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Dialectica Categories for the Lambek Calculus

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APA/ASL meeting Seattle, April 2017

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Dialectica Categories for the Lambek Calculus

  1. 1. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories Dialectica Categories for the Lambek Calculus Valeria de Paiva Nuance Communications, CA (joint work with Harley Eades III, Augusta, GA) April, 2017 Valeria de Paiva APA2017 – Seattle, WA
  2. 2. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories Introduction I want to talk to you about modeling the Lambek Calculus, using Dialectica Categories. (dedicated to Jim Lambek, 1922–2014) Valeria de Paiva APA2017 – Seattle, WA
  3. 3. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories Introduction Lambek Calculus (1958, 1988, 1993, 2012) Valeria de Paiva APA2017 – Seattle, WA
  4. 4. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories Introduction Lambek Calculus Dialectica Categories putting things together... Valeria de Paiva APA2017 – Seattle, WA
  5. 5. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories What is the Lambek Calculus? One of the several “type grammars”in use in Linguistics. A long history: Ajdukiewicz [1935], Bar-Hillel [1953], Lambek [1958, 1961], Ades-Steedman [1982], etc. It provides a syntactic account of sentencehood. Two classes of type grammars: 1. Combinatory Categorial Grammar: Szabolcsi [1992], Steedman-Baldridge [2011], etc.. 2. Categorial Type Logics: van Benthem, Morrill [1994], Moortgat [1994], etc.. Combinators/Lambda-calculus distiction. Both classes worked on nowadays Valeria de Paiva APA2017 – Seattle, WA
  6. 6. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories What is the Lambek Calculus? Here a purely a logical system, like usual propositional logic, but with no structural rules at all. Recall the basic logic ‘equation’: A → (B → C) ⇐⇒ A ∧ B → C ⇐⇒ B → (A → C) Now make your conjunction non-commutative, so that A ⊗ B = B ⊗ A Then you end up with two kinds of ‘implication’ ( , ): A → (B C) ⇐⇒ A ⊗ B → C ⇐⇒ B → (A C) Valeria de Paiva APA2017 – Seattle, WA
  7. 7. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories What is the Lambek Calculus? sequent calculus Valeria de Paiva APA2017 – Seattle, WA
  8. 8. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories Why Dialectica Categories? For G¨odel (1958): a way to prove consistency of higher order arithmetic For Girard (1987): a way to show that Linear Logic had serious pedigree For Hyland (1987): An intrinsic way modelling G¨odel’s Dialectica, Proof theory in the abstract (Hyland, 2002) Should produce a CCC, it wouldn’t. For me: a Swiss army knife... Valeria de Paiva APA2017 – Seattle, WA
  9. 9. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories Categorical Proof Theory Types are formulae/objects in appropriate category, Terms/programs are proofs/morphisms in the category, Logical constructors are appropriate categorical constructions. Most important: Reduction is proof normalization (Tait) Outcome: Transfer results/tools from Logic to Categories to Computing Valeria de Paiva APA2017 – Seattle, WA
  10. 10. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories Curry-Howard for Implication Natural deduction rules for implication (without λ-terms) A → B A B [A] · · · · π B A → B Natural deduction rules for implication (with λ-terms) M : A → B N : A M(N): B [x : A] · · · · π M : B λx.M : A → B function application abstraction Valeria de Paiva APA2017 – Seattle, WA
  11. 11. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories Valeria de Paiva APA2017 – Seattle, WA
  12. 12. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories Valeria de Paiva APA2017 – Seattle, WA
  13. 13. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories Valeria de Paiva APA2017 – Seattle, WA
  14. 14. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories The challenges of modeling Linear Logic Traditional categorical modeling of intuitionistic logic: formula A object A of appropriate category A ∧ B A × B (real product) A → B BA (set of functions from A to B) But these are real products, so we have projections (A × B → A) and diagonals (A → A × A) which correspond to deletion and duplication of resources. Not linear!!! Need to use tensor products and internal homs in Category Theory. Hard to define the “make-everything-as-usual”operator ”!”. Valeria de Paiva APA2017 – Seattle, WA
  15. 15. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories The simplest Dialectica Category The Dialectica category Dial2(Sets) objects are triples, an object is A = (U, X, R), where U and X are sets and R ⊆ U × X is a set-theoretic relation. A morphism from A to B = (V , Y , S) is a pair of functions f : U → V and F : Y → X such that uRFy → fuSy. Theorem 1: You just have to find the right structure. . . (de Paiva 1989) The category Dial2(Sets) has a symmetric mo- noidal closed structure, and involution which makes it a model of (exponential-free) multiplicative linear logic. Theorem 2 (Hard part): You still want usual logic. . . There is a comonad ! which models exponentials/modalities, hence recovers Intuitionistic and Classical Logic. Valeria de Paiva APA2017 – Seattle, WA
  16. 16. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories Can we give some intuition for these categories? Blass makes the case for thinking of problems in computational complexity. Intuitively an object of Dial2(Sets) (U, X, R) can be seen as representing a problem. The elements of U are instances of the problem, while the elements of X are possible answers to the problem instances. The relation R says whether the answer is correct for that instance of the problem or not. The morphisms between these problems have two components: while f maps instances of a problem to instances of another, F maps solutions ‘backwards’. Valeria de Paiva APA2017 – Seattle, WA
  17. 17. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories What’s different for the Lambek calculus? Need to have a non-commutative tensor ⊗. Need to have two (left and right) implications. Can we have these disturbing minimally the (admitedly) complicated structures? Valeria de Paiva APA2017 – Seattle, WA
  18. 18. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories The non-commutative Dialectica Category (de Paiva 1992, Amsterdam Colloquium) Category DialM(Sets), objects are A = (U, X, R), where U and X are sets and U × X → M is a M-valued relation. A morphism from A to B = (V , Y , S) is a pair of functions f : U → V and F : Y → X such that R(u, Fy) ≤M S(fu, y). Theorem 3: have the right strux. . . The category DialM(Sets) has a non-symmetric monoidal closed structure, hence it is a model of (exponential-free) non-commutative multiplicative linear logic. Theorem 4 (Hard part): You still want usual logic. . . There is a comonad ! which models exponentials/modalities, and a comonad κ (Yetter) that brings back commutativity. Putting the two together we recover Intuitionistic and Classical Logic. Valeria de Paiva APA2017 – Seattle, WA
  19. 19. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories Conclusions Introduced you to the Lambek calculus, as a relative of Linear Logic Introduced you to Dialectica categories (there’s much more to say...) Described one example of Dialectica categories DialM(Sets), a non-commutative case. Should’ve shown you the modalities that make it work. Advantages over previous work: 1. Proved syntax works as expected. 2. Working on implementation in Agda. Hinted at why one might want to use this system for PLs. To do: comparison with pregroups... Valeria de Paiva APA2017 – Seattle, WA
  20. 20. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories Thank you! Valeria de Paiva APA2017 – Seattle, WA
  21. 21. Introduction Lambek Calculus Categorical Proof Theory Dialectica Categories Some References J. Lambek, The Mathematics of Sentence Structure. American Mathematical Monthly, pages 154–170, 1958. de Paiva, The Dialectica Categories, Cambridge University DPMMS PhD thesis, Technical Report 213, 1991. de Paiva, A Dialectica Model of the Lambek Calculus, In Proc Eighth Amsterdam Colloquium, December 17–20, 1991. Proceedings edited by Martin Stokhof and Paul Dekker, Institute for Logic, Language and Computation, University of Amsterdam, 1992, pp. 445-462. Hyland, J. Martin E. Proof theory in the abstract, Annals of pure and applied logic 114.1-3, 2002, pp. 43-78. Valeria de Paiva APA2017 – Seattle, WA

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