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A Proof-Theoretic Analysis of Weak Crossover

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New Frontiers in Artificial Intelligence (JSAI-isAI 2021)

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Abstract

Dependent type semantics (DTS) is a framework for proof-theoretic semantics of natural language based on Martin-Löf type theory. In this paper, I first show that DTS has an empirical problem regarding the quantificational weak crossover. I then suggest that the definition of underspecified terms in DTS shall be replaced by the underspecified types. Finally, I claim that the suggested setting not only successfully solves the problem, but also allows us to naturally define the mechanism of local accommodation in DTS.

Daisuke Bekki: I sincerely thank the anonymous reviewers of LENLS18 for their insightful comments. This work was partially supported by Japan Science and Technology Agency (JST) CREST Grant Number JPMJCR20D2.22 and Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number JP18H03284.

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Notes

  1. 1.

    Much of the literature reports that (1b) is not completely unacceptable under the specified BVA reading and judges its acceptability as “??” or even “?”, rather than “*”, as in (1b). I assume, however, that the status of (1b) is unacceptable for carefully controlled informants, especially those who do not allow the quantifier chosen for the experiment (every, in the case of (1)) to take the inverse scope over the subject. See See Hoji (2015), in which chapter 6 is devoted to the interpretation and the controls of experiments on the acceptability of (1).

  2. 2.

    Another construction in the WCO paradigm is the wh-question variant of WCO, originating from Postal (1971), for which I do not provide analysis here. Because the analysis of WCO in DTS is purely semantic, we also need a purely semantic analysis of wh-questions prior to the analysis of the wh-question variant of WCO, which I prefer to leave as an open issue.

  3. 3.

    By DTS, I refer to the version of DTS adopted in Yana et al. (2019), Kubota et al. (2019), and Tanaka (2021), which is slightly different from the one adopted in Bekki and Mineshima (2017) that uses the context-passing mechanism.

  4. 4.

    The particular choice of a syntactic theory does not affect the following discussion; we may adopt any lexical grammar assuming it is homomorphic to the type system of DTS.

  5. 5.

    The abbreviation \({\textbf {e}}\) stands for \(\textsf {entity}\), the type for entities. Tense information is omitted for the sake of simplicity.

  6. 6.

    In this paper, I adopt a slightly different definition of UDTT (for the sake of simplicity) to that presented in Bekki and Sato (2015), where UDTT has its own type system that consists of a set of typing rules and a set of @-elimination rules, which are defined in a mutually recursive manner. See the Appendix for details.

  7. 7.

    Proof search in DTT is undecidable; therefore, anaphora resolution in DTS is undecidable. This may not be a useful result from the computational perspective, but it does not constitute an empirical problem either since there is no evidence that human anaphora resolution is a decidable process.

  8. 8.

    This is not a unique solution—the signature may provide proof constructions of other male persons as well.

  9. 9.

    I thank Kenichi Asai for pointing out this problem (p.c. July 19th, 2015).

  10. 10.

    See Appendix A for a formal presentation.

  11. 11.

    The square brackets for underspecified types are syntactically distinguished from those of the \(\varSigma \)-type notation, but when these square brackets nest, they may be omitted except for the outermost ones in the same way as brackets used in \(\varSigma \)-type notation.

  12. 12.

    Sentence (24) on p. 403.

  13. 13.

    This operation should not be applied to all types of anaphora and presupposition: for example, it is known that pronouns in general do not undergo local accommodation when their antecedents are missing. For this purpose, we may want to add a binary feature to underspecified types that tells us whether local accommodation is applicable to them. Alternatively, we may argue that the prohibition of local accommodation for pronouns is based rather on pragmatic factors. This is a controversial issue and I would like to leave it open. I thank the anonymous reviewer of LENLS18 for pointing it out.

  14. 14.

    \(\{\}\) is the enumeration type with no constructor.

  15. 15.

    The type inference and type checking rules for enumeration types, disjoint union types, intensional equality types, natural number types, wellordering types, and universes are omitted for brevity.

References

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Appendices

Appendix

A Dependent Type Theory

1.1 A.1 Syntax

Definition 3 (Preterms, Inferable terms and checkable terms)

The collection of preterms (notation \(\varLambda \)) are defined by the following BNF grammar, where \(x\in \mathcal {V}ar\) and \(c\in \mathcal {C}on\).

figure aq

The collection of inferable terms (notation \(\varLambda _{\uparrow }\)) and the collection of checkable terms (notation \(\varLambda _{\downarrow }\)) are simultaneously defined by the following BNF grammar where \(x\in \mathcal {V}ar\) and \(c\in \mathcal {C}on\).

figure ar

Free variables, substitutions, \(\beta \)-reductions are defined in the standard way. The full version of DTT also employs such types as enumeration types, disjoint union types, intensional equality types, natural number types, wellordering types, and universes. See Martin-Löf (1984) for details.

Definition 4

(Vertical/box notation for \(\varSigma \) -type).

figure as

Definition 5

(Implication, conjunction, and negation). Footnote 14

figure at

1.2 A.2 Type System

Definition 6

(Signature). A collection of signatures (notation \(\sigma \)) for an alphabet \((\mathcal {V}ar,\mathcal {C}on)\) is defined by the following BNF grammar:

$$\begin{aligned} \sigma :\,\!:= () \ |\ \sigma ,c:A, \end{aligned}$$

where () is an empty signature, \(c \in \mathcal {C}on\), and \(\sigma \vdash A:\textsf {type}\).

Definition 7

(Context). A collection of contexts under a signature \(\sigma \) (notation \(\varGamma \)) is defined by the following BNF grammar:

$$\begin{aligned} \varGamma :\,\!:= () \ |\ \varGamma ,x:A, \end{aligned}$$

where () is an empty context, \(x \in \mathcal {V}ar\), and \(\varGamma \vdash _\sigma \textsf {type}\).

Definition 8

(Judgment). A judgment of DTT is the following form

$$\begin{aligned} \varGamma \vdash _\sigma M:A, \end{aligned}$$

where \(\varGamma \) is a context under a signature \(\sigma \) and M and A are preterms, which states that there exists a proof diagram of DTT from the context \(\varGamma \) to the type assignment M : A. The subscript \(\sigma \) may be omitted when no confusion arises.

Definition 9

(Truth). The judgment of the form \(\varGamma '~ true \) states that there exists a term M that satisfies \(\varGamma \vdash M:A\).

Definition 10

(Structural rules).

figure au

Definition 11

( \(\varPi \) -types).

figure av

Definition 12

( \(\varSigma \) -types).

figure aw

1.3 A.3 Type Checking

A type checking of UDTT, which has the form

$$\begin{aligned} \llbracket {\varGamma \vdash _\sigma M:A}\rrbracket , \end{aligned}$$

where \(\varGamma \) is a context of DTT under a signature \(\sigma \), M is a checkable term of UDTT, and A is a term of DTT, denotes a (possibly empty) set of proof diagrams of DTT, recursively defined by the following set of rules.

Definition 13

(Type checking rules for checkable terms).

figure ax

1.4 A.4 Type Inference

A type inference of UDTT, which has the form

$$\begin{aligned} \llbracket {\varGamma \vdash _\sigma M:\mathord {?}}\rrbracket , \end{aligned}$$

where \(\varGamma \) is a context of DTT under a signature \(\sigma \) and M is a inferable term of UDTT, denotes a (possibly empty) set of proof diagrams of DTT, recursively defined by the following set of rules.Footnote 15

Definition 14

(Structural rules).

figure ay

Definition 15

( \(\varPi \) -types).

figure az

Definition 16

( \(\varSigma \) -types).

figure ba

Definition 17

(Underspecified types).

figure bb

where \(\llbracket {\varGamma \vdash \mathord {?}:A'}\rrbracket \) is a proof search in DTT.

Definition 18

(Annotated terms).

figure bc

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Bekki, D. (2023). A Proof-Theoretic Analysis of Weak Crossover. In: Yada, K., Takama, Y., Mineshima, K., Satoh, K. (eds) New Frontiers in Artificial Intelligence. JSAI-isAI 2021. Lecture Notes in Computer Science(), vol 13856. Springer, Cham. https://doi.org/10.1007/978-3-031-36190-6_16

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