Who's afraid of Categorical models?

Who’s Afraid of Categorical Models?

Valeria de Paiva

Logic in Rio 2012

August 2012

Valeria de Paiva Who’s Afraid of Categorical Models?

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”


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”


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 themselvesGentzen: inference rules

the way mathematicians thinkNatural Deduction and Sequent Calculus


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 ofproofs are given in terms of natural constructions in Category Theory).

Dummett as a champion


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 ofproofs are given in terms of natural constructions in Category Theory).Dummett as a champion


Category Theory: Unifying Mathematics since 1945...

Basic idea there’s an underlying unity of mathematical concepts andtheories. More important than the mathematical concepts themselvesis how they relate to each other.

Topological spaces come with continuous maps, while vector spacescome 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.


The quest for proofs...

Traditional proof theory, to the extent that it relies on models, usesalgebraic structures such as Boolean algebras, Heyting algebras orKripke models of several styles.These models lose one important dimension. In these models differentproofs are not represented at all.Provability, the fact that Γ a collection of premisses A1, . . . ,Ak entailsA, 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 collapsedinto the existence of this relation.


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 theproof we are thinking of. Thus we can observe and name andcompare different derivations. Which means that we can see subtledifferences in the logics.


Categorical Proof Theory: How?

Relating Computation to Proof TheoryRelating Computation to Categories


Computation: Lambda Calculus and Combinatory Logic

In the 30s Church introduced “the lambda calculus”, a formal system inmathematical logic for expressing computation using variable bindingand substitution.Curry developed “combinatory logic”, a way of dealing withcomputation without variables.


Curry-Howard Correspondence

Curry 1934: types of the combinators as axiom-schemes forintuitionistic implicational logic.Curry and Feys 1958: Hilbert-style deduction system coincides withtyped fragment of combinatory logic.Prawitz 1965: Natural Deduction normalizesHoward 1969: Intuitionistic natural deduction as a typed variant of thelambda calculus.⇒ the right leg of the triangle below


Curry-Howard Correspondence

A triangle of correspondences relating logic in Natural Deduction style(as shown well-behaved by Prawitz) to typed lambda-calculus (asproposed by Howard) to categories and morphisms (as done byLawvere) and shown to preserve reductions/rewriting (by Tait).

The correspondence:types as theorems as objects of a categorylambda terms as proofs as morphisms of the categorySimplification of proofs corresponds to lambda-terms reduction.



Is this a coincidence?

No! The correspondence works forMartin-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


Linear Logic

Jean-Yves Girard: “[...]linear logic comes from a proof-theoreticanalysis of usual logic.”

Linear logic is a resource-conscious logic, or a logic of resources.

The resources in Linear Logic are premises, assumptions andconclusions, as they are used in logical proofs.Resource accounting: each meaning used exactly once, unlessspecially marked by !Great win of Linear Logic: account for resources when you want.

only when you want.


Linear Logic

Jean-Yves Girard: “[...]linear logic comes from a proof-theoreticanalysis of usual logic.”

Linear logic is a resource-conscious logic, or a logic of resources.

The resources in Linear Logic are premises, assumptions andconclusions, as they are used in logical proofs.Resource accounting: each meaning used exactly once, unlessspecially marked by !Great win of Linear Logic: account for resources when you want.only when you want.


Resource Counting

• $1−◦ gauloisesIf 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


Linear Implication and (Multiplicative) Conjunction

Traditional implication: A,A→ B ` BA,A→ B ` A∧ B Re-use A

Linear implication: A,A−◦ B ` BA,A−◦ B 6` A⊗ B Cannot re-use A

Traditional conjunction: A∧ B ` A Discard B

Linear conjunction: A⊗ B 6` A Cannot discard B

Of course: !A ` A⊗!A Re-use!(A)⊗ B ` B Discard


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, −◦ )


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→ BWith proof terms

[x : A]i····

π

P : B→, i

λx .P : A→ B


Curry-Howard for Linear Logic?

Need linear lambda calculus and linear version of cartesian closedcategory or linear category.


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)


Linear Implication Introduction as Lambda Abstraction

Natural deduction rule for implication introduction

[A]i

πB−◦ , i

A−◦ B

Assuming A allows one to prove B.Therefore, discharging the assumption, [A]i , one proves A−◦ BWith proof terms

[x : A]i····

π

P : B−◦ , i

λx .P : A−◦ B


Categorical Models?

A category C consists of a set of objects and morphisms betweenobjects.

C

X

Y

Z

f g

Examples: category Group of mathematical groups andhomomorphismscategory Group of Haskell types and programs.


Categorical Models?

Fundamental idea:propositions interpreted as the objects of an appropriate category(natural deduction) proofs of propositions interpreted as morphisms ofthat 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.


Categorical Models?

Fundamental idea:propositions interpreted as the objects of an appropriate category(natural deduction) proofs of propositions interpreted as morphisms ofthat 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.


Categorical Models?

Say a notion of categorical model is complete if for any signature ofthe logic L there is a category C and an interpretation of the logic inthe category such that:If Γ ` M : A and Γ ` N : A are derivable in the system then M and Nare interpreted as the same map Γ→ A in the category C just whenM = N : A is provable from the equations of the typed equational logicdefining L.


Categorical Models of MILL?

Fragment of multiplicative intuitionistic linear logic (ILL) consisting onlyof linear implications and tensor products, plus their identity, theconstant I.Natural deduction formulation of the logic is uncontroversial: linearimplication 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-likeproducts had been named and investigated by category theoristsdecades earlier.They are called symmetric monoidal closed categories or smccs.


Categorical Models of the Modality !

The sequent calculus rules are intuitive: Duplicating and erasingformulae prefixed by “!” correspond to the usual structural rules ofweakening and contraction:

∆ ` B

∆, !A ` B

∆, !A, !A ` B

∆, !A ` B

The rules for introducing the modality are more complicated, butfamiliar from Prawitz’s work on S4.

!∆ ` B

!∆ `!B

∆,A ` B

∆, !A ` B

(Note that !∆ means that every formula in ∆ starts with a ! operator.)



Transform the rules above into Natural Deduction ones with a sensibleterm assignment

∆ ` M : !A

∆ ` derelict(M) : A

∆1 ` M : !A ∆2 ` N : B

∆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...)



The upshot of these rules: each object !A has morphisms of the former : !A→ I and dupl : !A→!A⊗!A, which allow us to erase andduplicate 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 amultiplication and unit).each object !A has morphisms of the form eps : !A→ A anddelta : !A→!!A that provide it with a coalgebra structure, induced by acomonad.

How should the comonad structure interact with the comonoidstructure? This is where the picture becomes complicated...



The upshot of these rules: each object !A has morphisms of the former : !A→ I and dupl : !A→!A⊗!A, which allow us to erase andduplicate 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 amultiplication and unit).each object !A has morphisms of the form eps : !A→ A anddelta : !A→!!A that provide it with a coalgebra structure, induced by acomonad.

How should the comonad structure interact with the comonoidstructure? This is where the picture becomes complicated...


Lafont Models

Lafont suggested (even before LL appeared officially) that one shouldmodel !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 commutativecomonoid generated by A.

Freeness (and co-freeness) of algebraic structures gives very elegantmathematics, but concrete models satisfying cofreeness are very hardto come by.None of the original models of Linear Logic satisfied this strongrequirement.

The notable exception being dialectica categories of yours truly...


Lafont Models

Lafont suggested (even before LL appeared officially) that one shouldmodel !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 commutativecomonoid generated by A.

Freeness (and co-freeness) of algebraic structures gives very elegantmathematics, but concrete models satisfying cofreeness are very hardto come by.None of the original models of Linear Logic satisfied this strongrequirement.The notable exception being dialectica categories of yours truly...


Seely Models

Model the interaction between linear logic and intuitonistic logic via thenotion of a comonad ! relating these systems.Seely’s definition requires the presence of additive conjunctions in thelogic 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 ‘!’ takethe comonoid structure of the cartesian product to the comonoidstructure of the tensor product.


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 amonoidal adjunction.


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 whosecomponents dA : !A→!A⊗!A and eA : !A→ 1 form acommutative 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...


Linear Categories

Definition (Hyland, Schalk, 1999)

A linear category is a symmetric monoidal closed category C, withfinite products, equipped with a linear exponential comonad.

A linear exponential comonad unpacks to the previous definition andthe proof of equivalence is long. More surprising (to me, at least) is thereformulation:

Definition (Maneggia, 2004)

A linear category is a symmetric monoidal closed category C togetherwith a symmetric monoidal comonad such that the monoidal structureinduced on the associated category of Eilenberg-Moore coalgebras isa finite product structure.


LinearNonLinear Models

All the notions of model so far have a basis, a symmetric monoidalclosed 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 themodality to make sure that we also have a model of intuitionistic logic,a cartesian closed category and whether we do (or do not) assumeproducts 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 speciallinear cats too.



Anyway: Having a monoidal comonad on a category C means that thiscomonad induces a spectrum of monoidal adjunctions spanning fromthe category of Eilenberg-Moore coalgebras to the co-Kleisli category.(this is basic category theory!)

A different proposal came from ideas discussed independently byBenton, Hyland, Plotkin and Barber: putting linear logic andintuitionistic logic on the same footing, making the monoidal adjunctionitself (between the linear category and the non-linear category) themodel.



Definition (Benton, 1996)

A linear-non-linear (LNL) category consists of a symmetric monoidalclosed category S, a cartesian closed category C and a symmetricmonoidal adjunction between them.

This is much simpler, it required the phd work of Barber to make thelambda-calculus associated (DILL) work. Different kinds of context(linear and nonlinear) in the lambda-calculus allow a smallsimplification.

Definition (Barber, 1996)

A dual intuitionistic linear logic (DILL) category is a symmetricmonoidal adjunction between S a symmetric monoidal closed categoryand C a cartesian category.


ILL vs DILL Models

Benton, Barber and Mellies have proved, independently, that given alinear category we obtain a DILL-category and given a DILL-categorywe obtain a linear category.Are all these notions of model equivalent then?

Maietti, Maneggia, de Paiva and Ritter (Maietti et al., 2005) set out toprove some kind of categorical equivalence of models but discoveredthat the situation was not quite as straightforward as expected.


ILL vs DILL Models

Benton, Barber and Mellies have proved, independently, that given alinear category we obtain a DILL-category and given a DILL-categorywe obtain a linear category.Are all these notions of model equivalent then?

Maietti, Maneggia, de Paiva and Ritter (Maietti et al., 2005) set out toprove some kind of categorical equivalence of models but discoveredthat the situation was not quite as straightforward as expected.


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 theoriesof DILL. (easy)Have two calculi, ILL and DILL, sound and complete with respect totheir models, whose categories of theories are equivalent. One wouldexpect their categories of models (linear categories and symmetricmonoidal adjunctions) to be equivalent too.BUT considering the natural morphisms of linear categories and ofsymmetric monoidal adjunctions (to construct categories Lin andSMA), we do not obtain a categorical equivalence. Only a retraction...somewhat paradoxical situation:calculi with equivalent categories of theories, whose classes of modelare not equivalent.


What gives?

Maietti et al: soundness and completeness of a notion of categoricalmodel are not enough to determine the most apropriate notion ofcategorical model.More than soundness and completeness need to say a class ofcategories is a model for a type theory when we can prove an internallanguage theorem relating the category of models to the category oftheories of the calculus.

Definition

Say that a typed calculus L provides an internal language for the classof models inM(L)if we can establish an equivalence of categoriesbetween the category of L-theories, Th(L) and the category ofL-modelsM(L).

Functors L :M(L)→ Th(L) and C : Th(L)→M(L) establish theequivalence. Have that M ∼= C(L(M)) and V ∼= L(C(V )) unlike Barrand Wells, who only require first equivalence.


A solution?

Maietti et al: Postulate that despite being sound and complete forDILL, symmetric monoidal adjunctions between a cartesian and asymmetric monoidal closed category (the category SMA) are not themodels for DILL.Instead take as models for DILL a subcategory of SMA, the symmetricmonoidal adjunctions generated by finite tensor products of freecoalgebras.(This idea originally due to Hyland, was expanded on and explained byBenton and Maietti et al.)price to pay for the expected result that equivalent categories oftheories imply equivalent categories of models is high: not only wehave to keep the more complicated notion of model of linear logic, butwe need also to insist that categorical modeling requires soundness,completeness and an internal language theorem.


Other solutions?

Mogelberg, Birkedal and Petersen(2005) call linear adjunctions thesymmetric monoidal adjunctions between an smcc and a cartesiancategory, say that DILL-models are the full subcategory of thecategory of linear adjunctions on objects equivalent to the objectsinduced by linear categories, when performing the product of freecoalgebras construction.

Mellies: not worry about the strange calculi with equivalent categoriesof theories, whose classes of model are not equivalent?...


Conclusions

Explained why we want categorical models and what are they.Surveyed notions of categorical model for intuitionistic linear logic andcompared them as categories.Linear categories (in various guises) and symmetric monoidaladjunctionsThe notion of a symmetric monoidal adjunction (SMA) (between asymmetric monoidal closed category and a cartesian category) is veryelegant and appealingBUT the category SMA is too big, has objects and morphisms that donot 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 arepervasive in logic.More research/more concrete models should clarify the criteria forcategorical modeling of modalities


Thanks!

References:


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Who's afraid of Categorical models?

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