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- 1. Who’s Afraid of Categorical Models? Valeria de Paiva Logic in Rio 2012 August 2012 Valeria de Paiva Who’s Afraid of Categorical Models?
- 2. Categorical Models? When studying logic one can concentrate on: its models (Model Theory) its proofs (Proof Theory) on foundations and its favorite version (Set Theory) on computability and its effective versions (Recursion Theory). In this talk: we’re interested in Proof Theory, in using categorical models to discuss it, and in modeling Linear Logic using categories. Valeria de Paiva Who’s Afraid of Categorical Models?
- 3. Categorical Models? When studying logic one can concentrate on: its models (Model Theory) its proofs (Proof Theory) on foundations and its favorite version (Set Theory) on computability and its effective versions (Recursion Theory). In this talk: we’re interested in Proof Theory, in using categorical models to discuss it, and in modeling Linear Logic using categories. More prosaically: “Categorical Semantics of Linear Logic for All” Valeria de Paiva Who’s Afraid of Categorical Models?
- 4. Proof Theory: Proofs as Mathematical Objects of Study Frege: quantiﬁers! but also ﬁrst to use abstract symbols to write proofs Hilbert: proofs are mathematical objects of study themselves Gentzen: inference rules the way mathematicians think Natural Deduction and Sequent Calculus Valeria de Paiva Who’s Afraid of Categorical Models?
- 5. Proofs as ﬁrst class objects? Programme: elevate proofs to “ﬁrst class” logical objects. Instead of asking ‘when is a formula A true’, ask ‘what is a proof of A?’ Using Frege’s distinction between sense and denotation: proofs are the senses of logical formulas, whose denotations might be truth values. Sometimes I call this programme Proof Semantics. sometimes I call it Categorical Proof Theory (because the semantics of proofs are given in terms of natural constructions in Category Theory). Valeria de Paiva Who’s Afraid of Categorical Models?
- 6. Proofs as ﬁrst class objects? Programme: elevate proofs to “ﬁrst class” logical objects. Instead of asking ‘when is a formula A true’, ask ‘what is a proof of A?’ Using Frege’s distinction between sense and denotation: proofs are the senses of logical formulas, whose denotations might be truth values. Sometimes I call this programme Proof Semantics. sometimes I call it Categorical Proof Theory (because the semantics of proofs are given in terms of natural constructions in Category Theory). Dummett as a champion Valeria de Paiva Who’s Afraid of Categorical Models?
- 7. Category Theory: Unifying Mathematics since 1945... Basic idea there’s an underlying unity of mathematical concepts and theories. More important than the mathematical concepts themselves is how they relate to each other. Topological spaces come with continuous maps, while vector spaces come with linear transformations, for example. Morphisms are how structures transform into others in a (reasonable) way to organize the mathematical ediﬁce. Detractors call CT “Abstract Nonsense” The language of CT is well-accepted in all branches of Math, the praxis and the philosophy less so. Valeria de Paiva Who’s Afraid of Categorical Models?
- 8. The quest for proofs... Traditional proof theory, to the extent that it relies on models, uses algebraic structures such as Boolean algebras, Heyting algebras or Kripke models of several styles. These models lose one important dimension. In these models different proofs are not represented at all. Provability, the fact that Γ a collection of premisses A1 , . . . , Ak entails A, is represented by the less or equal ≤ relation in the model. This does not give us a way of representing the proofs themselves. We only know if a proof exists Γ ≤ A or not. All proofs are collapsed into the existence of this relation. Valeria de Paiva Who’s Afraid of Categorical Models?
- 9. The quest for proofs... By contrast in categorical proof theory we think and write a proof as Γ →f A where f is the reason why we can deduce A from Γ, a name for the proof we are thinking of. Thus we can observe and name and compare different derivations. Which means that we can see subtle differences in the logics. Valeria de Paiva Who’s Afraid of Categorical Models?
- 10. Categorical Proof Theory: How? Relating Computation to Proof Theory Relating Computation to Categories Valeria de Paiva Who’s Afraid of Categorical Models?
- 11. Computation: Lambda Calculus and Combinatory Logic In the 30s Church introduced “the lambda calculus”, a formal system in mathematical logic for expressing computation using variable binding and substitution. Curry developed “combinatory logic”, a way of dealing with computation without variables. Valeria de Paiva Who’s Afraid of Categorical Models?
- 12. Curry-Howard Correspondence Curry 1934: types of the combinators as axiom-schemes for intuitionistic implicational logic. Curry and Feys 1958: Hilbert-style deduction system coincides with typed fragment of combinatory logic. Prawitz 1965: Natural Deduction normalizes Howard 1969: Intuitionistic natural deduction as a typed variant of the lambda calculus. ⇒ the right leg of the triangle below Valeria de Paiva Who’s Afraid of Categorical Models?
- 13. Curry-Howard Correspondence A triangle of correspondences relating logic in Natural Deduction style (as shown well-behaved by Prawitz) to typed lambda-calculus (as proposed by Howard) to categories and morphisms (as done by Lawvere) and shown to preserve reductions/rewriting (by Tait). The correspondence: types as theorems as objects of a category lambda terms as proofs as morphisms of the category Simpliﬁcation of proofs corresponds to lambda-terms reduction. Valeria de Paiva Who’s Afraid of Categorical Models?
- 14. Valeria de Paiva Who’s Afraid of Categorical Models?
- 15. Is this a coincidence? No! The correspondence works for ¨ Martin-Lof’s “Dependent Type Theory”, Girard/Reynolds “System F” and Coquand’s “Calculus of Constructions” too! But we’re interested in a simpler system Linear Logic Valeria de Paiva Who’s Afraid of Categorical Models?
- 16. Linear Logic Jean-Yves Girard: “[...]linear logic comes from a proof-theoretic analysis of usual logic.” Linear logic is a resource-conscious logic, or a logic of resources. The resources in Linear Logic are premises, assumptions and conclusions, as they are used in logical proofs. Resource accounting: each meaning used exactly once, unless specially marked by ! Great win of Linear Logic: account for resources when you want. Valeria de Paiva Who’s Afraid of Categorical Models?
- 17. Linear Logic Jean-Yves Girard: “[...]linear logic comes from a proof-theoretic analysis of usual logic.” Linear logic is a resource-conscious logic, or a logic of resources. The resources in Linear Logic are premises, assumptions and conclusions, as they are used in logical proofs. Resource accounting: each meaning used exactly once, unless specially marked by ! Great win of Linear Logic: account for resources when you want. only when you want. Valeria de Paiva Who’s Afraid of Categorical Models?
- 18. Resource Counting • $1 −◦ gauloises If I have a dollar, I can get a pack of Gauloises • $1 −◦ gitanes If I have a dollar, I can get a pack of Gitanes • $1 I have a dollar Can conclude: — Either: gauloises — Or: gitanes — But not: gauloises ⊗ gitanes I can’t get Gauloise and Gitanes with $1 Valeria de Paiva Who’s Afraid of Categorical Models?
- 19. Linear Implication and (Multiplicative) Conjunction Traditional implication: A, A → B B A, A → B A∧B Re-use A Linear implication: A, A −◦ B B A, A −◦ B A⊗B Cannot re-use A Traditional conjunction: A ∧ B A Discard B Linear conjunction: A⊗B A Cannot discard B Of course: !A A⊗ !A Re-use ! (A) ⊗ B B Discard Valeria de Paiva Who’s Afraid of Categorical Models?
- 20. Semantics of Proofs:Implication Elimination as Functional Application Natural deduction rule for (intuitionistic) implication elimination: A→B A B A → B: function f that takes a proof a of A to give a proof f (a) of B f :A→B a:A f (a ) : B (Also works for linear implication, −◦ ) Valeria de Paiva Who’s Afraid of Categorical Models?
- 21. Implication Introduction as Lambda Abstraction Natural deduction rule for implication introduction [A]i · ·π · · B →, i A→B Assuming A allows one to prove B. Therefore, discharging the assumption, [A]i , one proves A → B With proof terms [x : A]i · ·π · · P:B →, i λ x .P : A → B Valeria de Paiva Who’s Afraid of Categorical Models?
- 22. Curry-Howard for Linear Logic? Need linear lambda calculus and linear version of cartesian closed category or linear category. Valeria de Paiva Who’s Afraid of Categorical Models?
- 23. Semantics of Proofs:Linear Implication Elimination as Functional Application Natural deduction rule for linear implication elimination: A −◦ B A −◦ E B A −◦ B: function f that takes a proof a of A to give a proof f (a) of B f : A −◦ B a:A f (a ) : B (only difference is that −◦ consumes the only copy of a : A around) Valeria de Paiva Who’s Afraid of Categorical Models?
- 24. Linear Implication Introduction as Lambda Abstraction Natural deduction rule for implication introduction [A]i π B −◦ , i A −◦ B Assuming A allows one to prove B. Therefore, discharging the assumption, [A]i , one proves A −◦ B With proof terms [x : A]i · ·π · · P:B −◦ , i λx .P : A −◦ B Valeria de Paiva Who’s Afraid of Categorical Models?
- 25. Categorical Models? A category C consists of a set of objects and morphisms between objects. C Y f g X Z Examples: category Group of mathematical groups and homomorphisms category Group of Haskell types and programs. Valeria de Paiva Who’s Afraid of Categorical Models?
- 26. Categorical Models? Fundamental idea: propositions interpreted as the objects of an appropriate category (natural deduction) proofs of propositions interpreted as morphisms of that category. Valeria de Paiva Who’s Afraid of Categorical Models?
- 27. Categorical Models? Fundamental idea: propositions interpreted as the objects of an appropriate category (natural deduction) proofs of propositions interpreted as morphisms of that category. A category C is said to be a categorical model of a given logic L, if: 1. For all proofs Γ L M : A there is a morphism [[M ]] : Γ → A in C . 2. For all equalities Γ L M = N : A it is the case that [[M ]] =C [[N ]], where =C refers to equality of morphisms in the category C . Valeria de Paiva Who’s Afraid of Categorical Models?
- 28. Categorical Models? Say a notion of categorical model is complete if for any signature of the logic L there is a category C and an interpretation of the logic in the category such that: If Γ M : A and Γ N : A are derivable in the system then M and N are interpreted as the same map Γ → A in the category C just when M = N : A is provable from the equations of the typed equational logic deﬁning L. Valeria de Paiva Who’s Afraid of Categorical Models?
- 29. Categorical Models of MILL? Fragment of multiplicative intuitionistic linear logic (ILL) consisting only of linear implications and tensor products, plus their identity, the constant I. Natural deduction formulation of the logic is uncontroversial: linear implication are just like the rules for implication in intuitionistic logic (with the understanding that variables always used a single time) Structures consisting of linear-like implications and tensor-like products had been named and investigated by category theorists decades earlier. They are called symmetric monoidal closed categories or smccs. Valeria de Paiva Who’s Afraid of Categorical Models?
- 30. Categorical Models of the Modality ! The sequent calculus rules are intuitive: Duplicating and erasing formulae preﬁxed by “!” correspond to the usual structural rules of weakening and contraction: ∆ B ∆, !A, !A B ∆, !A B ∆, !A B The rules for introducing the modality are more complicated, but familiar from Prawitz’s work on S4. !∆ B ∆, A B !∆ !B ∆, !A B (Note that !∆ means that every formula in ∆ starts with a ! operator.) Valeria de Paiva Who’s Afraid of Categorical Models?
- 31. Categorical Models of the Modality ! Transform the rules above into Natural Deduction ones with a sensible term assignment ∆ M : !A ∆1 M : !A ∆2 N: B ∆ derelict(M ) : A ∆1 , ∆2 discard M in N : B ∆1 M : !A ∆ 2 , a : ! A, b : ! A N: B ∆1 , ∆2 copy M as a, b inN : B ∆1 M1 : !A1 , . . . , ∆k Mk : !Ak a1 : !A1 , . . . , ak : !Ak N: B ∆1 , ∆2 , . . . , ∆k promote Mi for ai in N : !B (controversial...) Valeria de Paiva Who’s Afraid of Categorical Models?
- 32. Categorical Models of the Modality ! The upshot of these rules: each object !A has morphisms of the form er : !A → I and dupl : !A →!A⊗!A, which allow us to erase and duplicate the object !A. These morphisms give !A the structure of a (commutative) comonoid (A comonoid is the dual of a monoid, intuitively like a set with a multiplication and unit). each object !A has morphisms of the form eps : !A → A and delta : !A →!!A that provide it with a coalgebra structure, induced by a comonad. Valeria de Paiva Who’s Afraid of Categorical Models?
- 33. Categorical Models of the Modality ! The upshot of these rules: each object !A has morphisms of the form er : !A → I and dupl : !A →!A⊗!A, which allow us to erase and duplicate the object !A. These morphisms give !A the structure of a (commutative) comonoid (A comonoid is the dual of a monoid, intuitively like a set with a multiplication and unit). each object !A has morphisms of the form eps : !A → A and delta : !A →!!A that provide it with a coalgebra structure, induced by a comonad. How should the comonad structure interact with the comonoid structure? This is where the picture becomes complicated... Valeria de Paiva Who’s Afraid of Categorical Models?
- 34. Lafont Models Lafont suggested (even before LL appeared ofﬁcially) that one should model !A via free comonoids. Deﬁnition A Lafont category consists of 1. A symmetric monoidal closed category C with ﬁnite products, 2. For each object A of C, the object !A is the free commutative comonoid generated by A. Freeness (and co-freeness) of algebraic structures gives very elegant mathematics, but concrete models satisfying cofreeness are very hard to come by. None of the original models of Linear Logic satisﬁed this strong requirement. Valeria de Paiva Who’s Afraid of Categorical Models?
- 35. Lafont Models Lafont suggested (even before LL appeared ofﬁcially) that one should model !A via free comonoids. Deﬁnition A Lafont category consists of 1. A symmetric monoidal closed category C with ﬁnite products, 2. For each object A of C, the object !A is the free commutative comonoid generated by A. Freeness (and co-freeness) of algebraic structures gives very elegant mathematics, but concrete models satisfying cofreeness are very hard to come by. None of the original models of Linear Logic satisﬁed this strong requirement. The notable exception being dialectica categories of yours truly... Valeria de Paiva Who’s Afraid of Categorical Models?
- 36. Seely Models Model the interaction between linear logic and intuitonistic logic via the notion of a comonad ! relating these systems. Seely’s deﬁnition requires the presence of additive conjunctions in the logic and it depends both on named natural isomorphims m : !A⊗!B ∼!(A&B ) and p : 1 ∼!T = = and on the requirement that the functor part of the comonad ‘!’ take the comonoid structure of the cartesian product to the comonoid structure of the tensor product. Valeria de Paiva Who’s Afraid of Categorical Models?
- 37. Seely Models Deﬁnition (Bierman) A new-Seely category, C, consists of 1. A symmetric monoidal closed category C, with ﬁnite products, together with 2. A comonad (!, ε, δ) to model the modality, and 3. Two natural isomorphism, n : !A⊗!B ∼!(A&B ) and p : I ∼!T, = = such that the adjunction between C and its co-Kleisli category is a monoidal adjunction. Valeria de Paiva Who’s Afraid of Categorical Models?
- 38. Linear Categories Deﬁnition (Benton, Bierman, de Paiva, Hyland, 1992) A linear category consists of a symmetric monoidal closed category C, with ﬁnite products, together with 1. A symmetric monoidal comonad (!, ε, δ) to model the modality, 2. Two monoidal natural transformations d and e whose components dA : !A →!A⊗!A and eA : !A → 1 form a commutative comonoid (A, dA , eA ) for all objects A such that: 3. The morphisms dA and eA are co-algebra morphisms, and 4. The co-multiplication of the comonad δ is a comonoid morphism. A mouthful indeed... Valeria de Paiva Who’s Afraid of Categorical Models?
- 39. Linear Categories Deﬁnition (Hyland, Schalk, 1999) A linear category is a symmetric monoidal closed category C, with ﬁnite products, equipped with a linear exponential comonad. A linear exponential comonad unpacks to the previous deﬁnition and the proof of equivalence is long. More surprising (to me, at least) is the reformulation: Deﬁnition (Maneggia, 2004) A linear category is a symmetric monoidal closed category C together with a symmetric monoidal comonad such that the monoidal structure induced on the associated category of Eilenberg-Moore coalgebras is a ﬁnite product structure. Valeria de Paiva Who’s Afraid of Categorical Models?
- 40. LinearNonLinear Models All the notions of model so far have a basis, a symmetric monoidal closed category S modeling the linear propositions and a functor ! : S → S modeling the modality of course!. The differences are which (minimal) conditions do we put on the modality to make sure that we also have a model of intuitionistic logic, a cartesian closed category and whether we do (or do not) assume products as part of the original set-up. To a certain extent a matter of taste: Lafont cats are very special linear cats new-Seely cats are special linear cats too. Valeria de Paiva Who’s Afraid of Categorical Models?
- 41. LinearNonLinear Models Anyway: Having a monoidal comonad on a category C means that this comonad induces a spectrum of monoidal adjunctions spanning from the category of Eilenberg-Moore coalgebras to the co-Kleisli category. (this is basic category theory!) A different proposal came from ideas discussed independently by Benton, Hyland, Plotkin and Barber: putting linear logic and intuitionistic logic on the same footing, making the monoidal adjunction itself (between the linear category and the non-linear category) the model. Valeria de Paiva Who’s Afraid of Categorical Models?
- 42. LinearNonLinear Models Deﬁnition (Benton, 1996) A linear-non-linear (LNL) category consists of a symmetric monoidal closed category S, a cartesian closed category C and a symmetric monoidal adjunction between them. This is much simpler, it required the phd work of Barber to make the lambda-calculus associated (DILL) work. Different kinds of context (linear and nonlinear) in the lambda-calculus allow a small simpliﬁcation. Deﬁnition (Barber, 1996) A dual intuitionistic linear logic (DILL) category is a symmetric monoidal adjunction between S a symmetric monoidal closed category and C a cartesian category. Valeria de Paiva Who’s Afraid of Categorical Models?
- 43. ILL vs DILL Models Benton, Barber and Mellies have proved, independently, that given a linear category we obtain a DILL-category and given a DILL-category we obtain a linear category. Are all these notions of model equivalent then? Valeria de Paiva Who’s Afraid of Categorical Models?
- 44. ILL vs DILL Models Benton, Barber and Mellies have proved, independently, that given a linear category we obtain a DILL-category and given a DILL-category we obtain a linear category. Are all these notions of model equivalent then? Maietti, Maneggia, de Paiva and Ritter (Maietti et al., 2005) set out to prove some kind of categorical equivalence of models but discovered that the situation was not quite as straightforward as expected. Valeria de Paiva Who’s Afraid of Categorical Models?
- 45. ILL vs DILL Models? Bierman proved linear categories are sound and complete for ILL. Barber proved DILL-categories are sound and complete for DILL. The category of theories of ILL is equivalent to the category of theories of DILL. (easy) Have two calculi, ILL and DILL, sound and complete with respect to their models, whose categories of theories are equivalent. One would expect their categories of models (linear categories and symmetric monoidal adjunctions) to be equivalent too. BUT considering the natural morphisms of linear categories and of symmetric monoidal adjunctions (to construct categories Lin and SMA), we do not obtain a categorical equivalence. Only a retraction... somewhat paradoxical situation: calculi with equivalent categories of theories, whose classes of model are not equivalent. Valeria de Paiva Who’s Afraid of Categorical Models?
- 46. What gives? Maietti et al: soundness and completeness of a notion of categorical model are not enough to determine the most apropriate notion of categorical model. More than soundness and completeness need to say a class of categories is a model for a type theory when we can prove an internal language theorem relating the category of models to the category of theories of the calculus. Deﬁnition Say that a typed calculus L provides an internal language for the class of models in M(L)if we can establish an equivalence of categories between the category of L-theories, Th(L) and the category of L-models M(L). Functors L : M(L) → Th(L) and C : Th(L) → M(L) establish the equivalence. Have that M ∼ C (L(M )) and V ∼ L(C (V )) unlike Barr = = and Wells, who only require ﬁrst equivalence. Valeria de Paiva Who’s Afraid of Categorical Models?
- 47. A solution? Maietti et al: Postulate that despite being sound and complete for DILL, symmetric monoidal adjunctions between a cartesian and a symmetric monoidal closed category (the category SMA) are not the models for DILL. Instead take as models for DILL a subcategory of SMA, the symmetric monoidal adjunctions generated by ﬁnite tensor products of free coalgebras. (This idea originally due to Hyland, was expanded on and explained by Benton and Maietti et al.) price to pay for the expected result that equivalent categories of theories imply equivalent categories of models is high: not only we have to keep the more complicated notion of model of linear logic, but we need also to insist that categorical modeling requires soundness, completeness and an internal language theorem. Valeria de Paiva Who’s Afraid of Categorical Models?
- 48. Other solutions? Mogelberg, Birkedal and Petersen(2005) call linear adjunctions the symmetric monoidal adjunctions between an smcc and a cartesian category, say that DILL-models are the full subcategory of the category of linear adjunctions on objects equivalent to the objects induced by linear categories, when performing the product of free coalgebras construction. Mellies: not worry about the strange calculi with equivalent categories of theories, whose classes of model are not equivalent?... Valeria de Paiva Who’s Afraid of Categorical Models?
- 49. Conclusions Explained why we want categorical models and what are they. Surveyed notions of categorical model for intuitionistic linear logic and compared them as categories. Linear categories (in various guises) and symmetric monoidal adjunctions The notion of a symmetric monoidal adjunction (SMA) (between a symmetric monoidal closed category and a cartesian category) is very elegant and appealing BUT the category SMA is too big, has objects and morphisms that do not correspond to objects and morphisms in DILL/ILL. Categorical modeling: soundness, completeness and (essentially) internal language theorems. Modality of course! of linear logic is like any other modality, these are pervasive in logic. More research/more concrete models should clarify the criteria for categorical modeling of modalities Valeria de Paiva Who’s Afraid of Categorical Models?
- 50. Thanks! References: Valeria de Paiva Who’s Afraid of Categorical Models?

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