linear type theory revisited (bacat feb 2014)

Linear Type Theory Revisited

Valeria de PaivaNuance Comms

February 21, 2014

Introduction

I Discuss very old work (early 90’s) on linear type theories

I based on Term Assignment for Intuitionistic Linear Logic.(Benton, Bierman, de Paiva and Hyland). Technical Report262, University of Cambridge Computer Laboratory 1992.

I available fromhttp://www.cs.bham.ac.uk/ vdp/publications/papers.html.

I then get to the state-of-the art...

Summary from Abstract

I Investigate the problem of deriving a term assignment systemfor Girard’s Intuitionistic Linear Logic

I both sequent calculus and natural deduction proof systemsI satisfying two important properties:

I the substitution property (the set of valid deductions is closedunder substitution) and

I subject reduction (reduction on terms is well-typed)

I a categorical model for Intuitionistic Linear Logic and how touse it to derive the term assignment

I long (57 pages) but slow and easy...

I both sequent calculus and natural deduction proof systems

I satisfying two important properties:I the substitution property (the set of valid deductions is closed

under substitution) andI subject reduction (reduction on terms is well-typed)

Introduction

I the problem of deriving a term assignment system for Girard’sIntuitionistic Linear Logic

I Previous approaches have simply annotated the sequentcalculus with terms and have given little or no justification fortheir choice

I Phil Wadler: There’s no substitute for LL

I substitution lemma does not hold for the term assignmentsystem in Abramsky’s ‘Computational Interpretations of LinearLogic’ (Cited by 546)

Introduction

Digression: Other old work...

I Linear types can change the world

I Is there a use for linear logic?

I A taste of linear logic

I Operational interpretations of linear logic

I Reference counting as a computational interpretation of linearlogic (Chirimar)

Introduction 2

I solving the problem of deriving a term assignment system forGirard’s Intuitionistic Linear Logic:

I Two waysI By considering the sequent calculus formulation of the logic

and using the underlying categorical constructions to suggest aterm assignment system

I By considering a linear natural deduction system

I two approaches produce equivalent term assignment systems

I BUT for equality (via reduction of terms) matters are moresubtle: natural equalities for category theory are stronger thanthose suggested by computational considerations

Introduction 2

Outline of TR

I Girard’s Intuitionistic Linear Logic:

I a simple categorical model of Intuitionistic Linear Logic(sequent calculus)

I a linear system of natural deduction

I how our two systems of Intuitionistic Linear Logic are related

I the process of proof normalisation within the linear naturaldeduction system

I model for Intuitionistic Linear Logic

I cut-elimination in the sequent calculus

I conclusions

Outline of TR

I conclusions

Outline of TR

I conclusions

Outline of TR

I conclusions

Outline of TR

I conclusions

Outline of TR

I conclusions

Outline of TR

I conclusions

Outline of TR

I conclusions

Intuitionistic Linear Logic

Recommended if sequent calculus/ND are not part of your vocab:Jean Gallier: Constructive Logics. Part I: A Tutorial on ProofSystems and Typed lambda-Calculi. Theoretical ComputerScience, 110(2), 249-239 (1993). fromhttp://www.cis.upenn.edu/ jean/gbooks/logic.html

I multiplicative fragment of Intuitionistic Linear Logic (ILL)

I a refinement of intuitionistic logic (IL) where formulae mustbe used exactly once

I Weakening and Contraction rules are removed

I To regain the expressive power, rules returned in a controlledmanner using operator “!”

I “!” similar to the modal necessity operator �

Intuitionistic Linear Logic Rules

Generic Categorical considerations

I sequent calculus: providing not proofs themselves but ameta-theory concerning proofs

I fundamental idea of categorical treatment of proof theoryI propositions interpreted as objects of a category (or

multicategory or polycategory)

I proofs interpreted as maps of the categoryI operations transforming proofs into proofs then correspond (if

possible) to natural transformations between appropriatehom-functors

multicategory or polycategory)I proofs interpreted as maps of the category

I operations transforming proofs into proofs then correspond (ifpossible) to natural transformations between appropriatehom-functors

multicategory or polycategory)I proofs interpreted as maps of the categoryI operations transforming proofs into proofs then correspond (if

possible) to natural transformations between appropriatehom-functors

Categorical considerationsI dealing with sequents Γ ` A in principle we should deal with

multicategories

I simplifying assumption: multicategorical structure representedby a tensor product .

I a sequent of form

C1,C2, . . . ,Cn ` A

will be represented by

C1 ⊗ C2 ⊗ . . .⊗ Cn → A

sometimes written as Γ→ AI (a coherence result is assumed)I We seek to enrich the sequent judgement to a term

assignment judgement of the form

x1 : C1, x2 : C2, . . . , xn : Cn ` e : A

where the xi are distinct variables and e is a term.

multicategoriesI simplifying assumption: multicategorical structure represented

by a tensor product .

I a sequent of form

C1,C2, . . . ,Cn ` A

C1 ⊗ C2 ⊗ . . .⊗ Cn → A

x1 : C1, x2 : C2, . . . , xn : Cn ` e : A

by a tensor product .I a sequent of form

C1,C2, . . . ,Cn ` A

C1 ⊗ C2 ⊗ . . .⊗ Cn → A

sometimes written as Γ→ A

I (a coherence result is assumed)I We seek to enrich the sequent judgement to a term

x1 : C1, x2 : C2, . . . , xn : Cn ` e : A

C1,C2, . . . ,Cn ` A

C1 ⊗ C2 ⊗ . . .⊗ Cn → A

sometimes written as Γ→ AI (a coherence result is assumed)

I We seek to enrich the sequent judgement to a termassignment judgement of the form

x1 : C1, x2 : C2, . . . , xn : Cn ` e : A

C1,C2, . . . ,Cn ` A

C1 ⊗ C2 ⊗ . . .⊗ Cn → A

x1 : C1, x2 : C2, . . . , xn : Cn ` e : A

Identity and Cut

I The sequent representing the Identity rule is interpreted as thecanonical identity arrow idA : A→ A

I The corresponding rule of term formation is x : A ` x : A

I The rule of Exchange we interpret by assuming that we have asymmetry for the tensor product

I We suppress Exchange and the corresponding symmetry,considering multisets of formulae, so no term formingoperations result from this rule (others do diff...)

I The cut rule is interpreted as a generalized form ofcomposition. if the maps f : Γ→ A and g : A,∆→ B are theinterpretations of hypotheses of the rule, then the compositeΓ⊗∆→f⊗1∆ A⊗∆→g B is the interpretation of theconclusion

Identity and Cut

Substitution in Terms...

I Assumption: any logical rule corresponds to an operation onmaps of the category which is natural in the interpretations ofthe components of the sequents which remain unchangedduring the application of a rule

I Composition corresponds to Cut: our operations commutewhere appropriate with Cut.

I Composition is interpreted by textual substitution thus thefree variables have to reflect the possibility for substitution.

I The rules for the constant I are parallel to the rules for thetensor product ⊗, which we describe next.

I The rules for (linear) implication are usual.

I The rules for the modality ! are the hard ones. these slides will

get us to page 12 of the report...

Text substitution Terms

use this and BB to get equation (2)

Tensor Terms

I Abramsky introduced a let constructor for tensor products, areasonable syntax to bind variables x and y in the term f .

I Naturality in C gives rise to the equation equation explaininghow lets interact with composition.

I For the tensoring on the right, we simply multiply (or tensor)the terms.

I since the constant I is the unity for the tensor operation, itsrules are similar.

Tensor Terms

Implication Terms

I Are just like usual implication (lambda) terms

I the difference is that the variable that your lambda binds hasto be present in the context and only once

I There is an explanation for why the Yoneda lemma isresponsible for the simplification that one can do in the syntax(and why Peter Schroeder-Heister is really right about hisformulation of Natural Deduction) but we don’t need to gothere.

Implication Terms

Modality ! Term Assignment

I The left rules are ok. The dereliction rules creates a binder, abit like the tensor and the I rules.

I The weakening and contraction rules discard and copyvariables, and have naturality conditions as the tensor rule.

I The problematic rule, ”promotion”

Multiplicative Term Assignment

Conclusions so Far...

I The categorical model can give you guidance for the shape of thetype theory rules.

I Categorical rules are about full beta and eta equivalence plus somenaturality conditions (not very liked by FPers)

I Newer models plus new term calculi via DILL (Dual IntuitionisticLinear Logic) and monoidal adjunction models (Benton, Barber,Mellies, who else?...)

I Which one to read about?

I Rewriting is a new ball game, which I would like to investigate toocf. Barney Hilken paper “Towards a proof theory of rewriting: thesimply-typed 2-[lambda] calculus” Theor. Comp. Sci. Vol. 170,1-2, pp 407-444. 1996.

linear type theory revisited (bacat feb 2014)

Education

linear types

linear type theories

introductionthe problem

sequent calculus formulation

subject reduction reduction

substitution property

old work

ll substitution lemma