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<Paper uid="P99-1007">
  <Title>Unifying Parallels</Title>
  <Section position="4" start_page="49" end_page="50" type="metho">
    <SectionTitle>
2 Representing discourse anaphors
</SectionTitle>
    <Paragraph position="0"> The main tenet of the DSP approach is that interpreting an elliptical clause involves recovering a relation from the source clause and applying it to the target elements. This leaves open the question of how this procedure relates to sentence level semantic construction and in particular to the semantic representation of VPellipsis. Consider for instance the following example: null (2) Jon runs but Peter doesn't.</Paragraph>
    <Paragraph position="1"> Under the DSP analysis, the unresolved semantics of (2) is (3)a and equation (3)b is set up. HOU yields the solution given in (3)c and as a result, the semantics of the target clause</Paragraph>
    <Paragraph position="3"> It is unclear how the semantic representation (3)a comes about. Under a Montague-type approach where syntactic categories map onto semantic types, the semantic type of a VP-Ellipsis is (et), the type of properties of individuals i.e. unary relations, not binary ones. And under a standard treatment of subject NPs and auxiliaries, one would expect the representation of the target clause to be neg(P(peter)) not P(neg)(peter). There is thus a discrepancy between the representation DSP posit for the target, and the semantics generated by a standard, Montague-style semantic construction module.</Paragraph>
    <Paragraph position="4"> Furthermore, although DSP only apply their analysis to VP-ellipsis, they have in mind a much broader range of applications: \[...\] many other elliptical phenomena and related phenomena subject to multiple readings akin to the strict and sloppy readings discussed here may be analysed using the same techniques (Dalrymple et al., 1991, page 450).</Paragraph>
    <Paragraph position="5"> In particular, one would expect the HOU-analysis to support a general theory of sloppy identity. For instance, one would expect it to account for the sloppy interpretation (I'll kiss you if you don't want me to kiss you) of (4).</Paragraph>
    <Paragraph position="6"> (4) I'll \[help you\] 1 if you \[want me tol\] 2. I'll kiss you if you don't2.</Paragraph>
    <Paragraph position="7"> But for such cases, the discrepancy between the semantic representation generated by semantic construction and the DSP representation of the target is even more obvious. Assuming help and kiss are the parallel elements, the equation generated by the DSP proposal is:</Paragraph>
    <Paragraph position="9"> and accordingly, the semantic representation of the target is -~R(k) which is in stark contrast with what one could reasonably expect from a standard semantic construction process namely: -~P(you) -+ k(i, you).</Paragraph>
    <Paragraph position="10"> What is missing is a constraint which states that the representation of the target must unify with the semantic representation generated by the semantic construction component. If we integrate this constraint into the DSP account, we get the following representations and constraints: null</Paragraph>
    <Paragraph position="12"> where T is the semantic representation generated for the target by the semantic construction module. The second equation requires that this representation T unifies with the representation of the target postulated by DSP.</Paragraph>
    <Paragraph position="13"> With this clarification in mind, example (2) is handled as follows. The semantic representation  of (2) is (6)a where the semantic representation of the target clause is the representation one would expect from a standard Montague-style semantic construction process. The equations are as given in (6)b-c where C represents the semantics shared by the parallel structures and P the VP-Ellipsis. HOU then yields the solution in (6)d: the value of C is that relation shared by the two structures i.e. a binary relation as in DSP. However the value of P (the semantic representation of the VPE) is a property - as befits a verbal phrase.</Paragraph>
    <Paragraph position="15"> In sum, provided one equation is added to the DSP system, the relation between the HOU-approach to VP-ellipsis and standard Montague-style semantic construction becomes transparent. Furthermore it also becomes immediately obvious that the DSP approach does indeed generalise to a much wider range of data than just VP-Ellipsis. The key point is that there is now not just one, but several, free variables coming into play; and that although the free variable C always represents the semantics shared by two parallel structures, the free variable(s) occuring in the semantic representation of the target may represent any kind of unresolved discourse anaphors - not just ellipsis. Consider the following example for instance: (7) Jon 1 took his1 wife to the station. No, BILL took his wife to the station.</Paragraph>
    <Paragraph position="16"> There is no ellipsis in the target, yet the discourse is ambiguous between a strict and a sloppy interpretation 2 and one would expect the HOU-analysis to extend to such cases. Which indeed is the case. The analysis goes as follows.</Paragraph>
    <Paragraph position="17"> ~I assume that in the target took his wife to the station is deaccented. In such cases, it is clear that the ambiguity of his is restricted by parallelism i.e. is a sloppy/strict ambiguity rather than just an ambiguity in the choice of antecedent.</Paragraph>
    <Paragraph position="18"> As for ellipsis, anaphors in the source are resolved, whereas discourse anaphors in the target are represented using free variables (alternatively, we could resolve them first and let HOU filter unsuitable resolutions out).</Paragraph>
    <Paragraph position="19"> Specifically, the target pronoun his is represented by the free variable X and therefore we have the following representation and equations:</Paragraph>
    <Paragraph position="21"> HOU yields inter alia two solutions for these equations, the first yielding a strict and the second, a sloppy reading:</Paragraph>
    <Paragraph position="23"> Thus the HOU-approach captures cases of sloppy identity which do not involve ellipsis.</Paragraph>
    <Paragraph position="24"> More generally, the HOU-approach can be viewed as modeling the effect of parallelism on interpretation. In what follows, I substantiate this claim by considering two such cases: first, the interaction of parallelism and sloppy identity and second, the interaction of parallelism and focus.</Paragraph>
  </Section>
  <Section position="5" start_page="50" end_page="53" type="metho">
    <SectionTitle>
3 Parallelism and Focus
</SectionTitle>
    <Paragraph position="0"> Since (Jackendoff, 1972), it is widely agreed that focus can affect the truth-conditions of a sentence 3. The following examples illustrate this, where upper-letters indicate prosodic prominence and thereby focus.</Paragraph>
    <Paragraph position="1"> (8) a. Jon only introduced MARY to Sue.</Paragraph>
    <Paragraph position="2"> b. Jon only introduced Mary to SUE.</Paragraph>
    <Paragraph position="3"> Whereas (8a) says that the only person introduced by Jon to Sue is Mary, (8b) states that the only person Jon introduced Mary to, is Sue.</Paragraph>
    <Paragraph position="4"> To capture this effect of focus on semantics, a focus value 4 is used which in essence, is the 3The term focus has been put to many different uses. Here I follow (Jackendoff, 1972) and use it to refer to the semantics of that part of the sentence which is (or contains an element that is) prosodically prominent. aThis focus value is defined and termed differently by different authors: Jackendoff (Jackendoff, 1972) calls it the presuppositional set, Rooth (Rooth, 1992b) the Alternative Set and Krifka (Krifka, 1992) the Ground.  set of semantic objects obtained by making an appropriate substitution in the focus position.</Paragraph>
    <Paragraph position="5"> For instance, in (Gaxdent and Kohlhase, 1996a), the focus value of (8a) is defined with the help of the equation: I Focus Value Equation I Sere = X(F) I where Sern is the semantic of the sentence without the focus operator (e.g. intro(j,m,s) for (8)), F represents the focus and X helps determine the value of the focus variable (written X) as follows: Definition 3.1 (Focus value) Let X = Ax.C/ be the value defined by the focus value equation and T be the type of x, then the Focus value derivable from X, written X, is {C/ J x wife}.</Paragraph>
    <Paragraph position="6"> Given (8a), the focus value equation is thus (9a) with solution (9b); the focus value derived from it is (9c) and the semantics of (8a) is (9d) which given (9c) is equivalent to (9e).</Paragraph>
    <Paragraph position="7">  (9) a. intro(j,m,s) = X(m) b. {X +-- Ax.intro(j,x,s)} c. --X = {intro(j, x, s) I x E wife} d. VP\[P E -X A P -+ P = intro(j,m,s)\] e. VP\[P E {intro(j, x, s) I x E wife} A</Paragraph>
    <Paragraph position="9"> In English: the only proposition of the form John introduced x to Sue that is true is the proposition John introduced Mary to Sue.</Paragraph>
    <Paragraph position="10"> Now consider the following example:  (10) a. Jon only likes MARY b. No, PETER only likes Mary.</Paragraph>
    <Paragraph position="11">  In a deaccenting context, the focus might be part of the deaccented material and therefore not prosodically prominent. Thus in (10)b, the semantic focus Mary is deaccented because of the partial repetition of the previous utterance. Because they all use focus to determine the focus value and thereby the semantics of sentences such as (8a), focus deaccenting is a challenge for most theories of focus. So for instance, in the HOU-analysis of both (Pulman, 1997) and (Gaxdent and Kohlhase, 1996a), the right-hand side of the focus equation for (10b) becomes FV(F) where neither FV (the focus value) nor F (the focus) are known. As a result, the equation is untyped and cannot be solved by Huet's algorithm (Huet, 1976).</Paragraph>
    <Paragraph position="12"> The solution is simple: if there is no focus, there is no focus equation. After all, it is the presence of a focus which triggers the formation of a focus value.</Paragraph>
    <Paragraph position="13"> But how do we determine the interpretation of (10b)? Without focus equation, the focus value remains unspecified and the representation of (10b) is: VP\[P E FV A P -+ P = like(p,m)\] which is underspecified with respect to FV. (Rooth, 1992a) convincingly argues that deaccenting and VP-ellipsis are constrained by the same semantic redundancy constraint (and that VP-ellipsis is additionally subject to a syntactic constraint on the reconstructed VP). Moreover, (Gaxdent, 1999) shows that the equational constraints defined in (5) adequately chaxacterise the redundancy constraint which holds for both VPE and deaccenting. Now example (10b) clearly is a case of deaccenting: because it repeats the VP of (10a), the VP only likes mary in (10b) is deaccented. Hence the redundancy constraint holding for both VPE and deaccenting and encoded in (5) applies5:</Paragraph>
    <Paragraph position="15"> These equations axe solved by the following substitution:</Paragraph>
    <Paragraph position="17"> so that the interpretation of (10b) is correctly fixed to: VP\[P E {like(p,x)} A P --+ P = like(p,m)\] Thus, the HOU approach to deaccenting makes appropriate predictions about the interpretation of &amp;quot;second occurrence expressions&amp;quot;  (SOE) 6 such as (10b). It predicts that for these cases, the focus value of the source is inherited by the target through unification. Intuitively, a sort of &amp;quot;parallelism constraint&amp;quot; is at work which equates the interpretation of the repeated material in an SOE with that of its source counterpart. null Such an approach is in line with (Krifka, 1992) which argues that the repeated material in an SOE is an anaphor resolving to its source counterpart. It is also partially in line with Rooth's account in that it similarly posits an initially underspecified semantics for the target; It is more specific than Rooth's however, as it lifts this underspecification by unification. The difference is best illustrated by an example: (11) ?? Jon only likes SARAH. No, PETER only likes Mary.</Paragraph>
    <Paragraph position="18"> Provided only likes Mary is deaccented, this discourse is ill-formed (unless the second speaker knows Sarah and Mary to denote the same individual). Under the HOU-analysis this falls out of the fact that the redundancy constraint cannot be satisfied as there is no unifying substitution for the following equations: null</Paragraph>
    <Paragraph position="20"> In constrast, Rooth's approach does not capture the ill-formedness of (11) as it places no constraint on the interpretation of PETER only likes Mary other than that given by the compositional semantics of the sentence namely: VP\[P E FV A P --+ P = like(p,m)\] where FV represents the quantification domain of only and is pragmatically determined. Without going into the details of Rooth's treatment of focus, let it suffice to say, that the first clause does actually provide the appropriate antecedent for this pragmatic anaphor so that despite its ill-formedness, (11) is assigned a full-fledged interpretation.</Paragraph>
    <Paragraph position="21"> ~The terminology is borrowed from (Krifka, 1995) and refers to expressions which partially or totally repeat a previous expression.</Paragraph>
    <Paragraph position="22"> Nonetheless there are cases where pragmatic liberalism is necessary. Thus consider Rooth's notorious example: (12) People who GROW rice usually only EAT rice This is understood to mean that people who grow rice usually eat nothing else than rice. But as the focus (RICE) and focus value (Ax.eat(pwgr, x)) that need to be inherited by the target VP only EAT rice are simply not available from the previous context, the redundancy constraint on deaccenting fails to predict this and hence, fails to further specify the underspecified meaning of (12). A related case in point is: (13) We are supposed to TAKE maths and semantics, but I only LIKE semantics.</Paragraph>
    <Paragraph position="23"> Again the focus on LIKE is a contrastive focus which does not contribute information on the quantification domain of only. In other words, although the intended meaning of the but-clause is o/ all the subjects that I like, the only subject I like is semantics, the given prosodic focus on LIKE fails to establish the appropriate set of alternatives namely: all the subjects that I like. Such cases clearly involve inference, possibly a reasoning along the following lines: the but conjunction indicates an expectation denial. The expectation is that if x takes maths and semantics then x likes maths and semantics. This expectation is thus made salient by the discourse context and provides in fact the set of alternatives necessary to interpret only namely the set {like(i, sem), like(i, maths)}. To be more specific, consider the representation of I only like semantics: VP\[P E FV A P --+ P = like(i, sem)\] By resolving FV to the set of propositions {like(i, sem),like(i, maths)}, we get the appropriate meaning namely: VP\[P E {like(i, sem), like(i, maths)} A P --+ P = like(i, sem)\] Following (Rooth, 1992b), I assume that in such cases, the quantification domain of both usually and only are pragmatically determined.</Paragraph>
    <Paragraph position="24">  The redundancy constraint on deaccenting still holds but it plays no role in determining these particular quantification domains.</Paragraph>
  </Section>
  <Section position="6" start_page="53" end_page="54" type="metho">
    <SectionTitle>
4 Sloppy identity
</SectionTitle>
    <Paragraph position="0"> As we saw in section 2, an important property of DSP's analysis is that it predicts sloppy/strict ambiguity for VP-Ellipsis whereby the multiple solutions generated by HOU capture the multiple readings allowed by natural language. As (Hobbs and Kehler, 1997; Hardt, 1996) have shown however, sloppy identity is not necessarily linked to VP-ellipsis. Essentially, it can occur whenever, in a parallel configuration, the antecedent of an anaphor/ellipsis itself contains an anaphor/ellipsis whose antecedent is a parallel element. Here are some examples.</Paragraph>
    <Paragraph position="2"> Jon 1 /took his1 wife to the station\] 2.</Paragraph>
    <Paragraph position="3"> No, BILL/took his wife to the station\]2.</Paragraph>
    <Paragraph position="4"> (Bill took Bill's wife to the station) Jon 1 spent /hisl paycheck\] 2 but Peter saved it2. (Peter saved Peter's paycheck) null I'll /help you\] 1 if you /want me to1\] 2. I'll kiss you if you don't2. (I'll kiss you if you don't want me to kiss you) Because the HOU-analysis reconstructs the semantics common to source and target rather than (solely) the semantics of VP-ellipses, it can capture the full range of sloppy/strict ambiguity illustrated above (and as (Gardent, 1997) shows some of the additional examples listed in (Hobbs and Kehler, 1997)). Consider for instance example (16). The ellipsis in the target has an antecedent want me to which itself contains a VPE whose antecedent (help you) has a parallel counterpart in the target. As a result, the target ellipsis has a sloppy interpretation as well as a strict one: it can either denote the same property as its antecedent VP want me to help you, or its sloppy copy namely want me to kiss you.</Paragraph>
    <Paragraph position="5"> The point to note is that in this case, sloppy interpretation results from a parallelism between VPs not as is more usual, from a parallelism between NPs. This poses no particular problem for the HOU-analysis. As usual, the parallel elements (help and kiss) determine the equational constraints so that we have the following equalitiesZ:</Paragraph>
    <Paragraph position="7"> Resolution of the first equation yields AR.wt(you, R(i, you)) --+ R(i, you) as a possible value for C and consequently, the value for C(k) is: C(k) = wt(you, k(i, you)) -+ k(i, ou) Therefore a possible substitution for P is: {P +-- x.wt(x,k(i,x))} and the VPE occurring in the target can indeed be assigned the sloppy interpretation x want me to kiss x.</Paragraph>
    <Paragraph position="8"> Now consider example (15). The pronoun it occurring in the second clause has a sloppy interpretation in that it can be interpreted as meaning Peter's paycheck, rather than Jon's paycheck. In the literature such pronouns are known as paycheck pronouns and are treated as introducing a definite whose restriction is pragmatically given (cf. e.g. (Cooper, 1979)). We can capture this intuition by assigning paycheck pronouns the following representation:</Paragraph>
    <Paragraph position="10"> are treated as definites whose restriction (P) is a variable of type (e --+ t). Under this assumption, (15) is assigned the following equationsS: C(j, sp) = 31x~)c_of(x, j) A sp(j, x)\] C(p, sa) = 31x\[P(x) A sa(p, x)\] Resolving the first equation yields</Paragraph>
    <Paragraph position="12"> as a value for C, and therefore we have that: C(p, sa) = 31xbc_of(x,p ) A sa(p, x)\] {P +-- )~y.pc_of(y, p)} That is, the target clause is correctly assigned the sloppy interpretation: Peter saved Peter's paycheck.</Paragraph>
    <Paragraph position="14"> Thus the HOU-treatment of parallelism can account for both paycheck pronouns and examples such as (16). Though lack of space prevents showing how the other cases of sloppy identity are handled, the general point should be clear: because the HOU-approach associates sloppy identity with parallelism rather than with VPellipsis, it can capture a fairly wide range of data providing some reasonable assumptions are made about the representations of ellipses and anaphors.</Paragraph>
  </Section>
  <Section position="7" start_page="54" end_page="54" type="metho">
    <SectionTitle>
5 Implementation
</SectionTitle>
    <Paragraph position="0"> It is known that for the typed lambda-calculus, HOU is only semi-decidable so that the unification algorithm need not terminate for unsolvable problems. Fortunately, the class of equations that is needed for semantic construction is a very restricted class for which much better results hold. In particular, the fact that free variables only occur on the left hand side of our equations reduces the problem of finding solutions to higher-order matching, a problem which is decidable for the subclass of third-order formulae (Dowek, 1992).</Paragraph>
    <Paragraph position="1"> These theoretical considerations have been put into practice in the research prototype CHoLI, a system which permits testing the HOU-approach to semantic construction.</Paragraph>
    <Paragraph position="2"> Briefly, the system can: parse a sequence of sentences and return its semantic representation, interactively build the relevant equations (parallel elements are entered by the user and the corresponding equations are computed by the system) and solve them by means of HOU.</Paragraph>
    <Paragraph position="3"> The test-suite includes approximately one hundred examples and covers the following phenomena: null * VP-ellipsis and its interaction with anaphora, proper nouns (e.g., Mary, Paul) and control verbs (i.e., verbs such as try whose subject &amp;quot;control&amp;quot; i.e., is co-referential with some other element in the verb complement).</Paragraph>
    <Paragraph position="4"> * Deaccenting and its interaction with anaphora, VP-ellipsis, context and sloppy/strict ambiguity.</Paragraph>
    <Paragraph position="5"> * Focus with varying and ambiguous foci. It is currently being extended to sentences with multiple foci and the interaction with deaccenting.</Paragraph>
    <Paragraph position="6"> As mentioned in section 2 the HOU-approach sometimes over-generates and yields solutions which are linguistically invalid. However as (Gardent et al., 1999) shows, this shortcoming can be remedied using Higher-Order Colored Unification (HOCU) rather than straight HOU.</Paragraph>
    <Paragraph position="7"> In CHOLI both an HOU and an HOCU algorithm can be used and all examples have been tested with and without colors. In all cases, colors cuts down the number of generated readings to exactly these readings which are linguistically acceptable.</Paragraph>
  </Section>
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