dependent types in natural language semantics

Dependent Types in Natural Language Semantics

Daisuke Bekki†‡�§

†Ochanomizu University‡CREST, Japan Science and Technology Agency

�National Institute of Informatics§National Institute of Advanced Industrial Science and Technology

4th July, 2015The Second International Workshop on Linguistics of BA

at Future University Hakodate

1. Proof-theoretic turn in discourserepresentation

“Contexts in DTS” Future University Hakodate, 4th July, 2015 /31

1.1. Donkey and E-type anaphora

Since 1980, the enterprise of dynamic semantics has pursued an alternative frame-work to Montagovian semantics, which compensates for the gap between syntacticstructures of natural language sentences involving dynamic binding. The difficultyof this pursuit implies that there is tension between dynamism and compositionality,which have not yet been unified in a coherent semantic theory that accounts for bothaspects.

This tension has been the driving force behind dynamic semantics, and in factsome theories have achieved partial success in unifying the two aspects. Thus, Ishould clarify what I mean by dynamism and compositionality. Dynamic semanticsexplores various empirical data concerning dynamic binding, whose nature is exem-plified by the two paradigms of donkey sentences in (1) by Geach (1962) and E-typeanaphora in (2) by Evans (1980).

(1) a. Every farmer who owns [a donkey]i beats iti.b. If [a farmer]i owns [a donkey]j , hei beats itj .

(2) [A man]i entered. Hei whistled.



As discussed elsewhere, (1a), for example, is problematic in terms of composition-ality. Compositional semantic theory is such that it provides a way to calculate anysemantic representation of any target sentence from the semantic representationsof its parts. The structural analogue of (1a) (and (1b)), which allows us to givea straightforward compositional analysis, is (3). However, it is not an appropriatesemantic representation for (1a) since variable y occurs as a free variable outside ofthe scope of ∃y.

(3) ∀x(Farmer(x) ∧ ∃y(Donkey(y) ∧ Own(x, y)) → Beat(x, y))

In the same way, the structural analogue of (2) is (4), which is not an appropriaterepresentation for (2) since variable x in Whistle(x) is not bound by ∃x.

(4) ∃x(Man(x) ∧Enter(x)) ∧ Whistle(x)



The first-order representations for sentences (1) and (2) necessary in order tocorrectly calculate their entailment relations are (5) and (6), respectively.

(5) ∀x(Farmer(x) → ∀y(Donkey(y) ∧ Own(x, y) → Beat(x, y)))

(6) ∃x(Man(x) ∧Enter(x)∧Whistle(x))

These represent proper information that the sentences (1) and (2) contain, ina sense that any proof system for first-order predicate logic will prove that theinferences in Example 1 and Example 2 are valid.

On the other hand, the structural similarity to the original sentences is lost in(5) and (6), so their direct decomposition does not lead to the respective lexicalizedrepresentations.



Example 1 (Donkey Syllogism).

Every farmer who owns [a donkey]i beats iti.John is a farmer.Bill is a donkey.John owns Bill.John beats Bill.

Example 2 (E-type Syllogisms).

[A man]i entered.Hei whistled.A man entered.

[A man]i entered.Hei whistled.A man whistled.


1.2. Proof-theoretic turn: Sundholm and Ranta

Sundholm (1986) noticed fairly early that dependent type theory provides se-mantic representations for donkey sentences whose structures are parallel to theirsyntactic structures in a different way from DRT Kamp (1981), Kamp and Reyle(1993), DPL Groenendijk and Stokhof (1991), and their successors.

(7) a. A man entered. He whistled.⎡⎢⎢⎣ u:

⎡⎣ x:entity[

man(x)enter(x)

] ⎤⎦

whistle(π1(u))

⎤⎥⎥⎦

(8) a. Every farmer who owns a donkey beats it.

(x:entity) →

⎛⎜⎜⎝u:

⎡⎢⎢⎣

farmer(x)⎡⎣ y:entity[

donkey(y)own(x, y)

] ⎤⎦

⎤⎥⎥⎦

⎞⎟⎟⎠ → beat(x, π1π2(u))


1.3. Key concepts

1. Proof-theoretic semantics (vs. Model-theoretic semantics)

2. Curry-Howard Correspondence (between logic type theory)

3. Dependent types (vs. Simple types)


2. Proof-theoretic Semantics,Curry-Howard Correspondence,and Dependent Types


2.1. Proof-theoretic vs. model-theoretic semantics

The ‘meaning’ of a given proposition φ:

The Proof-theoretic Semantics Model-theoretic Semantics

Provability, or {Γ | Γ � φ} Truth-condition, or {(M, g) | �φ�M,g = 1}

Inference rules Semantic Values(natural deduction): (classical logic):

A....B

i

A → B(→I ),i

A A → BB

(→E)

A BA ∧ B

(∧I )A1 ∧ A2

Ai(∧E),i=1,2

�A → B�M,g = 1⇐⇒ �A�M,g = 0 or �B�M,g = 1

�A ∧ B�M,g = 1⇐⇒ �A�M,g = 1 and �B�M,g = 1


2.2. Curry-Howard Correspondence

Propositional Logic Simply-Typed Lambda Calculus

A....B

i

A → B(→I ),i

x : A....B

i

λx.M : A → B(→I ),i

A A → BB

(→E)N : A M : A → B

MN : B(→E)

A BA ∧ B

(∧I )M : A N : B(M, N) : A × B

(∧I )

A1 ∧ A2

Ai(∧E),i=1,2

(M1, M2) : A1 × A2

πi(M) : Ai

(×E),i=1,2


2.3. Curry-Howard Isomorphism

The correspondence between the notions of logic and type theory:

Logic Type Theoryproposition typeproof termaxiom constant symbolassumption variableprovability inhabitancecut substitutionnormalization reduction. . . . . .

• A term is an encoding of a proof of a type(=proposition)

• A proposition can be regarded as a collection of proofs.

• φ is true under Γ iff φ is inhabited under Γ.


2.4. Dependent function/sum Types

• Dependent function type (x:A) → B is a generalized form of the function/implicationtype A → B:

A → Bdef≡ (x:A) → B where x /∈ fv(B).

∀xBdef≡ (x:entity) → B

• Dependent sum type[

x:AB

]is a generalized form of the product/conjunction

type A × B:

A ∧ Bdef≡

[x:AB

]where x /∈ fv(B).

∃xBdef≡

[x:entityB

]


2.4. Dependent function/sum Types

Natural deduction rules for dependent types:

(x:A) → B : s

x : A....M : B

i

λx : A.M : (x:A) → B(ΠI ),i

M : (x:A) → B N : A

MN : B[N/x](ΠE)

M : A N : B[M/x]

(M, N) :[

x:AB

] (ΣI )M :

[x:AB

]

π1(M) : A(ΣE)

M :[

x:AB

]

π2(M) : B[π1(M)/x](ΣE)


2.5. Dependent Types in Natural LanguageSemantics

(9) Donkey anaphora: Sundholm (1986)a. Every farmer who owns a donkey beats it.

b. (x:entity) →

⎛⎜⎜⎝u:

⎡⎢⎢⎣

farmer(x)⎡⎣ y:entity[

donkey(y)own(x, y)

] ⎤⎦

⎤⎥⎥⎦

⎞⎟⎟⎠ → beat(x, π1π2(u))

(10) E-type anaphora: Ranta (1994)a. A man entered. He whistled.

b.

⎡⎢⎢⎣ u:

⎡⎣ x:entity[

man(x)enter(x)

] ⎤⎦

whistle(π1(u))

⎤⎥⎥⎦

Recall that (x:A) → B is a type for functions from A to B[x], and[

x:AB

]is a

type for pairs of A and B[x].


2.6. Previous Works

Subsequently, the following three approaches have been proposed to obtain Sund-holmian representations:

1. Ahn and Kolb (1990) provides a set of translation rules from DRS to DependentType representations

2. Davila-Perez (1995) presented a reformulation of Montague Grammar Mon-tague (1973) in terms of MLTT.

3. Ranta (1994) proposed a generative theory of grammar based on MLTT, knownas Type-Theoretical Grammar (TTG).


2.6. Previous Works

• Relative and Implicational Donkey Sentences, Branching Quantifiers, Inten-sionality, Tense: Ranta (1994)

• Summation: Fox (1994a,b)

• Presupposition Binding and Accommodation, Bridging: Krahmer and Piwek(1999), Piwek and Krahmer (2000)

• Coercion: Luo (1997, 1999, 2010, 2012b), Asher and Luo (2012)

• Adverbs: Chatzikyriakidis (2014)

• New frameworks: Cooper (2005), Luo (2012a), Bekki (2014), Martin and Pol-lard (2014)


2.6. Previous Works

Recent works in our group:

• Generalized Quantifiers: Tanaka et al. (2013), Tanaka (2014)

• Type checking/inference in DTS and its implementation: Bekki and Sato(2015)

• Modal Subordination: Tanaka et al. (2014)

• Factive Presupposition: Tanaka et al. (2015)

• Conventional Implicature: Bekki and McCready (2014), Watanabe et al. (2014)

• Double-Negated Antecedents: Bekki (2013)


A. Appendix


A.1. Dependent Types in Mathematics and Logic

• The notion of dependent types originates from:

– Martin-Lof Type Theory (MLTT) (Martin-Lof (1975, 1984)), which wasproposed as a foundation of constructive mathematics.

– Calculus of Constructions (CoC) (Coquand and Huet (1988)), which wasproposed as a foundation of functional programming and mathematicalproofs.

• Lately, fragments of MLTT and CoC have been integrated into a general the-ory of the λ-cube (Barendregt (1992)) and Pure Type Systems (PTS) (Berardi(1990); Barendregt (1991)) with other important type theories, such as Gi-rard’s F (Girard et al. (1989)).

• Calculus of Inductive Constructions (CoIC) (=CoC with inductive types) isknown as an underlying language of proof assistants Coq (Bertot and Casteran(2004)) and Agda (Nordstrom et al. (1990), Bove and Dybjer (2008)).


A.1. Dependent Types in Mathematics and Logic

The term “Dependent Type Theory” usually covers:

• λP is an extention of Simple Type Theory (λ→) with dependent functionaltype Π. (cf. Barendregt Cube) λω λC

λ2

��λP2

��

λω λPω

λ→

��λP

��

• Martin-Lof/Constructive/Intuitionistic/Modern Type Theory is an extentionof λP with dependent sum type Σ, (dependent) record type, Equational type,and Natural Numbers, Inductive types, etc.


A.2. What is NOT Dependent Type Theory

• Type-Theoretic Semantics (ex. Montague (1973), Gallin (1975)), where aproposition is a term of type t, while in Dependent Type Theory, a propo-sition is a type (=a collection of proofs).

• Dependent Type Theory is proof-theoretic, but it has a denotational semantics(cf. fibred category theory: Jacobs (1999)) and other types of semantics (cf.game-theoretic semantics) as well.


References

References

Ahn, R. and H.-P. Kolb. (1990) “Discourse Representation meets Constructive Math-ematics”, In: L. Kalman and L. Polos (eds.): Papers from the Second Symposiumon Logic and Language. Akademiai Kiado.

Asher, N. and Z. Luo. (2012) “Formalisation of coercions in lexical semantics”, Inthe Proceedings of Sinn und Bedeutung 17. Paris, pp.63–80.

Barendregt, H. P. (1991) “Introduction to generalized type systems”, Journal ofFunctional Programming 1(2), pp.125–154.

Barendregt, H. P. (1992) “Lambda Calculi with Types”, In: S. Abramsky, D. M.Gabbay, and T. Maibaum (eds.): Handbook of Logic in Computer Science, Vol. 2.Oxford Science Publications, pp.117–309.

Bekki, D. (2013) “A Type-theoretic Approach to Double Negation Elimination inAnaphora”, In the Proceedings of Logic and Engineering of Natural LanguageSemantics 10 (LENLS 10). Tokyo.


References

Bekki, D. (2014) “Representing Anaphora with Dependent Types”, In the Proceed-ings of N. Asher and S. V. Soloviev (eds.): Logical Aspects of Computational Lin-guistics (8th international conference, LACL2014, Toulouse, France, June 2014Proceedings), LNCS 8535. Toulouse, pp.14–29, Springer, Heiderburg.

Bekki, D. and E. McCready. (2014) “CI via DTS”, In the Proceedings of LENLS11.Tokyo, pp.110–123.

Bekki, D. and M. Sato. (2015) “Calculating Projections via Type Checking”, In theProceedings of TYpe Theory and LExical Semantics (TYTLES), ESSLLI2015workshop. Barcelona, Spain.

Berardi, S. (1990) “Type Dependence and Constructive Mathematics”, Ph.d thesis,Mathematical Institute.

Bertot, Y. and P. Casteran. (2004) Interactive Theorem Proving and Program De-velopment. Springer.

Bove, A. and P. Dybjer. (2008) “Dependent Types at Work”.


References

Chatzikyriakidis, S. (2014) “Adverbs in a Modern Type Theory”, In: N. Asher andS. V. Soloviev (eds.): Logical Aspect of Computational Linguistics, 8th Interna-tional Conference, LACL2014, Toulouse, France, June 18-20, 2014 Proceedings.Springer.

Cooper, R. (2005) “Austinian truth, attitudes and type theory”, Research on Lan-guage and Computation 3, pp.333–362.

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Davila-Perez, R. (1995) “Semantics and Parsing in Intuitionistic Categorial Gram-mar”, Ph.d. thesis, University of Essex.

Evans, G. (1980) “Pronouns”, Linguistic Inquiry 11, pp.337–362.

Fox, C. (1994a) “Discourse Representation, Type Theory and Property Theory”, Inthe Proceedings of H. Bunt, R. Muskens, and G. Rentier (eds.): the International


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Gallin, D. (1975) Intensional and Higher-Order Modal Logic. With Application toMontague Semantics. Amsterdam, New York, North-Holland Publisher/ElsevierPublisher.

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References

Jacobs, B. (1999) Categorical Logic and Type Theory, Vol. 141 of Studies in Logicand the Foundations of Mathematics. North Holland, Elsevier.

Kamp, H. (1981) “A Theory of Truth and Semantic Representation”, In: J. Groe-nendijk, T. M. Janssen, and M. Stokhof (eds.): Formal Methods in the Study ofLanguage. Amsterdam, Mathematical Centre Tract 135.

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References

Luo, Z. (1999) “Coercive subtyping”, Journal of Logic and Computation 9(1),pp.105–130.

Luo, Z. (2010) “Type-theoretical semantics with coercive subtyping”, In the Pro-ceedings of Semantics and Linguistic Theory 20 (SALT 20). Vancouver.

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References

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References

Tanaka, R. (2014) “A Proof-Theoretic Approach to Generalized Quantifiers in De-pendent Type Semantics”, In the Proceedings of R. de Haan (ed.): the ESSLLI2014 Student Session, 26th European Summer School in Logic, Language and In-formation. Tubingen, Germany, pp.140–151.

Tanaka, R., K. Mineshima, and D. Bekki. (2014) “Resolving Modal Anaphora inDependent Type Semantics”, In the Proceedings of the Eleventh InternationalWorkshop on Logic and Engineering of Natural Language Semantics (LENLS11),JSAI International Symposia on AI 2014. Tokyo, pp.43–56.

Tanaka, R., K. Mineshima, and D. Bekki. (2015) “Factivity and Presupposition inDependent Type Semantics”, In the Proceedings of TYpe Theory and LExicalSemantics (TYTLES), ESSLLI2015 workshop.

Tanaka, R., Y. Nakano, and D. Bekki. (2013) “Constructive Generalized QuantifiersRevisited”, In the Proceedings of Logic and Engineering of Natural LanguageSemantics 10 (LENLS 10). Tokyo, pp.69–78.


References

Watanabe, N., E. McCready, and D. Bekki. (2014) “Japanese Honorification: Com-positionality and Expressivity”, In the Proceedings of S. Kawahara and M.Igarashi (eds.): FAJL 7: Formal Approaches to Japanese Linguistics, the MITWorking Papers in Linguistics 73. International Christian University, Japan,pp.265–276.


dependent types in natural language semantics

Science

dts future university

etype anaphora example

paradigms of donkey

variable y

semantic representations

donkey syllogism

y beatx

dynamic binding