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.


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 themselves
Gentzen: inference rules
the way mathematicians think
Natural 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 of
proofs are given in terms of natural constructions in Category Theory).


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


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.


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.

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.


Categorical Proof Theory: How?

Relating Computation to Proof Theory
Relating Computation to Categories


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.


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


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.


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


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.


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.

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


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


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 → B
With proof terms

[x : A]i
·
·π
·
·
P:B
→, i
λ x .P : A → B


Curry-Howard for Linear Logic?

Need linear lambda calculus and linear version of cartesian closed
category 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
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


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.


Categorical Models?

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


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 .


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.


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.


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.)



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...)



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.



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...


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.


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...


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.


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.


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...


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.


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.



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.



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.


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?


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.


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.


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.

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.


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?...


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

Thanks!

References:


Who's afraid of Categorical models?

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