analysis of the elliptic clause, stem from the analysis of the arguments of the relation in C.^{47} Let us make this precise.

3.1 Some preliminary concepts.

We need some preliminary concepts to be sure of what assumptions we make with regards to the kind of DRSs we are working on. The conditions introduced to a DRS will be of three types, trees and formulae, which are the simple conditions, and complex conditions.

Definition 1 A simple condition γ is a formula if it is an expression in the relational language^{48} consisting of the special symbols of L in F with discourse referents as variables. Formulae are not trees.

Only trees have leaves, but both formulae and trees may have some referents attached, called the arguments of the top node.

Definition 2 Let γ be a condition in a DRS. The referent x is an argument of the top node of γ if

(i) γ is a tree and the top node is on the form X(…, x, …) where X is a syntactic category or a lexicalised relation, or on the form (x) and γ has more than one node.^{49}

(ii) γ is a formula where all variables are discourse referents and x is the leftmost discourse referent of γ.

47Larger sources, that is, sources that consist of a larger part of discourse than a sentence, are beyond the scope of the present presentation. It is a matter for analyses of discourse structure to describe what kind of discourse parts that can constitute such a compound source. Nothing said here, except for the identification of the single clause that covers the source material, and perhaps the identification of parallel elements, hinges on there being one such source. Working on the basis of a collection of such sources, after they and the parallel elements have been identified, can proceed exactly in the same way as in the case of a single condition covering the source. See [PSvdB94] for an approach to VP ellipsis and parallelism on larger discourse units.

48A relational language is a first order language without quantification.

49Conditions with top node marked with a referent only are conditions on the form x ∈ X and X = A_{1} ⊕ ⋯ ⊕ A_{n}. See Definition 10 (iii) and the comments on Summation and CR.NP(and) in Section 3.4.

(iii) γ is a complex condition on the form

x =  x = ∑ w  [figure: Q; x]

Note that, e.g., coreference equations are formulae and not a tree on the specified form. Thus, coreference equations have no leaves.

What do completed DRSs look like?

Definition 3 Let C be a simple irreducible condition on the form R(x) or xRy where R is a relation and x, y are discourse referents, or on the form of a tree where exactly one leaf node is a relation R. Then R is called the relation of C, and the C is said to be of one relation.

We will assume that any DRS fragment we will consider, will obey the constraint that all simple, irreducible conditions are of one relation. This is a constraint on the syntax for the fragment together with the set of construction rules. It means that it must not be possible to have syntactic constructs with more than one relation as leaves, without there being some construction rule that can be applied to it. Let us call such a combination of syntax and construction rules exhaustive:

Definition 4 (i) F is a DRS construction system if it consists of a GPSG-style grammar G and a set of DRS construction rules CR. In addition, F may have a set of function symbols, constants and relation symbols L that are not a syntactic category or lexical item of G.

(ii) F is exhaustive if in any DRS resulting from repeated use of construction rules from CR on sentences from a discourse of sentences according to G that contains no irreducible conditions, all simple conditions are of one relation.^{50}

If the fragment considered is exhaustive, we know that the one clause that represents the full source clause, including the parallel elements, is a condition of one relation. This is because we assume that the source is fully analysed before we do elliptic copy. But how shall we find the material corresponding to the parallel elements, and how shall we find the material that is to be copied for the analysis of an elliptic clause? To answer this question, we must in general take the history of construction into account.

50Kamp and Reyle’s definitions of what it is to be a DRS (confined to V and R) and of embeddings from such DRSs into models only treat DRSs where all simple conditions are of one relation (see [KR93, pages 110-112]). The additional language L consists in [KR93] of the constants male and fem, the unary function symbol Gen, and the relation symbols =, <, ◯, ⊃⊂, . . ..

Definition 5 (i) If γ is a reducible condition in a DRS D, and γ is reduced by a construction rule with exactly one instruction that says ‘replace γ by γ_{1}, …, γ_{m}, then γ_{1}, …, γ_{m} are the direct heirs of γ.^{51}

(ii) The relation heir is the reflexive and transitive closure of the relation direct heir.

Definition 5 works on the full history of construction of DRSs. We would like to be able to extract the necessary information by examining only the DRS at hand, however. We must then look at the DRS construction system that we are using, and see what kind of construction rules make it possible to retrieve the information needed.

Definition 6 (i) Each simple condition that occur at some point in the construction of a DRS D is called an origin of D

(ii) The referents x_{1}…. , x_{k} and the conditions C_{1}, …, C_{n} in a DRS D stem directly from an origin/the same origin if they are introduced to D by the application of one construction rule. The condition being reduced in this application of the construction rule is called the direct origin of x_{1}…. , x_{k} and C_{1}, …, C_{n}.

(iii) The relation stem from an origin/the same origin is the reflexive and transitive closure of the relation stem directly from. The relation origin of is the reflexive and transitive closure of direct origin of.

If we consider the little discourse John loves his wife and Bill does too, Bill does too is the elliptic clause (including parallel elements, and John loves his wife is its source (also including parallel elements). In Figure 5 we see that all the conditions and referents there stem from the analysis of the condition we started out with, namely the syntactic analysis of John loves his wife. Only John(x) and Gen(x)=male stem directly from that condition. However, the condition with S as its top node covers in some sense the full sentence, and we would like to be able to speak about this as if, e.g., John(x) and Gen(x)=male stem from that condition. S is the result of replacing subtrees of the origin of all the conditions by discourse referents, and can be said in some sense to be what the origin is transformed to in the process of analysis.

51As long as each application of a construction rule in DRT only consider one triggering configuration, each construction rule can make do with at most one ‘replace γ’-instruction.

As long as all instructions to delete conditions are replacement instructions, we would either have no direct heir and the reducible condition left intact, or one or more direct heirs and the reducible condition deleted.

Our goal is to eliminate the need for inspection of the construction history of DRSs in order to extract the material we need for, e.g., elliptic copy. Before we can do that, we must study the history of construction carefully, and take great care in how the conditions and referents of the DRSs have come to be, and what material can be said to belong to them or stem from a condition of which they are a heir. We will also be careful about what kinds of conditions that may be introduced in connection with the reduction of different types of conditions.

3.2 Linear construction systems.

Definition 7 F is a linear DRS construction system if each construction rule in F has exactly one deletion instruction, and this instruction is on the form ‘replace γ by γ′’, where γ is simple and refers to the condition being reduced, and γ′ refers to the one condition replacing γ.

If all triggering configurations in a set of construction rules CR specify simple conditions, this definition can always be fulfilled by a series of changes of one construction rule at the time, thus obtaining a set of equivalent^{52} construction rules. This can be done by adding an identity replacement condition to each construction rule without replacement instructions, and by choosing one of the γ_{i}s as the one that replaces the condition being deleted and add an introduction for each of the other γ_{i}s. The last change may be felt to ruin the explanatory power of the construction rule, however, in a way that we will come back to further on.

52Two construction rules are equivalent if they yield the same results on any input.

Two sets of construction rules are weakly equivalent if they for any discourse have the same completed DRSs as results, and strongly equivalent if they from any DRS have the same direct results as well.

Proposition 1 If F is a linear DRS construction system and D a (possibly not completed) DRS of F, then for each origin o of a set of conditions and referents in D, o has exactly one heir in D.

The proof is obvious by the definition of linear and the construction of DRSs. It is important that we only replace simple conditions, since we then obtain an unbroken chain of origins to each condition. Note that each condition in a DRS is the heir of at least one condition, namely itself, but it may also be the heir of many other conditions that occurred at an earlier stage in the construction of the DRS.

What about triggering configurations that specify a DRS part? As long as these do not delete any old conditions, every condition they introduce may be taken to be fresh origins, just as the conditions being introduced apart from the conditions replacing the condition being reduced in a linear construction system.

Definition 8 If CR is a construction rule with a triggering configuration specifying a DRS part and without deletion instructions, CR is called operational. Let CR be an operational DRS construction rule with triggering configuration Tr.

(i) If Tr is empty, and CR introduces conditions that are syntactic (sub)trees, possibly fresh referents that are also put as arguments of the top node in one of the conditions introduced, CR is introductory.

(ii) If Tr is non-empty and all simple conditions introduced are copies of irreducible conditions in Tr, possibly modulo the substitution of referents, or formulae, then CR is reproductive.

The DRS construction algorithm utilises an introductory rule to introduce each syntactic tree for a sentence into the DRSs.

Definition 9 F is a linear^{+} DRS construction system if F is the extension of a linear construction system with operational construction rules that are either introductory or reproductive.

If we in a linear^{+} DRS construction systems speak of the heir, or the direct heir of a construction rule, we will always mean a construction rule with a simple triggering configuration.

Proposition 2 If F is a linear^{+} DRS construction system then for each origin o of a set of conditions and referents in D, o has exactly one heir in D.

The proof is also in this case standard.

Let the maximal origin of a simple condition C denote the origin of C that was not the heir of any condition but itself. Let us call all the conditions and referents that stem from some origin of a condition C in a DRS D of a DRS construction system F the heritage of C in D in F, in symbols HER_{C,D}^{F}, and say that C covers its heritage.^{53} When each origin o is represented in a DRS D with exactly one heir h, we will also call HER_{h,D}^{F} the heritage of o in D, in symbols HER_{o,D}^{F}. If material is copied from material already present in the DRS under construction, possibly under substitution of referents, the copy of the heritage relation under the same substitution of referents, restricted to the copied referents and conditions holds. Each time two origins have the same heir, what stems from one of the origins will be a subset of what stems from the other origin. (This is obvious since the origin with the smaller heritage did once in the history of construction stem from the origin with the larger heritage.) The heritage of a condition introduced by a reproductive construction rule is the copy of the heritage of its original, modulo the referent substitution of the copying, restricted to the set of conditions and referents introduced by the reproductive construction rule. The heritage of a condition C in D is thus exactly the conditions and referents that stem from its maximal origin, or the mimicry of that if the condition is introduced by a reproductive construction rule. If two conditions in a DRS have intersecting heritages, then the heritage of one of them must be a subset (not necessarily proper) of the heritage of the other. In linear^{+} systems, two different conditions in a DRS will never have the same heritage.

The important property that we wish to have, is of course that the heir represents the origins in a way that it can be said in some sense that everything that stems from the origins comes from the heir, and furthermore that it is possible to retrieve everything that stems from the origins of a heir from the information given in the DRS, if we know the heir.

53The superscript F and the subscript D may be omitted when this does not lead to confusion.

3.3 Co-syntactic construction systems.

For the concepts heritage and cover to be linguistically fruitful, the construction rules must be such that it is reasonable from a linguistic point of view that a direct heir covers the same material as its direct origin.

Definition 10 (i) If C is a syntactic tree with a subtree X with root node x and exactly one immediate subtree with root node y, y is of a subkind of x,^{54} and C′ is like C except that the two nodes x and y are unified, then C′ is a result of a collapse of identical nodes in C. The unification of two such nodes in the given configuration is called a collapse of identical nodes.^{55}

(ii) A tree γ′ is a pruned version of a tree with several nodes γ if γ′ is identical to γ except possibly for the substitution of some proper^{56} subtrees without referent proper leaves^{57} with discourse referents, and the deletion of some proper subtrees without referent leaves, or the result of a collapse of identical nodes in such a condition.

(iii) If γ is a one node subtree of a tree with a syntactic category at its root, with a referent X at the only node, and x another referent, then the tree with root node on the form (x) with one daughter node X, in symbols x ∈ X, is a pruned version of γ.

(iv) If γ′ is a pruned version of γ, then the result of adding some referents in parenthesis to the top node of γ′ is a pruned^{+} version of γ.

54That is, either the same category as x, a lexical word of the category of x, or a category that is defined as a subtype of the category of x.

55This happens, e.g., in the rule CR.LIN [KR93, page 124 or 338].

56By proper subtree of a tree we mean a subtree that is not the tree itself.

57By proper leaf of a tree we mean a leaf that is not also the root node of that tree. In [KR93] this is ensured one the one hand by always reducing by the triggering configuration with the highest top node, and on the other hand by the fact that when a condition has referent leaves, this is either the result of the substitution of a subtree as specified in the highest triggering configuration of an origin, and the triggering configurations work together in such a way this leaf will not be a part of another triggering configuration in any heir of this origin, or it is the result of CR.NRC, which substitutes a leaf in a similar position in one of the conditions introduced. That the leaf in this condition will be in a position so that it will never be in a subtree to be substituted away in any heir of the condition is ensured by the syntax of gaps together with the set of construction rules.

If γ has a referent leaf x, x may be substituted by some other referent, but no subtree of γ that has x as a proper leaf may be substituted, in order to obtain a pruned version of γ. Note that a referent itself is not a tree. Only as a subtree, we will say that a referent has a pruned version.

If the difference between the triggering configuration and the condition replacing it is only the substitution of subtrees by referents, this is clearly the case in construction rules with a simple triggering configuration. We will take special care if the subtree substituted contains referents.In all cases where these referents are proper leaves of the subtree, we will assume that the case can be covered by rules that only substitute trees without referent leaves. We will thus prohibit the substitution of subtrees with referent proper leaves. We will allow substitution of one-node subtrees with a referent leaf, however. Also if the subtree is not replaced by a referent, but simply deleted, it may be reasonable to claim that the heir covers the same material as its origin(s). In the case of construction rules with triggering configurations specifying a DRS part, we will restrict ourselves to introductory rules and reproductive rules where all the simple condtions introduced are irreducible.

Definition 11 (i) F is a heir co-syntactic DRS construction system if in each replacement instruction ‘replace γ by γ_{1}, …, γ_{n} in a construction rule of F, where γ refers to the simple condition being reduced, the γ_{i}s, 1 ≤ i ≤ n, are exactly the pruned versions of γ introduced, and all rules with triggering configurations specifying DRS parts, are either introductory or reproductive.

(ii) A DRS construction system F is condition co-syntactic if each simple condition introduced in a construction rule CR is either a formula or a pruned^{+} version of a syntactic (sub)tree of F if CR is operational, a pruned^{+} version of a subtree of the condition being reduced if CR is non-operational.

(iii) If F is a DRS construction system that is both heir co-syntactic and referent co-syntactic, then F is co-syntactic.

Definition 12 (i) A DRS construction system F is heir adequate if in a DRS D of F, each maximal origin o has exactly one heir, and the heir of o is a pruned version of o.

(ii) If every simple condition in a DRS D for a DRS construction system F is either a formula or a pruned^{+} version of a syntactic (sub)tree of F, F is condition adequate.

(iii) If F is a DRS construction system that is both heir adequate and condition adequate, then F is adequate.

Lemma 1 If F is a heir co-syntactic and linear^{+} DRS construction system, then F is heir adequate.

Proof: Since F is linear^{+}, each maximal origin has exactly one heir. Since F is heir co-syntactic, each heir is a pruned version of its origin, and since a pruned version of a pruned version is also a pruned version, the lemma follows by induction on the construction of DRSs. □

Lemma 2 If F is a condition co-syntactic and linear^{+} DRS construction system, then F is condition adequate.

Proof: By induction on the construction history for each condition. If γ′ comes from an introductory construction rule, then it is a pruned^{+} version of a syntactic (sub)tree. If γ′ comes from a construction step reducing the simple condition γ, then either γ′ is a formula or it is a pruned^{+} version of a subtree of γ, which is a pruned^{+} version of a subtree of a syntactic (sub)tree of F by hypothesis, since γ is then a simple condition that is not a formula or on the form x ∈ X. If γ′ comes from a construction step by a reproductive construction rule, γ′ is a copy, except possibly for the substitution of some referents, of a condition that is a pruned^{+} version of an element of D by hypothesis, and thus γ′ must also be a pruned^{+} version of an element of D. □

From Lemma 1 and 2 Proposition 3 follow:

Proposition 3 If F is a co-syntactic and linear^{+} DRS construction system, then F is adequate.

3.4 Linear^{+} and co-syntactic rules in [KR93].

Most of the construction rules given in Part I and Chapter 4, Part II of [KR93] give rise only to construction systems that are both co-syntactic and linear^{+}. The exceptions are CR.NEG, CR.NRC, CR.COND CR.EVERY, CR.OR(≠ NP), CR.OR(NP), CR.AND, Optional Distribution, CR.NP[Quant= +] and CR.Fl.Qu (each), and also Summation and CR.DA.^{58} Also the construction rule of Footnote 45 on page 32 is co-syntactic and linear.^{59}

The construction rule CR.NRC for relative clauses is fine from the point of view of linear construction systems. A condition γ described by the triggering configuration with an N(u) as the top node, having the two subtrees of the kind N and RC, is replaced by a condition γ′ exactly like the subtree with S as its top node of the RC subtree, with the referent u substituting the gap-subtree of the S tree, and a condition γ′′ is introduced which is exactly like the N subtree except that u is added as an argument to the top node. There is only one replacement instruction, and the condition being reduced is simple. The rule gives rise to construction systems that are not co-syntactic, however, since γ′ is not similar to γ in the required way. The condition γ′′, however, is similar to γ in the required way. Which of the conditions introduced in a construction rule is mentioned as the replacement of the condition being reduced is inessential for the purpose of [KR93], and CR.NCR could as well have been formulated by introducing γ′ and replacing γ by γ′′, which is a co-syntactic rule. We will in the following read CR.NRC in this way, and then it is fine in linear^{+} and co-syntactic construction systems.^{60}

In the case of quantification, negation, and Optional Distribution, the triggering configuration will be replaced by a complex condition in which there is a copy of the triggering configuration, except for the substitution of a referent for a subtree, and in the case of floating quantifiers (like each) and negation, the deletion of subtrees with one leaf, without any corresponding introduction of referents. We may in these cases say that even if the complex condition is the direct heir, the condition in its lowest^{61} DRS is in turn a direct heir to the complex condition. In this way, however, we loose the property of having unique heirs even in linear construction systems. This is a problem that we can easily solve by defining the simple condition that would have been the heir in the corresponding non-quantificational/non-negation rule, to be the heir also in the quantificational/negation cases:

58In the tensed fragment, matters are a bit more complicated, but it is beyond the scope of the present presentation to explain how this can be sorted out.

59If conditions confirm to the specifications of DRS construction systems that are defined by the formulation of their rules, we will take the liberty to say that they are of the type of the relevant DRS construction system.

60In this formulation, CR.NCR is a one-step formulation of a two-step construction process with exactly the same resulting DRSs from any input, by means of two rules: one rule exactly like CR.NRC except that γ is replaced by a condition γ′′′ that is just like γ except for the deletion of the RC-subtree, and one rule for the collapsing of the two top nodes in γ′′′, yielding γ′′. (The latter of these rules is CR.LIN on a non terminal level).

61By lowest we mean lowest in the weak subordination relation on DRSs, ≤, [KR93, page 118f].

Notation 1 Let Instr be an instruction on the form ‘replace γ with C in which the DRS K contains γ′ and …’, where γ′ is simple and a copy of γ except for the substitution of subtrees with referents/deletion of subtrees. Instr is a shorthand formulation of the two instructions Instr′ and Instr′′, where Instr′ is on the form ‘Introduce C′ and …’ such that C′ is just like C except that it lacks γ′ in K, and Instr′′ is on the form ‘replace γ by γ′ in K’.

If we read the replacement instructions in CR.NEG, CR.EVERY^{62}, Optional Distribution, CR.NP[Quant= +] and CR.Fl.Qu as the two instructions that they by Notation 1 are shorthand formulations of, also these rules are fine in co-syntactic and linear^{+} construction systems.

Summation, Abstraction, and CR.DA are special rules in that they are not reducing simple conditions. They are operations on larger parts of a DRS. Both Summation and Abstraction are operational rules, having no replacement instruction. Abstraction is a reproductive construction rule, and is thus fine in linear^{+} and co-syntactic systems.^{63}

CR.DA on the other hand, has a replacement instruction. Its effect is to add some referents and conditions which are copies of conditions from a condition introduced by Abstraction, except for uniform substitution of a referent, to the first DRS of a duplex condition of the triggering configuration. Thus this replacement instruction may be seen as a notational variant of a set of introduction instructions. We will therefore claim that CR.DA has no replacement instruction. Since all the simple conditions that CR.DA introduces are copies of conditions introduced by Abstraction, they are all irreducible and copies of conditions of the DRS under construction, possibly modulo substitution of referents. CR.DA on this interpretation is a reproductive construction rule that work fine in co-syntactic and linear^{+} construction systems.

62CR.EVERY is problematic for our purposes because its complex condition has no argument of the top node. In connection with the development of CR.NP[Quant = +], Kamp and Reyle write that CR.NP[Quant = +] “applies also to the singular determiner every and thus allows us to do away with the rule CR.EVERY” [KR93, page 318f]. We will therefore not continue to consider CR.EVERY in the following.

63The restriction that the conditions introduced cannot be further reduced is not given in the formulation of Abstraction on page 344 in [KR93], but on page 355f.

Summation tells us to choose referents v_{1}, …, v_{k}, k ≥ 2, accessible from some position in the DRS, and introduce the referent Z and the condition Z = v_{1} ⊕ ⋯ ⊕ v_{k} into the subDRS of this position. But the condition Z = v_{1}⊕⋯⊕v_{k} is not a syntactic subtree, not even one with a referent introduced in parenthesis to the root node. However, the condition of Figure 6 is a syntactic subtree.^{64}

Figure 6: Syntactic configuration for NP conjunction.

[figure: NP; NP_{1}; ⋯; and; NP_{k}]

Notation 2 Suppose Z, v_{1}, …, v_{k} are discourse referents, k ≥ 2. Then Z = v_{1} ⊕ ⋯ ⊕ v_{k} is a shorthand formulation of a pruned^{+} version of a syntactic subtree as described by the configuration in Figure 6, with Z as the only argument of the top node, and each NP_{i} replaced by v_{i}.

By Notation 2, Summation can be read as a one-step combination of an application of an introductory rule introducing at some position in the DRS the referent Z and a condition as described in Figure 6 with Z as the argument of the top node, and k applications of a linear rule choosing an accessible referent and substituting an NP subtree with it. These rules are all fine in linear^{+} and co-syntactic systems.

CR.NP(and) is a construction rule with a simple triggering configuration introducing a condition on the form X = ξ_{1} ⊕ ⋯ ⊕ ξ_{k}. In this rule it is even more appropriate to regard that condition as an abbreviation according to Notation 2, since the subtree being substituted by a referent, in this case X, is exactly on the form as described in Figure 6. CR.NP(and) is then a one step combination of the linear and co-syntactic construction rule introducing X and X = NP_{1}, …, NP_{k}^{65}, and replacing the NP subtree as described in Figure 6 with X, together with a series of construction rules very similar to CR.PN or CR.NP[Quant= ind/Num = plur], one for each NP_{i}, introducing ξ_{i} and replacing the NP_{i} with it. These rules are linear and co-syntactic.

64The indices on the NPs are not part of the syntactic tree, but only marks to make it possible to refer to the different nodes in an easy way.

65This is an abbreviation of a tree like in Figure 6. Note in particular that the top node is marked with (X), and is on the second form as described in Definition 2 (i).

We are left with the rules for conjunction, except NP conjunction, disjunction, and conditionals.^{66} These construction rules are co-syntactic (if they are read as shorthand formulations according to Notation 1), but they are not linear. Several pruned versions of the conditions being reduced is introduced, however, and if we define one of them to be the heir, the linearity is preserved. This may be felt to ruin the explanatory power of the construction rule, though. We conclude, however, that also these rules can work fine in co-syntactic and linear^{+} construction systems, even though there is a small question with respect to the linearity.

3.5 Narrow construction systems.

Let us sum up our results: If we have a linear^{+} and co-syntactic DRS construction system, each origin will have exactly one heir by Proposition 1, and moreover, this heir is a pruned version of the origin by Lemma 1. This is good; when the system is linear^{+} we can refer to the heirs, which are conditions of the DRS at hand, instead of the mysterious origins that existed sometime in the past.When the system is adequate, a heir is such that it is reasonable to claim that it covers the heritage of its maximal origin. We thus have a good overview of the passage from the origins to the heirs. But what about the referents and conditions apart from the heir that is introduced in each construction step?

Definition 13 A DRS construction system F is narrow if it is linear^{+} and co-syntactic and for each construction rule CR in F that contains a replacement instruction ‘replace γ by γ′’, γ′ differs from γ in that at most one referent-leaf-free or single-node subtree is deleted or substituted by a referent, or it is a result of collapse of identical nodes in such a condition. If CR is operational, at most one original condition ^{67} is introduced, if CR is nonoperational, at most one pruned^{+} version of the subtree that is substituted is introduced, exactly one if a referent leaf is substituted.

66NP conjunction is not a problem, however, since this is analysed by a rule with a replacement condition which substitute subtrees of the conditions being reduced by referents, one for each conjunct. [KR93, page 435].

67By original condition or referent introduced in an operational rule, we mean a condition or referent that is not introduced as part of the copying of earlier material from the DRS under construction.

A narrow DRS construction system is thus a system in which a simple condition is reduced only by concentrating on one of its subtrees in each construction step, and conditions from “nowhere” is introduced one at the time. What is the gain of having a narrow system? The difference in what stems from an origin o and what stems from its immediate heir o′ is then the heritage of each of the conditions, except o′, introduced in reducing o, together with the referents introduced in reducing o. If F is narrow, we can thus identify all the referents and conditions introduced in a non-operational construction rule, except o′, in the step from o to o′, as the analysis of the subtree being substituted by a referent or deleted if the subtree is without referent leaves. If the subtree being substituted is a referent leaf, a subtree with the root node at that position must have been substituted from an earlier origin, and we will then add the conditions and referents introduced in the current step to the analysis of that subtree. Let us use ST _{o,D}^{F}, where o is an origin of a DRS D in the DRS construction system F, to denote what stems from o in D in F. We may omit the superscript F and the subscript D if it is clear from the context which F or D is meant.

Definition 14 Let F be a narrow DRS construction system, o an origin of a DRS D and o′ its direct heir.

(i) ST′ = ST _{o,D}^{F} − ST _{o′,D}^{F} is said to belong to the analysis of T in o in F, where T is the subtree deleted or substituted in the passage from o to o′ in D in F.

(ii) Suppose moreover that T was substituted by the referent x. Either T is without referent leaves, and T is like it was in the maximal origin of o, or T is a referent leaf X, where some subtree T′ of some origin of o has been substituted by X in the construction history of o, say in the step from o′′ to o′′′. Let ST_{T} = ST _{o′′′,D}^{F} −ST _{o′′,D}^{F} if T = is X, and ST_{T} = ∅ otherwise. Then ST′ ∪ ST_{T} is said to belong to the relevant occurrence of x in the heir h of o in D in F, and we write ST _{x,h,D}^{F} for ST′ ∪ ST_{T}) if this x has not been substituted with another referent in an origin of h that has o as an origin.^{68}

(iii) If furthermore x ∈ ST′, ST′ is said to belong to x in D in F, and we may write ST _{x,D}^{F} for ST _{x,γ,D}^{F}. If a leaf X of o is substituted with a referent x, ST _{X,δ,D}^{F} = ST _{X,D}^{F} and ST _{δ,D}^{F} = ST _{X,δ,D}^{F} ∪ {δ}, where δ is the pruned^{+} version of the subtree X.

68Note that ST_{T} = ST _{X,o,D}^{F} if T is the subtree consisting of one node marked with X.

(iv) The set of conditions and the corresponding referents introduced by a reproductive rule, inherits the belong structure of their originals, with the proviso of uniform substitution of referents as in the condition copies, restricted to those conditions that are copied and the corresponding referents.

Note that since each referent that is introduced to a DRS is fresh, no referent is introduced twice. Furthermore, x will only be introduced in connection with the reduction of at most one origin, and the ST _{x,D}^{F} is thus well defined.

Definition 15 Let P be a property of construction rules. Let CR_{1}, …, CR_{n}, 2 ≤ n, be P construction rules. Suppose that the corresponding triggering configurations, Tr_{CR_{i}} are such that each triggering configuration, Tr_{i+1}, is a triggering configuration for the heir of the CR_{i} if CR_{i} is non-operational, and for the original condition introduced if CR_{i} is operational. Let CR^{∗} be the construction rule obtained by having Tr_{1} as triggering configuration, and having as heir the condition resulting from performing all the replacement operations of CR_{1}, …, CR_{n} successively on the heir of CR_{i−1}, starting by a condition satisfying Tr_{1}, and all other instructions of each of the CR_{i}s. Then CR^{∗} is said to be a P combination rule, the sequence (CR_{i})_{1≤i≤n} is said to be a factorisation of CR, and each CR_{i} is called a factor construction rule of CR. If all the CR_{i}s are narrow, the belong structure of the resulting set of conditions and referents is as if the conditions and referents had come from the application of the factor rules in the given order.

The reason for using combination rules may be either that it gives fewer and more intuitively correct construction rules, or that some of the factor rules would overgenerate in combination with other rules of the system. If all the factor rules are narrow, the belong structure is the same as if the factor rules were used instead:

Lemma 3 Let CR be a narrow combination rule with the factorisation (CR_{1})_{1≤i≤n}, and F a DRS construction system. Let F^{+} be F extended with CR and F^{∗} F extended with CR_{i}, 1 ≤ i ≤ n. Then for all x, γ, D of F^{+},

ST _{γ,D}^{F^{+}} = ST _{γ,D}^{F^{∗}}
ST _{x,γ,D}^{F^{+}} = ST _{x,γ,D}^{F^{∗}}
ST _{x,D}^{F^{+}} = ST _{x,D}^{F^{∗}}.

Proof: By induction on the number of applications of CR from a maximal origin to its heir in the construction history of D in F^{+}. We compare the construction history of D in F^{+} with the construction history in F^{∗} which is exactly the same, except that each application of CR in F^{+} is replaced by successive applications of each of the factor rules of CR in the appropriate order in F^{∗}. If there are no applications of CR the construction history of D in F^{+} is the same as the construction history in F^{∗}, so the belong structure is the same, too. If the lemma holds for all D where there are less than k applications of CR in the construction history of D in F^{+}, and D′ is a DRS k applications of CR in its history of construction, then suppose that the step from D′′ to D′′′ in the construction history of D′ in F^{+}was the last step where CR was used. By hypothesis, for all γ, x in D′′

ST _{γ,D′′}^{F^{+}} = ST _{γ,D′′}^{F^{∗}}
ST _{x,γ,D′′}^{F^{+}} = ST _{x,γ,D′′}^{F^{∗}}
ST _{x,D′′}^{F^{+}} = ST _{x,D′′}^{F^{∗}}.

By the definition of the belong structure of narrow combination rules, exactly the same referents and conditions are introduced by CR as would be if reducing by its factorisation in F^{∗}, and all the conditions and referents introduced by CR belong to the same subtrees of the heir of the condition reduced by CR as they do of the heir of the condition reduced by the factorisation, so

ST _{γ,D′′′}^{F^{+}} = ST _{γ,D′′′}^{F^{∗}}
ST _{x,γ,D′′′}^{F^{+}} = ST _{x,γ,D′′′}^{F^{∗}}
ST _{x,D′′′}^{F^{+}} = ST _{x,D′′′}^{F^{∗}}.

The steps from D′′′ to D′ must be the same in the construction history in F^{+} as in F^{∗}, so we also have that

ST _{γ,D′}^{F^{+}} = ST _{γ,D′}^{F^{∗}}
ST _{x,γ,D′}^{F^{+}} = ST _{x,γ,D′}^{F^{∗}}
ST _{x,D′}^{F^{+}} = ST _{x,D′}^{F^{∗}}.
□

Proposition 4 Let F be a narrow system where every heir in a construction rule differs from its origin either in that a subtree is substituted by a referent, or the construction rule that introduced it introduces no other conditions, and γ a condition in a DRS D of F. Then

HER_{γ,D} = {γ} ∪ ⋃{ST_{x,γ,D} | x is a leaf of γ}.

Proof: For every condition γ in D, HER_{γ,D} = ST_{o,D} for the maximal origin o of γ. We will prove the proposition by induction on the construction history. If D is the first DRS in a history, D has no conditions and the proposition holds. Let D and D′ be DRSs of F such that D′ comes from D in one construction step and HER_{C,D} = {C}∪⋃{ST_{x,C,D} | x is a leaf of C} for all conditions C in D. The step from D to D′ must be of one of five types:

(i) D′ comes from D by an introductory rule introducing γ, γ has neither leaves nor heritage apart from γ itself.

(ii) D′ comes from D by reducing the condition C, and no subtree of C is substituted by a referent. Let γ denote the immediate heir of C. Then the heritage of C is exactly the same as the heritage of γ, except that C is replaced by γ, and C and γ have exactly the same referent leaves, so the induction hypothesis must hold for γ in D′ if it holds for C in D.

(iii) D′ comes from D by reducing the condition C, and some subtree of C without referent leaves is substituted by a referent. Then D′ must be like D except that C is deleted and a set of new conditions γ_{1}, …, γ_{n} and fresh referents y_{1}, … y_{k} are introduced. Since F is narrow, the direct heir γ_{j} of C differs from C by having exactly one subtree without leaves substituted by a referent, say z, and possibly also some identical nodes collapsed. All the other γ_{i}s are without referent leaves and constitute their own heritage. Then the referents that are leaves of γ_{j} are exactly the referents that are leaves of C with the addition of z, and ST_{z,γ_{j},D′} = {y_{1}, … y_{k}} ∪ {γ_{1}, …, γ_{j−1}, γ_{j+1}, …, γ_{n}}, since F is narrow.

HER_{γ_{j},D′} = HER_{C,D′}
= (HER_{C,D} − {C}) ∪ ST_{z,γ_{j},D′} ∪ {γ_{j}}
= ⋃{ST_{x,C,D} | x is a leaf of C} ∪ ST_{z,γ_{j},D′} ∪ {γ_{j}}
= {γ_{j}} ∪ ⋃{ST_{x,γ_{j},D′} | x is a leaf of γ_{j}}.

(iv) D′ comes from D by reducing the condition C, and a single-node subtree of C with the referent leaf Z is substituted by a referent z . Then D′ must be like D except that C is deleted and a set of new conditions γ_{1}, …, γ_{n}, z ∈ Z and fresh referents y_{1}, … y_{k}, z are introduced. Since F is narrow, the direct heir γ_{j} of C differs from C by having exactly one single-node subtree with the referent leaf Z substituted by z, and possibly also some identical nodes collapsed. The other γ_{i}s are without referent leaves and constitute their own heritage. The referents that are leaves of γ_{j} are the referents that are leaves of C except Z, and with the addition of z, and

ST_{z,γ_{j},D′} = {y_{1}, … y_{k}, z} ∪ {γ_{1}, …, γ_{j−1}, γ_{j+1}, …, γ_{n}, z ∈ Z} ∪ ST_{Z,D}

since F is narrow.

HER_{γ_{j},D′} = HER_{C,D′}
= (HER_{C,D} − {C}) ∪ {y_{1}, … y_{k}, z} ∪
∪{γ_{1}, …, γ_{j−1}, γ_{j+1}, …, γ_{n}, z ∈ Z} ∪ {γ_{j}}
= ⋃{ST_{x,C,D} | x is a leaf of C} ∪ {y_{1}, … y_{k}, z} ∪
∪{γ_{1}, …, γ_{j−1}, γ_{j+1}, …, γ_{n}, z ∈ Z} ∪ {γ_{j}}
= ⋃{ST_{x,C,D} | x is a leaf of γ_{j}, x ≠ Z} ∪
∪ST_{z,γ_{j},D′} ∪ {γ_{j}}
= {γ_{j}} ∪ ⋃{ST_{x,γ_{j},D′} | x is a leaf of γ_{j}}.

(v) D′ comes from D by a reproductive rule introducing γ, and

HER_{γ,D} = {γ} ∪ ⋃{ST_{x,γ,D} | x is a leaf of γ}

holds by definition of belong and heritage for reproductively introduced conditions and referents.

If C is a condition of D such that D′ comes from D by reducing the condition C′ ∉ HER_{C,D}, then HER_{C,D′} = HER_{C,D} and ST_{x,C,D′} = ST_{x,C,D} for all leaves x of C. Let C be a condition of D such that D′ comes from D by reducing the condition C′ in ST_{y,C,D} for some leaf y of C. But since ‘stem from’ is the (reflexive) and transitive closure of ‘stem directly from’,

C′ ∈ ST_{y,C,D} ⊆ HER_{C,D} ⟹ ST_{C′,D′} ⊆ ST_{y,C,D′} ⊆ HER_{C,D′}, and the only change from D to D′ is that C′ is replaced by ST_{C′,D′}, so HER_{C,D′} = {C} ∪ ⋃{ST_{x,C,D′} | x is a leaf of C}. □

Definition 16 Let F be a DRS construction system where all construction rules are narrow or narrow combination rules. Then F is a narrow^{+} DRS construction system.

Corollary 1 follows directly from Lemma 3 and Proposition 4.

Corollary 1 If F is a narrow^{+} DRS construction system where in no construction rule, conditions or referents belong to subtrees that are deleted, then for all γ a heir in a DRS D of F,

HER_{γ,D} = {γ} ∪ ⋃{ST_{x,γ,D} | x is a leaf of γ}.

But we still cannot decide what the heritage or belong structure is just by examining the DRS. We have to appeal to the construction history. In the systems we are after, we must be able to detect the heritage of the conditions, at least for the conditions we are interested in, and to detect what corresponds to the parallel elements, directly from the DRSs:

Definition 17 A DRS construction system F is referent detectable if there is an algorithm that with input D and x in D for every DRS D of F and discourse referent x in D, decides what belongs to x in D, only by examining D.

Definition 18 (i) Let C be a (sub)set of the possible simple conditions of a DRS construction system F. F is C condition detectable if for every DRS D of F and every simple condition γ in {γ | γ a condition in D} ∩ C, the heritage of γ in D can be decided by an algorithm with input D and γ in D only by examining D.

(ii) A DRS construction system F is condition detectable if F is C condition detectable for C = the set of all possible simple conditions of F.

Note that complex conditions are not reduced, and thus their heritage consists only in the complex condition itself.

Definition 19 (i) Let C be a (sub)set of the possible simple conditions of a DRS construction system F. F is C detectable if F is referent detectable and C condition detectable.

(ii) A DRS construction system F is detectable if F is referent detectable and condition detectable.

Lemma 4 Let the DRS construction system F be such that for any condition γ in any DRS of D, γ ∈ C for a C such that there is an algorithm for deciding whether the conditions of a D in F are in C, and

HER_{γ,D} = ⋃{ST_{x,D} | x a leaf of γ} ∪ {γ}.

If F is referent detectable then F is C detectable.

Proof: F is condition detectable by for given D and γ in D, by collecting the ST_{x,D} for each x ∈ X_{γ} by an algorithm by which F is referent detectable, and forming ⋃{ST_{x,D} | x a leaf of γ} ∪ {γ}. □

Lemma 5 Let F be a DRS construction system, and C and C′ be two sets of conditions such that the set of conditions in a DRS of F is a subset of C ∪ C′. Suppose F is both C condition detectable by the algorithm A and C′ condition detectable by A′. Suppose further that there is an algorithm B that for any condition γ in a DRS of F decides if γ ∈ C on the input D and γ in D. Then F is condition detectable.

Proof: F is condition detectable by for given D and γ in D, use B to decide if γ is in C, and if it is, then use A, otherwise use A′ to determine the heritage of γ in D. □

Lemma 6 If CR is a narrow combination rule with factorisation CR_{1}, …, CR_{n}, and F extended with CR_{1}, …, CR_{n}, is referent detectable / C condition detectable for some set of conditions C by an algorithm A, then F extended with CR is referent detectable / C condition detectable by A

Proof: Let F^{∗} be F extended with CR_{1}, …, CR_{n}, F^{+} F extended with CR. Let D be a DRS in F^{+}. Then D is a DRS in F^{∗}, and thus A decides what belongs to x in D for every referent x in D as a DRS in F^{∗} /the heritage of every condition γ in D as a DRS in F^{∗} such that γ ∈ C. But by Lemma 3 each condition and referent in D belong to the same referents in D as a DRS in F^{+} and as a DRS in F^{∗}, and the heritage of each condition of D is the same as a DRS in F^{+} as in D as a DRS in F^{∗}. This holds for every DRS D of F^{+}, so F^{+} is C condition detectable by A. □

3.6 Restricted construction systems.

There may be many ways to ensure that we are working with systems where we can detect the information needed, just as there may be many algorithms and many ways to code the history of the construction of a DRS into the DRS. We will here build on a coding that is used in [KR93], namely that when we substitute a subtree of a referent, and introduce conditions which are the analysis of this subtree, the referent is always used as an argument to these conditions. To be able to do that, we must ensure that the encoding is taken care of in a way that is appropriate for our purpose.

Definition 20 A co-syntactic and linear^{+} DRS construction system F is substitution restricted if for each construction rule CR in F that contains a replacement instruction, ‘replace γ by γ′’, CR also obeys

(i) x is introduced by CR if and only if γ′ differs from γ by the substitution of a referent x for a subtree of γ, and x is not an argument of the top node of γ.

(ii) Each condition introduced, except the heir, has exactly one of the referents that are substituted for a subtree of γ to obtain γ′, or is an argument of the top node of γ, as the only argument of the top node, and no referent leaves, except possibly for leaves with the referent that is the argument of the top node of that condition.

(iii) If a tree is substituted that has a referent leaf, X, then a referent x and exactly one condition on the form x ∈ X is introduced.

If CR is an operational construction rule, CR introduces all fresh referents that occurs in one of the conditions it introduces, CR reuses referents only in formulae, and then not as arguments of the top node, and if CR has a fresh referent x in a leaf in a condition, there is exactly one condition γ introduced by CR that has x as a leaf but not as an argument of the top node.

Note that in the case of substitution restricted systems, coreference equations (or something equivalent) must be used to link referents in the case of coreference. Only in some cases, when the argument is passed down from above in the tree, like in NPs that have relative clauses or are constructed by a genitive construction, the same referent can be reused. In a substitution restricted system it is only possible to use the same referent for several leaves of one condition if the referent is an argument of the top node, or if the construction system is not narrow, and the referent has been substituted for several subtrees in one construction step. If the system is both narrow and substitution restricted we know that each referent only occurs in the analysis of subtrees that it is substituted for, or in the analysis of a condition introduced by an introductory rule, in which case they are arguments to the top node of that condition.

Definition 21 Let F be a co-syntactic and linear^{+} DRS construction system where in each construction rule CR reducing a simple condition, if CR introduces other conditions than the direct heir, then CR introduces a referent. Then F is called referent restricted. If F is both substitution and referent restricted, then F is restricted.

We will also be interested in narrow and restricted combination rules.

Definition 22 (i) Let F be a narrow^{+} DRS construction system in which each narrow combination rule is also a substitution restricted combination rule by the same factorisation. Then F is a substitution restricted^{+} DRS construction system.

(ii) Let F be a narrow^{+} DRS construction system in which each narrow combination rule is also a referent restricted combination rule by the same factorisation. Then F is a referent restricted^{+} DRS construction system.

(iii) Let F be a narrow^{+} DRS construction system in which each narrow combination rule is also a restricted combination rule by the same factorisation. Then F is a restricted^{+} DRS construction system.

If F is restricted^{+}, we know that conditions are introduced only in connection with the introduction of a referent (that is also substituted for a subtree in the condition being reduced). This means that ST_{x,γ,D} = ∅ for all x such that [γ](x). We also know that if a referent x occurs as a leaf in a condition where it is an argument of the top node, that condition belongs to the analysis of x.

Lemma 7 Let F be substitution restricted and each construction rule have at most one referent substitution, and let D be a DRS of F and x a leaf in a condition in D. Then there is exactly one condition γ such that x is a leaf of γ but not an argument of the top node of γ. For all referents x introduced in D, all conditions with x as an argument of the top node belong to the analysis of x in γ.

Proof: x must be introduced either in a construction rule CR where a simple condition is being reduced, or CR is operational. If a simple condition γ is being reduced, and F is substitution restricted, CR replaces γ by a condition that is a pruned version of γ where x has been substituted for a subtree. x is not an argument of the top node of γ, since x is a fresh referent. Since F is co-syntactic and linear^{+}, the one heir in D of this condition will have x as a leaf but not as an argument of the top node. Since F is substitution restricted and each construction rule has at most one referent substitution, all the other conditions introduced in C having x as a leaf must also have x as an argument of the top node, and must belong to the analysis of x in the direct heir of γ. If CR is operational, the lemma holds after the use of CR by Definitions 14 and 20. Since only fresh referents are introduced, and in each construction step other than the step where x is introduced, we either replace a condition with x as a leaf but not as an argument of the top node by exactly one other condition with x as a leaf but not as an argument of the top node and introduce no conditions with x as their top node, or we substitute a subtree of a condition with x as its top node by x, and x is an argument of the top node of any condition introduced, all of which stem from the condition being reduced, or we substitute a subtree with a fresh referent, and introduce only conditions without x at all, except possibly from the heir of the condition being reduced, or we use an operational rule preserving the desired properties. Thus the lemma holds by induction on the construction history of DRSs. □

Lemma 8 Let F be a narrow and substitution restricted DRS construction system. Then F is referent detectable.

Proof: Let x be a referent that is a leaf of a condition γ of a DRS D. The following algorithm will yield ST_{x,D}:

Let

St_{1} = {γ | [γ](x)} ∪ {x}
St_{i+1} = St_{i} ∪
{γ | [γ](z), z a leaf, but not [γ′](z) for a γ′ ∈ St_{i}} ∪
{z | z a leaf, but not [γ′](z) for a γ′ ∈ St_{i}}

Since there are only a finite set of simple conditions in D, there is a final j such that St_{j} = St_{j+1}, and then St_{j} = St_{j+k} for any k ≥ 1. Then ST_{x,D} = St_{j}:

St_{j} ⊆ ST_{x,D} by the construction of St_{j}, since, by Lemma 7, we start out with a subset of ST_{x,D}, and in each step, we only add conditions and referents that are in the heritage of conditions that were there before, or belong to conditions that were there before by transitivity of belong.

ST_{x,D} ⊆ St_{j}, by Definition 14 and since if γ is a condition that was introduced by an operational rule, the construction process ensures that we get exactly what belongs to γ, which is the same as its heritage since γ is its own origin and there is no further reduction on conditions introduced by operational rules. Otherwise, if x is a referent or condition in ST_{x,D}, it must either be in St_{1}, or it must stem from a condition in St_{1}. Since F is restricted, everything that stem from a condition c will either have the same argument as the argument of the top node as c, or it will have as an argument of the top node a referent that is a leaf, but not an argument of the top node, of a condition that stem from c (including c). □

Lemma 9 Let F be a narrow and restricted DRS construction system. Then for any D in F and any γ in D,

HER_{γ,D} = {γ} ∪ ⋃{ST_{x,D} | x is a leaf of γ}.

Proof: Since F is narrow and restricted, every direct heir in a construction rule differs from its origin either in that one subtree is substituted by a referent, and that referent is introduced by the construction rule, or the construction rule introduces no other conditions than the direct heir. Every other condition introduced except heirs, has either no referent leaves, or they are on the form x ∈ X. Thus, ST_{x,γ,D} = ∅ for all x such that [γ](x). By Proposition 4,

HER_{γ,D} = {γ} ∪ ⋃{ST_{x,γ,D} | x is a leaf of γ}
= {γ} ∪ ⋃{ST_{x,D} | x is a leaf of γ}.

□

Proposition 5 Let F be a narrow and restricted DRS construction system. Then F is detectable.

Proof: By Lemma 8, F is referent detectable. By Lemma 4 and 9, F is then also detectable. □

Corollary 2 is a corollary of Lemma 6 and Lemmas 8 and 9 and Proposition 5.

Corollary 2 (i) Let F be a substitution restricted^{+} DRS construction system. Then F is referent detectable.

(ii) Let F be a restricted^{+} DRS construction system. Then

HER_{γ,D} = {γ} ∪ ⋃{ST_{x,D} | x is a leaf of γ}.

(iii) Let F be a restricted^{+} DRS construction system. Then F is detectable.

3.7 Narrow^{+} and restricted^{+} rules in [KR93].

Our problem now is that we are interested in systems that are linear^{+}, cosyntactic, and narrow^{+}, but not necessarily referent- or substitution restricted^{+}. Table 1 shows which rules in Part I and Chapter 4, Part II of [KR93] are narrow combination rules, referent restricted^{+}, and substitution restricted^{+}, if we base our judgements following the suggestions on pages 43 - 47.

The main goal is, if given a condition or referent, to be able to detect the heritage of the condition, and the analysis that belongs to the referent. For this we need detectable systems, or at least systems that are detectable for the maximal origin of each condition we would like to find the heritage of. We know that systems that are substitution restricted^{+}, and without substitution of referent leaves and introductions of complex conditions are referent detectable, and that if the systems are also referent restricted^{+}, they are detectable. But we can see from Table 1 that many of the rules we are interested in including in our fragments, are not referent restricted^{+} or even substitution restricted^{+}.

Table 1: Types of construction rules.

Rule Type  nar-ow subst. ref.
ow  restr. restr.
Construction Rule  comb.  +  +
CR.PN (121)  ×  ×  ×
CR.ID (122)  ×  ×  ×
CR.NEG (123)  ×
CR.NRC (123) × ×
CR. COND (156)  if lin.
CR.OR(NP) (202)  if lin.
CR.OR(≠NP) (203)  if lin.
CR.AND (221)  if lin.
CR.PRO[Refl= −] (238)  ×  ×  ×
CR.PRO[Refl= +] (239)  ×  ×  ×
CR.α’s β (243)  ×  ×  ×
CR.α’s (245)  ×  ×  ×
CR.DD (254)  ×  ×  ×
CR.NP[Quant=def/Num=plur](337)  ×  ×  ×
CR.NP[Quant=ind/Num=plur](337)  ×  ×  ×
CR.LIN (124/338)  ×  ×  ×
Summation (341)  ×  ×
Abstraction (344)  ×  ×
Optional Distribution (348)  ×  ×  ×
CR. NP[Quant=+] (349]  ×  ×  ×
CR.NP(Dep) (388)  ×  ×  ×
CR.DA (389)  ×  ×
CR.PRO[Num=plur] (390)  ×  ×  ×
CR.NP(and) (435)  ×  ×  ×
CR.Fl Qu(each) (445)  ×  ×  ×
Rule of Footnote 45, page 32  ×  ×

Number in parentheses after constr. rule indicates page in [KR93].

We will not continue to study the rules CR.COND, CR.OR(NP), CR.OR- (≠NP), CR.AND. These rules fall outside the scope of the present presentation with respect to the analysis of ellipsis, and we will leave it to further study to find out the detectability in systems with these rules. Suffice it here to notice that they are all co-syntactic, and may be understood as linear combination rules by splitting off one conjunct/disjunct at the time, and letting the rest be the heir. In that case, they can also be taken as narrow combination rules. Conditionals are hard to imagine as sources for elliptic clauses, and sentence conjunctions are analysed as if it was only a sequence of sentences. I regard it as a question for analyses of discourse structure above the sentence level to be able to find sources that consist of discourse parts larger than one sentence, and this will not be studied here.^{69} Disjunction is analysed similarly, split up in a series of conditions corresponding to sentences that each are added in a disjunct DRS of a complex disjunction condition. In the following, we will refer to the set of rules in Table 1 as if these rules were deleted.

The problem with respect to Summation, Abstraction and CR.DA not being substitution restricted^{+}, is that they may reuse discourse referents. Summation does this by substituting subtrees with referents chosen from the DRS. The standard way to do substitution of subtrees with referents when coreference with earlier referents is intended, is to introduce a fresh referent, substitute the subtree with this referent, and introduce a coreference equation coreferring with the chosen referent from the DRS with the fresh referent. Summation is the only place where substitution of subtrees is not done this way, and I can’t see any harm in replacing Summation with Summation′, given in Figure 7. Summation′ is a restricted^{+} combination rule. We will in the following mean Summation′ whenever we mention Summation.

Abstraction and CR.DA reuse referents by copying existing conditions, possibly without substituting some of the referents. There is no harm in following the line of using coreferenceequations in this case, and substitute uniformly with fresh referents also all the referents of the conditions being copied that will not already be substituted by Abstraction or CR.DA as they are, introduce the fresh referents in the same DRS as the other fresh referents of Abstraction or CR.DA, and in the same DRS add coreference equations for these extra introduced referents, coreferring each of them with Figure 7: Summation′. Summation′

69See [PSvdB94] for an interesting study of discourse grammar, ellipsis and parallelism constraints.

Choose:  K′ a sub-DRS of the DRS K (possibly K itself) and discourse referents v_{1}, …, v_{k} (k ≥ 2) occur- ring in K and accessible from K′.

Introduce to Con_{K}:

A fresh non-individual discourse referent Z and fresh referents u_{1}, …, u_{k} of the same type as v_{1}, …, v_{k}, respectively, and conditions

Z = u_{1} ⊕ ⋯ ⊕ u_{k} and u_{i} = v_{i}

for 1 ≤ i ≤ k into U_{K′}

the referent it was substituted for, just as we did in Summation′. In this reformulation, Abstraction and CR.DA are substitution restricted^{+}, and we will in the following assume that Abstraction and CR.DA has this reformulation whenever they are mentioned.

CR.NEG is a narrow construction rule that is neither substitution nor referent restricted. The problem is that a condition, ¬□, is introduced without belonging to a referent. Without this condition, CR.NEG would have been restricted. We can easily prove that restricted^{+} DRS construction systems extended with CR.NEG are C detectable for all conditions in C except those with a ¬□ in their heritage. The detectability problem for negation may then be reduced to detect when ¬□ belongs to the heritage of a condition, and we will not discuss this further here.

Two rules are substitution restricted^{+}, but not referent restricted^{+}; CR.NRC and the rule of Footnote 45 on page 32. They can be constructed as the combination of two narrow rules at the cost of one of them not being referent restricted. Substitution restricted^{+} systems are referent detectable by Corollary 2. Depending on the rest of the systems that CR.NCR and the rule of Footnote 45 are construction rules in, these systems may be detectable. For our purposes, it must suffice that we can show that rules like CR.NRC and the rule of Footnote 45 can be extended to restricted^{+} systems to yield C′ detectable systems for a subset C′ of the set of possible conditions in DRSs of F.

The problem with rules that are not referent restricted^{+}, but have all the other properties mentioned in Table 1, is that we lose track of the heritage of some conditions with arguments of the top node, when conditions except the heir are being introduced in a (factor) rule without an introduction of a referent.

Proposition 6 Let F be a restricted^{+} DRS construction system. Let CR be a set of narrow and substitution restricted but not referent restricted construction rules, and let F^{+} be the extension of F with CR. Let C_{CR} be the set of conditions that are heirs of an origin reduced by means of CR, and let C denote the set of conditions of a DRS of F. Then F^{+} is referent detectable and C − C_{CR} detectable.

Proof: All we have to prove is that the set of conditions γ for which no origin is reduced by a construction rule in CR have the property that

HER_{γ,D} = ⋃{ST_{x,D} | x a leaf of γ} ∪ {γ}.

Then the lemma follows from Lemma 4.

Let γ be a condition in a DRS of F such that no origin of γ was reduced by a construction rule from CR, and suppose that o_{1}, …, o_{n} are the origins in the history of construction of γ in D, such that o_{1} is the maximal origin of γ and for each o_{i+1}, o_{i+1} is the direct heir of o_{i} in the construction history of D. Then o_{i+1} differs from o_{i} either in that one subtree is substituted by a referent, and that referent is introduced by the construction rule, or the construction rule introduces no other conditions than the direct heir and ST_{x,γ,D} = ∅ for all x such that [γ](x). By Proposition 4,

HER_{γ,D} = {γ} ∪ ⋃{ST_{x,γ,D} | x is a leaf of γ}
= {γ} ∪ ⋃{ST_{x,D} | x is a leaf of γ}.
□

Corollary 3 then follows from Proposition 6 by Lemma 6.

Corollary 3 Let F be a referent^{+} DRS construction system, and let F^{+} be F extended with a set of substitution restricted^{+} but not referent restricted^{+} construction rules, CR. Let C_{CR} be the set of conditions that are heirs of an origin reduced by means of a rule of CR, and let C denote the set of conditions of a DRS of F. Then F^{+} is referent detectable and C − C_{CR} detectable.

If there is an algorithm for determining a set that contains all the conditions that are heirs of an origin reduced by means of a rule from CR of Corollary 3, say the set C^{∗}, and the system F of Corollary 3 is C^{∗} detectable, then F is also detectable by Lemma 5.

4 Extending DRT with ellipsis.

In this section, we will present a definition of a mechanism building on the framework of DRT, as it is defined in [KR93], with the small changes suggested in Sections 3.4 and 3.7. This is not to say that it is not possible to define how to analyse ellipsis obeying exactly the same requirements within some other framework^{70}.

When analysing ellipsis in DRT, we mean to analyse on the level of DRS construction. One advantage in analysing on the DRS level, is that as long as the source is fully reduced before we do elliptic copy, the pronouns are resolved and the information about which other elements in the source they corefer with is in the DRS.

In giving an analysis of elliptic clauses we have to solve several problems. I will assume that these problems can be divided in three main groups:

• The identification of an elliptic clause

• The identification of the source and parallel elements

• The copying or reconstruction of the relevant material in the source, which yields an analysis of the elliptic clause.

We concentrate here on the last group of problems. We will also look at the two first problems, but we will not try to give an account of the semantic and syntactic clues with respect to parallelism that may help us to choose one possible source rather than another. We will only treat single sentence sources, and leave it to discourse analysis to provide a framework for determining sources beyond the sentence level.^{71} As soon as the set of sentences have been determined as a discourse source, the retrieving of the material belonging to the sources, and the elliptic copying process should proceed similarly as in the single sentence cases, however.

4.1 Identification of an elliptic clause.

In DRT, this amounts to on the one hand have syntax building elliptic clauses, and on the other hand have construction rules with triggering configurations corresponding to the elliptic syntactic constructs. The syntactic rules shown in Figure 8 are suggestions for syntactic rules that can be added to the syntax on [KR93, pages 53-56]. The rules cover VP-deletion, Stripping, Gapping, Pseudogapping and NP ellipsis such as in John’s drawings are terrible, but Mary’s are just wonderful. They are to be analysed by the triggering configurations given in Figure 9.

70A description within a DRT framework as it is defined in [Kam81] is given in [Sem93].

71For a discourse approach, see [PSvdB94].

Figure 8: Syntactic rules for ellipsis.

VP′ ⟶ ∅
VP ⟶ ∅
V′ ⟶ ∅
N ⟶ ∅

4.2 The identification of source and parallel elements.

A source clause for an elliptic clause must provide the material that the elliptic clause lack, and it must be possible to determine the parallelism. Given one of the triggering configurations of Figure 9 in a DRS, the job must be to find a corresponding irreducible condition, that covers the parallel elements and the material to be copied. From this condition, we must be able to determine the parallelism and decide the set of (irreducible) conditions and referents, and the structural relations that hold between them, of the material that shall be retrieved, and let this material undergo elliptic copy.

Both the structure of the elliptic condition in comparison with the condition that is a candidate for a source, and the internal structure of the corresponding parallel elements, can be factors that may enable us to choose one candidate for a source instead of another. It is beyond the scope of this presentation to study the factors that may be important in the choice between candidates for elliptic clauses. We will just try and sketch some minimal requirements that the choice of source may obey.

One possible demand is that the source must have a pruned^{+} version that is identical to the core information of the elliptic clause, except for a substitution of discourse referents. Tentatively, we will assume that this core information is everything in the elliptic clause except the AUX and negation subtrees.

The parallelism is given by comparing the elliptic clause condition and a candidate for a source condition. All the nodes with a discourse referent are

Figure 9: Triggering configurations for ellipsis

[figure: u; VP′; AUX; VP; ∅; S; u; VP′; ∅; S; NP; ∅; VP′; VP; V; u; ∅; S]

VP-deletion Stripping

[figure: u; VP′; VP; V; u; ∅; S; u; VP′; AUX; VP; V; v; ∅; S; NP; NP; ∅; Det; ’s; u]
Gapping  Pseudogapping  NP-ellipsis

parallel elements of the elliptic clause condition, γ, and the corresponding referents are the corresponding parallel elements in the source condition, δ. Given that the substitution [x_{1}, …, x_{n}/y_{1}, …, y_{n}] performed on the parallel elements of the source condition would give us the elliptic condition, except for the additional material of the source, and possibly AUX and negation subtrees, x_{i} is parallel to y_{i} for i = 1, …, n. The analysis of the parallel elements in the source and elliptic clause in a restricted^{+} system, will then be ST_{y_{i}} and ST_{x_{i}}, respectively. The material that shall undergo elliptic copy, is HER_{δ} − ∑_{i=1}^{n} ST_{y_{i}} Figure 10: Incomplete DRS for Bill believed that he loved his wife, and Harry did, too

x K u

Bill(x)

K=  y z w y=x z=y wife(w)

Harry(u)

[figure: VP′; x; VP; V; K; S; believed; VP′; y; VP; V; w; S(K); loved; NP(w); w; Det; ’s; z; u; VP′; AUX; did; VP; ∅; S]

In Figure 10, we see a DRS for Bill believed that he loved his wife, and Harry did, too, where the analysis of Bill believed that he loved his wife is completed, and Harry did, too is completed as far as we can get without a construction rule for ellipsis.^{72}

Both of the other conditions with S at their root node are possible sources. If we delete the subtrees with root node V and K/w, we obtain trees that are pruned^{+} versions of the elliptic condition with the AUX subtree deleted, except that the referent u is replaced by x/y. Both the clauses Bill believed that he loved his wife and he loved his wife are possible as sources for Harry did, too. We will just study the case that Bill believed that he loved his wife is the source clause.

4.3 The dominance relation.

In Section 1.1 we discovered that in order to explain the pattern of missing readings, we need a hierarchy between (some of) the coreferring elements. The dominance structure inducing this hierarchy gives almost the same results as induced by c-command, but deviates from the results given by using c-command if this is in conflict with the demand that the argument of a relation that has the perspective dominates a c-commanding argument. It is a matter for future research to work out the exact conditions under which one argument dominates another. We will restrict ourselves to sketch two alternative ways of working the domination structure into DRT.

72The fragment of [KR93] does not comprise verbs with sentence complements. We will assume that the syntactic and lexical insertion rules

VP_{[ \begin{matrix}Num & = & α \\ Fin & = & β \end{matrix} ]}  ⟶ V_{[ \begin{matrix}Num & = & α \\ Fin & = & β \\ Scomp & = & + \end{matrix} ]} that S V_{[ \begin{matrix}Num & = & sing/plur \\ Scomp & = & + \\ Fin & = & − \end{matrix} ]} ⟶ believe, say, … are added to, e.g., the fragment on page 53ff in [KR93]. In addition to that, we assume a narrow and restricted combination rule with the illustrated triggering configuration as a part of a condition. This construction rule must introduce a fresh DRS discourse [figure: V; S; that; VP]

referent, K, to the DRS of the condition under reduction, delete the that-subtree, replace the S-subtree with K, and introduce the condition K = □ with the S-subtree with K as an argument of the top node as the only condition in the fresh subDRS.

[Eng90] shows that this domination relation has consequences for bound pronouns and backwards anaphora. The dominance relation between the arguments of a predicate is not determined only by the predicate itself, but is sensitive also to other parts of the sentence, like aspect and factors like agentivity and controllability. In DRT, the order of reduction determines the scope of quantifiers and what referents can be available as antecedents. For sentences like (16.c,d) to be treated correctly in DRT, an analysis order determined by the reduction of the triggering configuration with the highest node will not do the job, whereas an analysis reducing by the triggering configuration for the object position before the triggering configuration for the subject configuration would. In connection with the analysis of tense and aspect, Kamp and Reyle write “We will assume that these aspects can “percolate up” the tree so that they are available at the nodes where they are needed and that they are given in the form of “feature assignments”.”^{73} Under the assumption that the appropriate syntactic information can percolate up the syntactic tree, and that it is possible to determine by the syntactic structure and the different factors what order of reduction is appropriate, an S or VP node could be marked for the order of reduction that the arguments should be reduced in. We will in the following assume that this has been worked out.

Which discourse elements that have got their reference through which other discourse referents can then be determined in the following way. Let [x]_{D} denote the class of discourse referents identified with a given referent x, relative to a DRS D.^{74} In a restricted^{+} DRS construction system, we can define that a referent y ∈ [x]_{D} gets its reference through another referent z ∈ [x]_{D} if and only if there is a referent w and a condition γ in D such that z and w are leaves but not arguments of the top node of γ, z dominates w in γ, and y ∈ ST_{w}.

This can be carefully encoded in the coreference equations, or checked before making the elliptic copy. If we want to code it in the coreference equations, we can corefer each referent that is to corefer, with the lowest referent that it gets its reference through. A referent x is lower than another referent y in this respect if x gets its reference through y. In Figure 10 the coreference equations conform with this encoding. Assuming that the features that have percolated up for the sentences in Bill believed that he loved his wife are such that both of them are marked for the analysis of the subject argument before the object argument, the referent x will be introduced and accessible at the point of analysis of the subject subtree of the embedded S. Thus, x must dominate K, and the condition under reduction is an element of ST_{K}. In the reduction step where z is introduced, both x and y are accessible and dominate a referent, K, and w, respectively, such that the condition under reduction is an element in what belongs to this referent, ST_{K} and ST_{w}, respectively. But ST_{w} ⊂ ST_{K}, so y is chosen for the coreference equation.

73[KR93, page 512].

74See [KR93, page 325f]. We will omit the subscript D in [x]_{D} when this does not lead to confusion.

What if no element in [x] is such that there is a referent w and a condition γ in D such that z and w are leaves but not arguments of the top node of γ, z dominates w in γ, and y ∈ ST_{w}? Then it doesn’t matter which referent in [x] is chosen. Each element in [x] must then be from another sentence, that cannot be in the same source clause, or an element in the same sentence, but such that it cannot possibly be a parallel element in the same source clause as the referent that is to be coreferred with it.

The advantage of this approach, is that if the reduction goes in the order given by the dominance, we do not need to have the dominance encoded for each condition in the completed DRSs. The information that one subtree, T, shall be reduced before another, can be deleted when we reduce T. We would then have that z dominates w in γ for a w such that y ∈ ST_{w} if and only if z is a leaf in γ at the time the coreference equation in question is to be introduced. As exhibited in connection with the psychological predicates, we would need to have a way to tell us the correct order of analysis anyway, so part of the mechanism needed for this encoding would have to be developed even for fragments without ellipsis. Another advantage is that it is easy to work in that a strict copy element gets its reference through its original; it comes automatically if it is coreferred with its original.

A possible disadvantage may be if there are cases in which there is no dominance. If there are more than one argument to a relation, they have to be analysed in some order, and if the dominance follows the order of reduction, we may have a dominance relation induced in cases where it is wrong to have any. More research is needed to show if and in that case when, there are no dominance between arguments of a relation.

Another approach would be to check the dominance structure between parallel elements in the source just before the elliptic copy is made. An advantage may be that we would need to calculate what belongs to which referents in fewer cases, and that a dominance structure need not be imposed in cases where it is not appropriate. A disadvantage is that a way to represent the dominance structure between the arguments in each condition would then be necessary. This would impose extra structure on the completed DRSs. In addition, the information that a strict copy referent gets its reference through its original must be specifically encoded.

For the time being, we opt for the first alternative; to code the hierarchy by means of the coreference equations. Future research may show that this is not a viable alternative, but for now, there are no serious objections, the coding is easy to define, and the missing readings of nested ellipsis can be handled in a way which is uniform to the solution for missing readings of other sentences.

4.4 Construction rule for elliptic copy.

In this section a DRS construction rule for elliptic constructions will be proposed, building on the reference hierarchies being encoded in the coreference equations as described above. The construction rule will be a combination rule, between a simple triggering configuration rule and a reproductive rule. There is a small obstacle. We would want our ellipsis construction rule to be a narrow and restricted combination rule. This includes that it must be co-syntactic, and to be co-syntactic, a simple triggering configuration rule must obey the demand that the condition under reduction is replaced by a pruned version of itself. We have not yet defined what a pruned version of ∅ nodes are. We have already decided that the source must be such that it can be pruned to match the core of the elliptic clause. If we make a copy of the source clause condition, and substitute the corresponding parallel elements in the elliptic condition for the parallel elements in the source condition copy, and add a fresh referent for each of the rest of the referents of the source, we obtain a condition that is sufficient for the proofs for detectability to go through; no referents are removed or deposisioned, we just get some fresh referents at the yet unanalysed parts of the condition. Let us call this a pruned^{∗} version of the elliptic condition. The rule CR.ellipsis is given in Figure 11.

Note that there is a slight difficulty here with respect to the scope of referents, since different members of [x] for a referent x could be introduced to different (sub)DRSs. This means that some of the elements in [x] may be inaccessible to the elliptic clause conditions even though the parallel element in [x] is accessible. This is an artificial impediment, imposed on the DRSs by allowing different members of an [x] to be introduced independently of the referents they are chosen to corefer with. A natural way to overcome

Figure 11: Construction rule for ellipsis.

CR.ellipsis

Triggering configurations: γ ∈ Con_{K}:

One of the configurations in Figure 9 on page 65.

Choose source condition:

A condition δ such that a copy under substitution of referents is a pruned^{∗} version of γ, where x (and y) is parallel to u (and v). Let P be the set of parallel elements of the source, and P′ the set of parallel elements of the elliptic condition.

Introduce into K:

A copy of HER_{δ} − Coref(HER_{δ}) − ⋃_{x∈P} ST_{x}, where all referents have been uniformly substituted, x (and y) by u (and v) and all other referents with fresh referents, and Coref(HER_{δ}) denotes the set of coreference equations of HER_{δ}. Assume that the source referents (x_{i})_{i=1,...,n} are substituted with (y_{i})_{i=1,...,n}, respectively. Go through the elements in Coref(HER_{δ}) in some order, let [z]_{i} denote the set of referents that are coreferent with z after performing the operations described for the first i coreference equations in Coref(HER_{δ}), and do: (Step k) If x_{i} = x_{j} ∈ Coref(HER_{δ}), y_{i} = y_{j} may be added to K in the subDRS K′ of the copy corresponding to the subDRS of x_{i} = x_{j}, if y_{j} ∈ [z]_{k−1} for a z ∈ P′, otherwise add y_{i} = x_{i} to K′. The copy of δ replaces γ. this, is to change all rules introducing fresh referents and coreferring them with old ones, in such a way that the referents are introduced to the same DRS as the referents they are coreferred with. In Figure 10 this would mean that y and z should be introduced to the main DRS rather than to K, y since x is introduced to the main DRS, and then in turn z since y is.

Figure 12 is a DRS for (6) showing an intermediate stadium in the processing of CR.ellipsis on the DRS of Figure 10 (with y and z introduced to the main DRS). The copy of HER_{δ} − Coref(HER_{δ}) − ⋃_{x∈P} ST_{x} has been introduced, and P = {x} and P′ = {u}, and x, K, y, z, w are substituted by u, K′, y′, z′, w′ respectively. We are about to go through Coref(HER_{δ}) = {y = x , z = y}. CR.ellipsis says that this can be done in any order. All orders will yield only possible readings, but not necessarily all of them. Orders where a coreference equation v′′ = v′ never comes before a coreference equation v′ = v will yield all possible readings.

Starting with the coreference equation y = x, we must find out if we may add its copy under the given substitution , y′ = u, to K′. u ∈ [u]_{0} = {u} ∈ P′ so it is possible, and we may choose the sloppy copy reading and add y′ = u, or the strict copy reading and add y′ = y.

If we choose y′ = u, the sloppy copy z′ = y′ is a possibility for the copy of z = y, since [y′]_{1} = [u]_{1} = {u, y′} and u ∈ P′. We may in this case choose the sloppy copy reading and add z′ = y′ to K′, or the strict copy reading and add z′ = z.

If we choose the strict copy reading for y′, y′ = y, z′ = y′ is not an option, since in this case [y′]_{1} = {y′}, and y′ ∉ P′, and we must choose the strict copy reading adding z′ = z.

Thus, we obtained exactly the three wanted readings: the consistently sloppy, the consistently strict, and the sloppy-strict reading.

Figure 12: Intermediate DRS for Bill believed that he loved his wife, and Harry did, too

x K y z u K′ y′ z′

Bill(x)

K=  w y=x z=y wife(w)

Harry(u)

K′= \begin{aligned}& w′ \\ wife(w′) \end{aligned}

[figure: VP′; x; VP; V; K; S; believed; VP′; y; VP; V; w; S(K); loved; NP(w); w; Det; ’s; z; VP′; u; VP; V; K′; S; believed; VP′; y′; VP; V; w′; S(K′); loved; NP(w′); w′; Det; ’s; z′]

Figure 13: Syntactic structure for ‘S_{1} before S_{2}’.

[figure: S_{1}; Adv; Conj; before; S′_{2}; S_{2}; S′_{1}]

Let us have a brief look at how to analyse the sentence (36):

(36) John_{i} revised his_{i} paper before the teacher did and Bill did too.

The large source clause of (36) is outside the scope of the syntactic fragment that we have considered so far, since it involves a before-clause. Adverbial adjuncts are covered by the syntactic fragment for tense and aspect in [KR93, pages 682ff], however. In order to preserve the desired detectability of the DRS construction system involved, our analysis below must make some rudimentary use of temporal analysis. We will not here give a full analysis for tense and aspect, but only sketch and give hints as to how to solve the problems necessary for describing the pattern of possible readings with respect to the pronoun in (36) in a detectable DRS construction system.

We will assume that the large source in (36) has the syntactic structure as given in Figure 13, a structure of the fragment of [KR93, pages 682ff] (extended with the rules for ellipsis given in Figure 8).

A configuration as in Figure 13 can be reduced according to CR.S′ in [KR93, page 610], introducing a time referent t and replacing the S′ condition by its two subtree configurations with t as an argument to the top node. This is, however, not exactly what we need, as we cannot see which condition is subordinate from this picture and thus lose detectability. We will therefore take the liberty to use a slightly changed version of CR.S′ where instead of introducing the S(t) tree, we introduce a S′ tree with S and (t) as subtrees, the (t) subtree replacing the Adv subtree if there is an Adv subtree in the triggering configuration.^{75} The result of applying the construction rule Figure 14: DRS after reducing the syntactic configuration for ‘S_{1} before S_{2}’.

75The interpretation rules from DRSs into models must be changed accordingly, so that the rules that take S(t) condition arguments must now take S′ conditions with the two subtrees S and t as arguments instead.

t

[figure: Adv(t); Conj; before; S′_{2}; S_{2}; S_{1}; t; S′_{1}]

resulting from this change, ignoring the bookkeeping instruction relating t to the temporal perspective point TPpt, to the configuration in Figure 13 in an otherwise empty DRS would then be the DRS given in Figure 14.

CR.S′ changed as described is a narrow and restricted rule except in the case of a sentence without adverbial. For such sentences, a single-node subtree (consisting of a referent leaf) is added in a place where there was no subtree before, and the heir is not a pruned version of its direct origin. We can get around this problem in two ways; by changing the syntax in such a way that there is an empty subtree to replace with a referent, or to change the definition of ‘pruned’ to allow for such adding of referent nodes, at least in some specia= l cases. We will here sketch a solution with an empty node in the cases where there is no adverbial arguments, and trust that the reader can fill in the necessary details for syntax and triggering configuration for construction rules. We will not look at how the temporal analysis for sentences can be done beyond the time referent argument for sentences. We must take a short look at the next step in the analysis, however, as there is no construction rule covering this case in [KR93]. We will assume that there is a construction rule with a triggering configuration matching the Adv condition in Figure 14, resulting in the introduction of a fresh time referent t′, the substitution of the S′ subtree with t′, and the introduction of a condition exactly like the S′ subtree except for the addition of t′ as an argument of the top node. The result of applying this to the DRS in Figure 14 is shown in Figure 15, where we have also added the empty node under S′_{2}.^{76}

We will leave the Adv condition of Figure 15 as it is, trusting that this condition can be interpreted as before(t, t′). The S′_{2}(t′) condition must be Figure 15: DRS after reducing the Adv subtree in Figure 14.

76This is not to indicate that this node comes into the condition in the step from Figure 14 to Figure 15, it is supposed to have been there right from the start.

t t′

[figure: Adv(t); Conj; before; t′; S_{2}; ∅; S′_{2}(t′); S_{1}; t; S′_{1}]

given further attention, though. We would like to replace it with a condition that is exactly like it, except for the substitution of t′ for the ∅ subtree. This case is exactly parallel to the case of the construction rule described in Footnote 45 on page 32. Except for the necessary substitution of syntactic category labels to the triggering configuration, the rule of Footnote 45 can be used for this step. Figure 16, shows a DRS for the large source in (36) with a strict copy reading of he for the inner ellipsis. Note that this analysis can go through only if t′ of the elliptic condition for the teacher did is seen as parallel to the t of the source condition. This is both reasonable and necessary, since at the point of reducing by CR.ellipsis on the condition γ covering the inner ellipsis, ST_{t′} ≠ {t},^{77} and moreover, γ ∈ ST_{t′}, since t′ is an argument of the top node of γ.

In the reduction of the reducible condition in the DRS, say D, of Figure 16 by CR.ellipsis, we want the time discourse referent t′′ to act as a non-parallel element, in contrast to what we wanted for the t′ of the inner ellipsis reduction. We see, however, that ST_{t′′,D} = {t′′},^{78} and that t′′ is not an argument of the top node of the elliptic condition in this case. We will thus assume that we can give an analysis in which t′′ is taken as a non-parallel element in this case, in such a way that it is used as the referent substituting the corresponding referent of the source clause condition. Under these assumptions, the reduction of the elliptic condition of D is straight forward, and the copying of the coreference equations follow the same pattern as the copying of the coreference equations in the analysis of (6) described in connection with the completion of the DRS in Figure 12, thus giving exactly the three wanted readings in case of a strict copy reading of the inner ellipsis.

77ST_{x,D} = {x} is the smallest possibility for a referent in a DRS D.

78This will not be correct in a construction system with the bookkeeping instructions relating such referents to the temporal perspective point. A thorough discussion of these matters must be deferred to the time when the analysis of tense and aspect is discussed in full.

Figure 16: DRS for strict copy reading analysis of (36).

t x y z t′ u y′ z′ t′′ v

John(x)

paper(y) z=x teacher(u) paper(y′) z′=z Bill(v)

[figure: VP′; x; t; VP; V; y; S; S′; revised; Adv(t); Conj; before; t′; NP(y); y; Det; ’s; z; VP′; u; t′; VP; V; y′; S; S′(t′); revised; NP(y′); y′; Det; ’s; z′; v; VP′; AUX; did; VP; ∅; S; S′; t′′]

5 Final remarks.

In Section 1 we came up with a series of specifications for what a framework an analysis of ellipsis must obey to cover the patterns of readings for examples where elements corefer with the parallel element(s). These requirements were:

1 Source internal coreference and non-coreference for the pronouns in the source is given information for elliptic copy.

2 On some level, there is a (partial) dominance order between the arguments of each predicate. This dominance order coincides with the dominance order given by the c-command hierarchy unless this conflicts with the demand that an argument that has the internal perspective dominates other arguments.

3 The semantics of a predicate, and aspectual properties and properties like agentivity and controllability decide which of its arguments have the internal perspective over elements in other arguments of this predicate.

4 If an element is subordinate to a coreferent element according to the dominance order, then the subordinate element gets its reference through the element that dominates it.

5 Elements that are recognised as coreferent with a parallel element gets a sloppy copy reading, other elements get a strict copy reading.

6 Elements need not be recognised as coreferent with parallel elements even if they are.

7 An element cannot be recognised as coreferent with a parallel element unless all elements that it gets its reference through are.

8 A controlled argument in the source of an elliptic clause does not have an independent status with respect to ellipsis. The controlled argument must have the same reading as its controller also in the elliptic clause.

9 A strict copy element gets its reference through its original.

Requirement 1 is solved by demanding that the analysis of the source clause is completed before analysing the elliptic clause.^{79}

The problems connected with requirements 2 and 3 must be solved in order to analyse the scope properties and anaphoric properties connected with psychological predicates. More research is needed in order to work into the syntax the features necessary and the conditions under which they percolate up through the syntactic tree. Our discussion has indicated that if these problems can be solved, the missing readings pattern for elliptic clauses with psychological predicates in their source follows for free.

Requirements 4 -7 and 9 have been solved in our proposal by the encoding by coreference equations. Requirement 8 may be solved by reuse of discourse referents. This must be worked out carefully in more detail, in order to preserve the intended properties of the belong structure and the detectability of the DRS construction systems. Future work is needed also with respect to the of tensed fragments.

79This may be a too strong demand in some marginal cases. One example is the Hirschbühler sentence

A Canadian flag hangs outside every embassy, and an American flag does too.

from [Hir82]. The problem comes out even more clearly if every embassy is substituted with most embassies. The problematic reading is when each quantifier only has scope over its own conjunct, while having scope over the parallel element of that conjunct. In the case of most embassies, the problematic reading will be true if both Outside most embassies hangs a Canadian flag and Outside most embassies hangs an American flag hold, even if it doesn’t hold that Outside most embassies hang a Canadian flag and an American flag. A possible way to obtain this reading is by delaying the analysis of the parallel elements, and allowing substitution of the full parallel element subtree instead of only the substitution of their referents. In any case, there are no pronouns, and thus requirement 1 holds in any case.

References

[Chi83] Gennaro Chierchia. Outline of a semantic theory of (obligatory) control. In M. Barlow, D. Flickinger, and M. Wescoat, editors, Proceedings of the West Coast Conference on Formal Linguistics, volume 2, pages 19–31, Stanford University, 1983. Stanford Linguistic Association.

[Chi84] Gennaro Chierchia. Anaphoric properties of indefinitives and gerunds. In M. Cobler, S. MacKaye, and M. Wescoat, editors, Proceedings of the West Coast Conference on Formal Linguistics, volume 4, pages 28–39, Stanford University, 1984. Stanford Linguistics Association.

[Dow87] David Dowty. Thematic proto roles, subject selection, and lexical semantic defaults. LSA Colloquium papaer, 1987.

[Dow89] David Dowty. On the semantic content of the notion of “thematic role”. In G. Chierchia, B. H. Partee, and R. Turner, editors, Properties, Types and Meaning, Vol II: Semantic Issues. Kluwer, Dordrecht, 1989.

[DSP91] Mary Dalrymple, Stuart M. Shieber, and Fernando C. N. Pereira. Ellipsis and higher-order unification. Linguistics and Philosophy, 14(4):399 – 452, August 1991.

[Eng90] Elisabet Engdahl. Argument roles and anaphora. In Robin Cooper, Kuniaki Mukai, and John Perry, editors, Situation Theory and its Applications, Volume 1, number 22 in Lecture Notes, pages 379–393. CSLI, 1990.

[FM92] Robert Fiengo and Robert May. The eliminative puzzles of ellipsis. In Steve Berman and Arild Hestvik, editors, Proceedings of the Stuttgart Ellipsis Workshop 1992, number 29 in Sprachteoretische Grundlagen für die Computerlinguistik, 1992.

[GKPS85] Gerald Gazdar, Ewan Klein, Geoffrey K. Pullum, and Ivan Sag. Generalized Phrase Structure Grammar. Blackwell, Oxford, 1985.

[GP90] M. Gawron and S. Peters. Anaphora and Quantification in Situation Semantics. Number 19 in CSLI Lecture Notes. CSLI, Stanford, CA., 1990.

[Hel88] L. Hellan. Anaphora in Norwegian and the Theory of Grammar. Foris, Dordrecht, 1988.

[Hes92] Arild Hestvik. Subordination and strict identity interpretation of reflexives. In Steve Berman and Arild Hestvik, editors, Proceedings of the Stuttgart Ellipsis Workshop 1992, number 29 in Sprachteoretische Grundlagen für die Computerlinguistik, 1992.

[Hir82] Paul Hirschbühler. VP deletion and across-the-board quantifier scope. In Proceedings of NELS 12. GLSA, Amherst, 1982. University of Massachusetts.

[Jac72] Ray S. Jackendoff. Semantic Interpretation in Generative Grammar. The MIT Press, Cambridge, MA, 1972.

[Jac87] Ray S. Jackendoff. The status of thematic relations in linguistic theory. Linguistic Inquiry, 18:369–412, 1987.

[Kam81] Hans Kamp. A theory of truth and semantic representation. In Theo Janssen Jeroen Groenendijk and Martin Stokhof, editors, Formal Methods in the Study of Language,Part I, number 135 in Mathematical Centre Tracts. Mathematisch Centrum, 1981.

[Kit91] N. Kitagawa. Copying identity. Natural Language and Linguistic Theory, 9:497– 536, 1991.

[KR93] Hans Kamp and Uwe Reyle. From Discourse to Logic. Number 42 in SLAP. Kluwer Academic Publishers, 1993.

[OD73] Östen Dahl. On so-called sloppy identity. Synthese, 26:81–112, 1973.

[OD74] Östen Dahl. How to open a sentence: Abstraction in natural language. Logical Grammar Reports, (12), 1974.

[PSvdB94] Hub Prüst, Remko Scha, and Martin van den Berg. Discourse grammar and verb phrase anaphora. Linguistics and Philosophy, 17(3):261–327, 1994.

[Ros67] J. R. Ross. Constraints on Variables in Syntax. PhD thesis, MIT, 1967.

[Sag76] Ivan A. Sag. Deletion and Logical Form. PhD thesis, MIT, 1976. Reprinted in 1980 by Garland Publishing Inc., New York and London.

[Sch73] T. Schiebe. Zum problem der grammatisch relevanten identität. In F. Kiefer and N. Ruwet, editors, Generative Grammar in Europe. Reidel, 1973.

[Sem93] Helle Frisak Sem. A proposal for ellipsis in DRT. Report series in Language, Logic and Information, Department of Linguistics, University of Oslo, July 1993.

[Wil77] Edwin Williams. Discourse and logical form. Linguistic Inquiry, 8(1):101–139, 1977.

[ZS74] A. Zwicky and J. Sadock. Ambiguity tests and how to fail them. In J. Kimball, editor, Syntax and Semantics 4. Academic Press, New York, 1974.