Going Without: a modality and its role

Introduction
Categorical Proof Theory
Linear Type Theory
Going Without: a modality and its role
Valeria de Paiva
BACAT
Santa Clara University
Feb, 2019
Valeria de Paiva BACAT 2019

Introduction
Linear Type Theory
Thanks Vlad, for keeping BACAT going!
I have talked at BACAT many times.
Feb 2014, I talked about Linear Type Theory.
(https://logic-forall.blogspot.com/2014/07/bacat-yes.html)
Dec 2014, I talked about Constructive Modal Logic.
(https://logic-forall.blogspot.com/2014/12/k-for-kripke.html)
Both make an appearance today!

Introduction
Linear Type Theory
Curry-Howard Enablers

Introduction
Linear Type Theory
Proof Theory using Categories...
Category: a collection of objects and of morphisms, satisfying
obvious laws
Functors: the natural notion of morphism between categories
Natural transformations: the natural notion of morphisms between
functors
Constructors: products, sums, limits, duals....
Adjunctions: an abstract version of equality
How does this relate to logic?
Where are the theorems?
A long time coming:
Schoenﬁnkel (1908), Curry and Feys(1958), Howard (1969,
published in 1980), etc.

Introduction
Linear Type Theory
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

Introduction
Linear Type Theory
Categorical Proofs are Programs
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 CT to CSci, etc

Introduction
Linear Type Theory

Introduction
Linear Type Theory
Model derivations/proofs, not whether theorems are true or
not
Proofs deﬁnitely ﬁrst-class citizens
How? Uses extended Curry-Howard correspondence
Why is it good? Modeling derivations useful where you need
proofs themselves, in linguistics, functional programming,
compilers, etc.
Why is it important? Widespread use of logic/algebra in CS
means new important problems to solve with our favorite
tools.
Why so little impact on maths, CS or logic?

Introduction
Linear Type Theory
How many Curry-Howard Correspondences?
Many!!!
Intuitionistic Propositional Logic, System F, Dependent Type
Theory (Martin-L¨of), Linear Logic, Constructive Modal Logics,
various versions of Classical Logic since the early 90’s.
The programs corresponding to these logical systems are
‘futuristic’ programs.
The logics inform the design of new type systems, that can be used
in new applications.

Introduction
Linear Type Theory
Linear Logic: a tool for semantics
A proof theoretic logic described by Girard in 1986.
Basic idea: assumptions cannot be discarded or duplicated.
They must be used exactly once – just like dollar bills (except
when they’re marked by a modality !)
Other approaches to accounting for logical resources before.
Relevance Logic!
Great win of Linear Logic:
Account for resources when you want to, otherwise fall back
to traditional logic via translation A → B iﬀ !A −◦ B

Introduction
Linear Type Theory
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

Introduction
Linear Type Theory
The challenges of modeling Linear Logic
Traditional 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) and
they relate via adjunction
A ∧ B → C ⇐⇒ A → Hom(B, C) = CB
These are real products, have projections (A × B → A)
and diagonals (A → A × A) corresponding to deletion and
duplication of resources.
Not linear!!!
Need to use tensor products and internal homs.
Hard to deﬁne the “make-everything-usual”operator ”!”.

Introduction
Linear Type Theory
Varieties of Linear Type Theory
Girard and Lafont Combinators (1988)
Abramsky (1993) and Mackie Lilac (JFP 1994)
Wadler’s There’s no substitute for linear logic (1991)
Benton et al Linear Lambda Calculus (1992)
Barber’s DILL (1996)
Benton LNL (Linear-Non-Linear Calculus 1995)
Mints (1998), Roversi and Della Rocca (1994), Lincoln and
Mitchell (1992), Negri, Pfenning (2001), Martini and Masini
(1995), etc.

Introduction
Linear Type Theory
Linear Type Theories with Categorical Models
Linear λ-calculus
https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-
262.html
DILL system
http://www.lfcs.inf.ed.ac.uk/reports/96/ECS-LFCS-96-347/
LNL system
https://www.researchgate.net/publication/22155807AMixedLinearan
LinearLogicProofsTermsandModels

Introduction
Linear Type Theory
Linear λ-calculus

Introduction
Linear Type Theory
DILL

Introduction
Linear Type Theory
LNL

Introduction
Linear Type Theory
Categorical Models
Linear λ-calculus (ILL)
A linear category is a symmetric monoidal closed (smc)
category equipped with a linear monoidal comonad, such that
the co-Kleisli category of the comonad is a Cartesian closed
Category (CCC). (loads of conditions)
DILL system
A DILL-category is a pair, a symmetric monoidal closed
category and a cartesian category, related by a monoidal
adjunction.
LNL system
An LNL-model is a monoidal adjunction between a smcc and
a ccc.

Introduction
Linear Type Theory
In pictures

Introduction
Linear Type Theory
Linear Curry-Howard Isos
“Relating Categorical Semantics for Intuitionistic Linear
Logic”(with P. Maneggia, M. Maietti and E. Ritter), Applied
Categorical Structures, vol 13(1):1–36, 2005.
Take home: Categorical models need to be more than sound and
complete. They need to provide internal languages for the theories
they model.

Introduction
Linear Type Theory
Why a new calculus?
(ILT, Fossacs 2000)
Choosing between the three type theories:
DILL is best, less verbose than ILL, but closer to what we
want to do than LNL.
Easy formulation of the promotion rule
Contains the usual lambda-calculus as a subsystem
however, most FPers would prefer to use a variant of DILL
where instead of !, one has two function spaces, → and −◦
But then what’s the categorical model?

Introduction
Linear Type Theory
Intuitionistic and Linear Type Theory
(ILT, Fossacs 2000)
If we have two function spaces, but no modality !, how can we
model it? All the models discussed before have a notion of !,
created by the adjunction.
Well, we need to use deeper mathematics, i.e. ﬁbrations or
indexed categories.

Introduction
Linear Type Theory
Have ILT MODELS
(Maietti et al, 2000)
Idea goes back to Lawvere’s hyperdoctrines satisfying the
comprehension axiom.
Model of ILT should modify this setting to capture the separation
between intuitionistic and linear variables.
A base category B, which models the intuitionistic contexts of ILT,
objects in B model contexts (Γ|). Each fibre over an object in B
modelling a context models terms Γ|∆ M : A for any context ∆
The fibres are now symmetric monoidal closed categories with
finite products and model the linear constructions of ILT

Introduction
Linear Type Theory
Have ILT MODELS
and have full theorems
Missing an understanding of what is essential, what can be
changed, and limitations of the methods.

Introduction
Linear Type Theory
Speculation
Can we do the same for Modal Logic S4?
Can we do the same for Relevant Logic?
I hope so.
Meanwhile pictures do help to remember
what we did manage to see.

Introduction
Linear Type Theory
Intuitionistic Type Theory

Introduction
Linear Type Theory
Linear Type Theory

Introduction
Linear Type Theory
Thank you!

Going Without: a modality and its role

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