Introduction to linear logic Emmanuel Beffara IML, CNRS & - - PowerPoint PPT Presentation

Introduction to linear logic

Emmanuel Beffara

IML, CNRS & Université d’Aix-Marseille

Summmer school on linear logic and geometry of interaction Torino – 27th August 2013 Lecture notes are available at http://iml.univ-mrs.fr/~beffara/intro-ll.pdf

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 1 / 84

The proof-program correspondence Linear sequent calculus A bit of semantics A bit of proof theory Proof nets

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 2 / 84

Plan

The proof-program correspondence The Curry-Howard isomorphism Denotational semantics Linearity in logic Linear sequent calculus A bit of semantics A bit of proof theory Proof nets

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 3 / 84

What are we doing here?

Proof theory in 3 dates: 1900 Hilbert: the question of foundations of mathematics 1930 Gödel: incompleteness theorem Gentzen: sequent calculus and cut elimination 1960 Curry-Howard correspondence The central question: consistency logic: is my logical system degenerate? computation: can my program go wrong? Implies a search for meaning: semantics.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 4 / 84

What are we doing here?

Proof theory in 3 dates: 1900 Hilbert: the question of foundations of mathematics 1930 Gödel: incompleteness theorem Gentzen: sequent calculus and cut elimination 1960 Curry-Howard correspondence The central question: consistency logic: is my logical system degenerate? computation: can my program go wrong? Implies a search for meaning: semantics.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 4 / 84

Curry-Howard: the setting

Definition

Formulas of propositional logic: 𝐵, 𝐶 ∶= 𝛽 propositional variables 𝐵 ⇒ 𝐶 implication 𝐵 ∧ 𝐶 conjunction

Definition

Terms of the simply-typed 𝜇-calculus with pairs: 𝑢, 𝑣 ∶= 𝑦 variable 𝜇𝑦𝐵.𝑢 abstraction, i.e. function (𝑢)𝑣 application ⟨𝑢, 𝑣⟩ pairing 𝜌𝑗𝑢 projection, with 𝑗 = 1 or 𝑗 = 2

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 5 / 84

Curry-Howard: statics

Identity: ax Γ, 𝑦 ∶ 𝐵 ⊢ 𝑦 ∶ 𝐵 Implication: Γ, 𝑦 ∶ 𝐵 ⊢ 𝑢 ∶ 𝐶 ⇒I Γ ⊢ 𝜇𝑦𝐵.𝑢 ∶ 𝐵 ⇒ 𝐶 Γ ⊢ 𝑢 ∶ 𝐵 ⇒ 𝐶 Γ ⊢ 𝑣 ∶ 𝐵 ⇒E Γ ⊢ (𝑢)𝑣 ∶ 𝐶 Conjunction: Γ ⊢ 𝑢 ∶ 𝐵 Γ ⊢ 𝑣 ∶ 𝐶 ∧I Γ ⊢ ⟨𝑢, 𝑣⟩ ∶ 𝐵 ∧ 𝐶 Γ ⊢ 𝑢 ∶ 𝐵 ∧ 𝐶 ∧E1 Γ ⊢ 𝜌1𝑢 ∶ 𝐵 Γ ⊢ 𝑢 ∶ 𝐵 ∧ 𝐶 ∧E2 Γ ⊢ 𝜌2𝑢 ∶ 𝐶

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 6 / 84

The typed 𝜇-calculus

Definition

Evaluation is the relation generated by the pair of rules (𝜇𝑦.𝑢)𝑣 ⇝ 𝑢[𝑣/𝑦] and 𝜌𝑗⟨𝑢1, 𝑢2⟩ ⇝ 𝑢𝑗 for 𝑗 = 1 or 𝑗 = 2

Theorem (Subject reduction)

If Γ ⊢ 𝑢 ∶ 𝐵 holds and 𝑢 ⇝ 𝑣 then Γ ⊢ 𝑣 ∶ 𝐵 holds.

Theorem (Termination)

A typable term has no infinite sequence of reductions.

Theorem (Confluence)

For any reductions 𝑢 ⇝∗ 𝑣 and 𝑢 ⇝∗ 𝑤, there is a term 𝑥 such that 𝑣 ⇝∗ 𝑥 and 𝑤 ⇝∗ 𝑥.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 7 / 84

The typed 𝜇-calculus

Definition

Evaluation is the relation generated by the pair of rules (𝜇𝑦.𝑢)𝑣 ⇝ 𝑢[𝑣/𝑦] and 𝜌𝑗⟨𝑢1, 𝑢2⟩ ⇝ 𝑢𝑗 for 𝑗 = 1 or 𝑗 = 2

Theorem (Subject reduction)

If Γ ⊢ 𝑢 ∶ 𝐵 holds and 𝑢 ⇝ 𝑣 then Γ ⊢ 𝑣 ∶ 𝐵 holds.

Theorem (Termination)

A typable term has no infinite sequence of reductions.

Theorem (Confluence)

For any reductions 𝑢 ⇝∗ 𝑣 and 𝑢 ⇝∗ 𝑤, there is a term 𝑥 such that 𝑣 ⇝∗ 𝑥 and 𝑤 ⇝∗ 𝑥.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 7 / 84

The typed 𝜇-calculus

Definition

Evaluation is the relation generated by the pair of rules (𝜇𝑦.𝑢)𝑣 ⇝ 𝑢[𝑣/𝑦] and 𝜌𝑗⟨𝑢1, 𝑢2⟩ ⇝ 𝑢𝑗 for 𝑗 = 1 or 𝑗 = 2

Theorem (Subject reduction)

If Γ ⊢ 𝑢 ∶ 𝐵 holds and 𝑢 ⇝ 𝑣 then Γ ⊢ 𝑣 ∶ 𝐵 holds.

Theorem (Termination)

A typable term has no infinite sequence of reductions.

Theorem (Confluence)

For any reductions 𝑢 ⇝∗ 𝑣 and 𝑢 ⇝∗ 𝑤, there is a term 𝑥 such that 𝑣 ⇝∗ 𝑥 and 𝑤 ⇝∗ 𝑥.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 7 / 84

Curry-Howard: dynamics

What does evaluation mean, when considering proofs?

Theorem

A proof in natural deduction is normal iff there is never an introduction rule followed by an elimination rule for the same connective.

Theorem (Subformula property)

In a normal proof, any formula occurring in a sequent at any point in the proof is a subformula of one of the formulas in the conclusion. Normal proofs are direct, explicit.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 8 / 84

Curry-Howard: dynamics

What does evaluation mean, when considering proofs?

Theorem

A proof in natural deduction is normal iff there is never an introduction rule followed by an elimination rule for the same connective.

Theorem (Subformula property)

In a normal proof, any formula occurring in a sequent at any point in the proof is a subformula of one of the formulas in the conclusion. Normal proofs are direct, explicit.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 8 / 84

Curry-Howard: dynamics

What does evaluation mean, when considering proofs?

Theorem

A proof in natural deduction is normal iff there is never an introduction rule followed by an elimination rule for the same connective.

Theorem (Subformula property)

In a normal proof, any formula occurring in a sequent at any point in the proof is a subformula of one of the formulas in the conclusion. Normal proofs are direct, explicit.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 8 / 84

Denotational semantics

The search for invariants of reduction: models of the 𝜇-calculus (as a theory of functions) structures for defining the value of proofs The kind of objects we want is: logic computation

• bject

formula type space proof term morphism normalization evaluation equality

Example

Sets for types, arbitrary functions for terms. It works but there are way too many functions!

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 9 / 84

Coherence spaces

Definition

A coherence space 𝐵 is a set |𝐵| (the web), a symmetric and reflexive binary relation ¨𝐵 (the coherence). A clique 𝑏 ∈ 𝐷 ℓ(𝐵) is a subset of |𝐵| of points pairwise related by ¨𝐵. Intuition: points are bits of information about objects of 𝐵, cliques are consistent descriptions of objects

Example

A coherence space for words could have bits to say “at position 𝑗 there is a letter 𝑏” “at position 𝑗 there is the end-of-string symbol”

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 10 / 84

Stable functions

A definable function maps information about an object in 𝐵 to information about an object of 𝐶.

Definition

A stable function from 𝐵 to 𝐶 is a function 𝑔 ∶ 𝐷 ℓ(𝐵) → 𝐷 ℓ(𝐶) that is continuous: for a directed family (𝑏𝑗)𝑗∈𝐽 in 𝐷 ℓ(𝐵), 𝑔 (⋃𝑗∈𝐽 𝑏𝑗) = ⋃𝑗∈𝐽 𝑔 (𝑏𝑗); stable: for all 𝑏, 𝑏′ ∈ 𝐷 ℓ(𝐵) such that 𝑏 ∪ 𝑏′ ∈ 𝐷 ℓ(𝐵), 𝑔 (𝑏 ∩ 𝑏′) = 𝑔 (𝑏) ∩ 𝑔 (𝑏′). Implies monotonicity. The value for an arbitrary input is deduced from finite approximations, For every bit of output, there is a minimum input needed to get it.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 11 / 84

Stable functions – traces

Definition

The trace of a stable function 𝑔 ∶ 𝐷 ℓ(𝐵) → 𝐷 ℓ(𝐶) is 𝑈𝑠􏿵𝑔 􏿸 ∶= 􏿻(𝑏, 𝛾) 􏿗 𝑏 ∈ 𝐷 ℓ(𝐵), 𝛾 ∈ 𝑔 (𝑏), ∀𝑏′ ⊊ 𝑏, 𝛾 ∉ 𝑔 (𝑏′)􏿾. Remarkable facts: Each stable function is uniquely defined by its trace. Traces are the cliques in a coherence space 𝐵 ⇒ 𝐶.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 12 / 84

Stable functions – linearity

Definition

A stable function 𝑔 is linear if for all (𝑏, 𝛾) ∈ 𝑈𝑠􏿵𝑔 􏿸, 𝑏 is a singleton. For one bit of output, you need one bit of input. The function uses its argument exactly once.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 13 / 84

Linearity in logic

Classical sequent calculus has weakening and contraction of formulas, which allows using any hypothesis any number of times: Γ ⊢ Δ wL Γ, 𝐵 ⊢ Δ Γ ⊢ Δ wR Γ ⊢ 𝐵, Δ Γ, 𝐵, 𝐵 ⊢ Δ cL Γ, 𝐵 ⊢ Δ Γ ⊢ 𝐵, 𝐵, Δ cR Γ ⊢ 𝐵, Δ These make the following rules equivalent: Γ ⊢ 𝐵, Δ Γ ⊢ 𝐶, Δ ∧Ra Γ ⊢ 𝐵 ∧ 𝐶, Δ Γ ⊢ 𝐵, Δ Γ′ ⊢ 𝐶, Δ′ ∧Rm Γ, Γ′ ⊢ 𝐵 ∧ 𝐶, Δ, Δ′ additive multiplicative And similarly for other connectives, left rules, etc. In the absence of weakening and contraction, these become different.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 14 / 84

Sequent calculi

Sequents in intuitionistic logic: 𝐵1, …, 𝐵𝑜 ⊢ 𝐶 “From hypotheses 𝐵1, …, 𝐵𝑜 deduce 𝐶.” A proof of this is interpreted as a way to make a proof of 𝐶 from proofs of the 𝐵𝑗 a function from 𝐵1 × ⋯ × 𝐵𝑜 to 𝐶 Contraction and weakening are allowed on the left.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 15 / 84

Sequent calculi

Sequents in classical logic: 𝐵1, …, 𝐵𝑜 ⊢ 𝐶1, …, 𝐶𝑞 “From hypotheses 𝐵1, …, 𝐵𝑜 deduce 𝐶1 or … or 𝐶𝑞 .” Contraction and weakening are allowed on both sides.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 15 / 84

Sequent calculi

Sequents in linear logic: 𝐵1, …, 𝐵𝑜 ⊢ 𝐶1, …, 𝐶𝑞 “From hypotheses 𝐵1, …, 𝐵𝑜 deduce 𝐶1 or … or 𝐶𝑞 linearly.” A proof of this is interpreted as a way to make a proof of 𝐶 from proofs of the 𝐵𝑗 using each 𝐵𝑗 exactly once a linear map from 𝐵1 ⊗ ⋯ ⊗ 𝐵𝑜 to 𝐶1 & ⋯ & 𝐶𝑞 Contraction and weakening are not allowed.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 15 / 84

Plan

The proof-program correspondence Linear sequent calculus Multiplicative linear logic One-sided presentation Full linear logic The notion of fragment A bit of semantics A bit of proof theory Proof nets

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 16 / 84

Formulas and sequents

In this talk we focus on the propositional structure: formulas 𝐵, 𝐶 ∶= 𝛽 propositional variable 𝐵⊥ linear negation 𝐵 ⊗ 𝐶, 𝐵 & 𝐶, 1, ⊥ multiplicatives 𝐵 & 𝐶, 𝐵 ⊕ 𝐶, ⊤, 0 additives ! 𝐵, ?𝐵 exponentials sequents Γ, Δ, Θ ∶= 𝐵1, …, 𝐵𝑜 ⊢ 𝐶1, …, 𝐶𝑞 with 𝑜, 𝑞 ≥ 0 We focus on MLL, the subsystem made only of multiplicative connectives and negation.

Definition

𝐵 ⊸ 𝐶 is a notation for 𝐵⊥ & 𝐶.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 17 / 84

Formulas and sequents

In this talk we focus on the propositional structure: formulas 𝐵, 𝐶 ∶= 𝛽 propositional variable 𝐵⊥ linear negation 𝐵 ⊗ 𝐶, 𝐵 & 𝐶, 1, ⊥ multiplicatives 𝐵 & 𝐶, 𝐵 ⊕ 𝐶, ⊤, 0 additives ! 𝐵, ?𝐵 exponentials sequents Γ, Δ, Θ ∶= 𝐵1, …, 𝐵𝑜 ⊢ 𝐶1, …, 𝐶𝑞 with 𝑜, 𝑞 ≥ 0 We focus on MLL, the subsystem made only of multiplicative connectives and negation.

Definition

𝐵 ⊸ 𝐶 is a notation for 𝐵⊥ & 𝐶.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 17 / 84

MLL – the deductive structure

The order of formulas is irrelevant: Γ, 𝐵, 𝐶, Δ ⊢ Θ exL Γ, 𝐶, 𝐵, Δ ⊢ Θ Γ ⊢ Δ, 𝐵, 𝐶, Θ exR Γ ⊢ Δ, 𝐶, 𝐵, Θ Axiom and cut rules: ax 𝐵 ⊢ 𝐵 Γ ⊢ 𝐵, Δ Γ′, 𝐵 ⊢ Δ′ cut Γ, Γ′ ⊢ Δ, Δ′ Linear negation: Γ ⊢ 𝐵, Δ ⊥L Γ, 𝐵⊥ ⊢ Δ Γ, 𝐵 ⊢ Δ ⊥R Γ ⊢ 𝐵⊥, Δ

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 18 / 84

MLL – the connectives

Multiplicatives: Γ ⊢ Δ, 𝐵 Γ′ ⊢ Δ′, 𝐶 ⊗R Γ, Γ′ ⊢ Δ, Δ′, 𝐵 ⊗ 𝐶 Γ, 𝐵, 𝐶 ⊢ Δ ⊗L Γ, 𝐵 ⊗ 𝐶 ⊢ Δ Γ, 𝐵 ⊢ Δ Γ′, 𝐶 ⊢ Δ′ & L Γ, Γ′, 𝐵 & 𝐶 ⊢ Δ, Δ′ Γ ⊢ Δ, 𝐵, 𝐶 & R Γ ⊢ Δ, 𝐵 & 𝐶 Additives: Γ ⊢ Δ, 𝐵 ⊕R1 Γ ⊢ Δ, 𝐵 ⊕ 𝐶 Γ ⊢ Δ, 𝐶 ⊕R2 Γ ⊢ Δ, 𝐵 ⊕ 𝐶 Γ, 𝐵 ⊢ Δ Γ, 𝐶 ⊢ Δ ⊕L Γ, 𝐵 ⊕ 𝐶 ⊢ Δ Γ, 𝐵 ⊢ Δ &L1 Γ, 𝐵 & 𝐶 ⊢ Δ Γ, 𝐶 ⊢ Δ &L2 Γ, 𝐵 & 𝐶 ⊢ Δ Γ ⊢ Δ, 𝐵 Γ ⊢ Δ, 𝐶 &R Γ ⊢ Δ, 𝐵 & 𝐶

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 19 / 84

MLL – provability

Example

The following sequents are provable in MLL: multiplicative excluded middle: ⊢ 𝐵 & 𝐵⊥ semi-distributivity of tensor over par: 𝐵 ⊗ (𝐶 & 𝐷) ⊢ (𝐵 ⊗ 𝐶) & 𝐷 However, 𝐵 ⊢ 𝐵 ⊗ 𝐵 is not provable. Exercise: Prove that!

Definition

𝐵 and 𝐶 are linearly equivalent if 𝐵 ⊢ 𝐶 and 𝐶 ⊢ 𝐵 are provable, write this 𝐵 ˛ 𝐶. Simplest example: 𝐵 ⊗ 𝐶 ˛ 𝐶 ⊗ 𝐵.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 20 / 84

Symmetries

Let us see if we can simplify the system a bit.

Theorem (De Morgan laws)

For all formulas 𝐵 and 𝐶, the following equivalences hold: 𝐵 𝐵⊥⊥, (𝐵 ⊗ 𝐶)⊥ 𝐵⊥ & 𝐶⊥, (𝐵 & 𝐶)⊥ 𝐵⊥ ⊗ 𝐶⊥. Exercise: Prove this.

Theorem

A sequent 𝐵1, …, 𝐵𝑜 ⊢ 𝐶1, …, 𝐶𝑞 is provable if and only the sequent ⊢ 𝐵⊥

1 , …, 𝐵⊥ 𝑜 , 𝐶1, …, 𝐶𝑞 is provable.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 21 / 84

Symmetries

Let us see if we can simplify the system a bit.

Theorem (De Morgan laws)

For all formulas 𝐵 and 𝐶, the following equivalences hold: 𝐵 ˛ 𝐵⊥⊥, (𝐵 ⊗ 𝐶)⊥ ˛ 𝐵⊥ & 𝐶⊥, (𝐵 & 𝐶)⊥ ˛ 𝐵⊥ ⊗ 𝐶⊥. Exercise: Prove this.

Theorem

A sequent 𝐵1, …, 𝐵𝑜 ⊢ 𝐶1, …, 𝐶𝑞 is provable if and only the sequent ⊢ 𝐵⊥

1 , …, 𝐵⊥ 𝑜 , 𝐶1, …, 𝐶𝑞 is provable.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 21 / 84

Symmetries

Let us see if we can simplify the system a bit.

Theorem (De Morgan laws)

For all formulas 𝐵 and 𝐶, the following equivalences hold: 𝐵 ˛ 𝐵⊥⊥, (𝐵 ⊗ 𝐶)⊥ ˛ 𝐵⊥ & 𝐶⊥, (𝐵 & 𝐶)⊥ ˛ 𝐵⊥ ⊗ 𝐶⊥. Exercise: Prove this.

Theorem

A sequent 𝐵1, …, 𝐵𝑜 ⊢ 𝐶1, …, 𝐶𝑞 is provable if and only the sequent ⊢ 𝐵⊥

1 , …, 𝐵⊥ 𝑜 , 𝐶1, …, 𝐶𝑞 is provable.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 21 / 84

One-sided presentation

Redefine the language of formulas: formulas 𝐵, 𝐶 ∶= 𝛽 propositional variable 𝛽⊥ negated variable 𝐵 ⊗ 𝐶, 𝐵 & 𝐶, 1, ⊥ multiplicatives 𝐵 & 𝐶, 𝐵 ⊕ 𝐶, ⊤, 0 additives ! 𝐵, ?𝐵 exponentials sequents Γ, Δ, Θ ∶= ⊢ 𝐵1, …, 𝐵𝑜 with 𝑜 ≥ 0

Definition

Negation is the operation on formulas defined as (𝐵 ⊗ 𝐶)⊥ ∶= 𝐵⊥ & 𝐶⊥ (𝐵 ⊕ 𝐶)⊥ ∶= 𝐵⊥ & 𝐶⊥ (!𝐵)⊥ ∶= ?(𝐵⊥) (𝐵 & 𝐶)⊥ ∶= 𝐵⊥ ⊗ 𝐶⊥ (𝐵 & 𝐶)⊥ ∶= 𝐵⊥ ⊕ 𝐶⊥ (?𝐵)⊥ ∶= !(𝐵⊥) (𝛽⊥)⊥ ∶= 𝛽 1⊥ ∶= ⊥ 0⊥ ∶= ⊤ ⊥⊥ ∶= 1 ⊤⊥ ∶= 0 By construction, 𝐵⊥⊥ = 𝐵.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 22 / 84

One-sided sequent calculus

Axiom and cut rules: ax ⊢ 𝐵⊥, 𝐵 ⊢ Γ, 𝐵 ⊢ Δ, 𝐵⊥ cut ⊢ Γ, Δ Multiplicatives: ⊢ Γ, 𝐵 ⊢ Δ, 𝐶 ⊗ ⊢ Γ, Δ, 𝐵 ⊗ 𝐶 ⊢ Γ, 𝐵, 𝐶 & ⊢ Γ, 𝐵 & 𝐶 Additives: ⊢ Γ, 𝐵 ⊢ Γ, 𝐶 & ⊢ Γ, 𝐵 & 𝐶 ⊢ Γ, 𝐵 ⊕1 ⊢ Γ, 𝐵 ⊕ 𝐶 ⊢ Γ, 𝐶 ⊕2 ⊢ Γ, 𝐵 ⊕ 𝐶 Units: 1 ⊢ 1 ⊤ ⊢ Γ, ⊤ ⊢ Γ ⊥ ⊢ Γ, ⊥

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 23 / 84

One-sided sequent calculus

Axiom and cut rules: ax ⊢ 𝐵⊥, 𝐵 ⊢ Γ, 𝐵 ⊢ Δ, 𝐵⊥ cut ⊢ Γ, Δ Multiplicatives: ⊢ Γ, 𝐵 ⊢ Δ, 𝐶 ⊗ ⊢ Γ, Δ, 𝐵 ⊗ 𝐶 ⊢ Γ, 𝐵, 𝐶 & ⊢ Γ, 𝐵 & 𝐶 Additives: ⊢ Γ, 𝐵 ⊢ Γ, 𝐶 & ⊢ Γ, 𝐵 & 𝐶 ⊢ Γ, 𝐵 ⊕1 ⊢ Γ, 𝐵 ⊕ 𝐶 ⊢ Γ, 𝐶 ⊕2 ⊢ Γ, 𝐵 ⊕ 𝐶 Units: 1 ⊢ 1 ⊤ ⊢ Γ, ⊤ ⊢ Γ ⊥ ⊢ Γ, ⊥

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 23 / 84

One-sided sequent calculus

Axiom and cut rules: ax ⊢ 𝐵⊥, 𝐵 ⊢ Γ, 𝐵 ⊢ Δ, 𝐵⊥ cut ⊢ Γ, Δ Multiplicatives: ⊢ Γ, 𝐵 ⊢ Δ, 𝐶 ⊗ ⊢ Γ, Δ, 𝐵 ⊗ 𝐶 ⊢ Γ, 𝐵, 𝐶 & ⊢ Γ, 𝐵 & 𝐶 Additives: ⊢ Γ, 𝐵 ⊢ Γ, 𝐶 & ⊢ Γ, 𝐵 & 𝐶 ⊢ Γ, 𝐵 ⊕1 ⊢ Γ, 𝐵 ⊕ 𝐶 ⊢ Γ, 𝐶 ⊕2 ⊢ Γ, 𝐵 ⊕ 𝐶 Units: 1 ⊢ 1 ⊤ ⊢ Γ, ⊤ ⊢ Γ ⊥ ⊢ Γ, ⊥

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 23 / 84

Additives vs multiplicatives

Example: distributivity of ⊗ over ⊕. ax ⊢ 𝐵⊥, 𝐵 ax ⊢ 𝐶⊥, 𝐶 ⊗ ⊢ 𝐵⊥, 𝐶⊥, 𝐵 ⊗ 𝐶 ⊕1 ⊢ 𝐵⊥, 𝐶⊥, (𝐵 ⊗ 𝐶) ⊕ (𝐵 ⊗ 𝐷) ax ⊢ 𝐵⊥, 𝐵 ax ⊢ 𝐷⊥, 𝐷 ⊗ ⊢ 𝐵⊥, 𝐷⊥, 𝐵 ⊗ 𝐷 ⊕2 ⊢ 𝐵⊥, 𝐶⊥, (𝐵 ⊗ 𝐶) ⊕ (𝐵 ⊗ 𝐷) & ⊢ 𝐵⊥, 𝐶⊥ & 𝐷⊥, (𝐵 ⊗ 𝐶) ⊕ (𝐵 ⊗ 𝐷) & ⊢ 𝐵⊥ & (𝐶⊥ & 𝐷⊥), (𝐵 ⊗ 𝐶) ⊕ (𝐵 ⊗ 𝐷) Hence 𝐵 ⊗ (𝐶 ⊕ 𝐷) ⊸ (𝐵 ⊗ 𝐶) ⊕ (𝐵 ⊗ 𝐷), equivalently (𝐵⊥ & 𝐶⊥) & (𝐵⊥ & 𝐷⊥) ⊸ 𝐵⊥ & (𝐶⊥ & 𝐷⊥).

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 24 / 84

Additives vs multiplicatives

Example: distributivity of ⊗ over ⊕. ax ⊢ 𝐵⊥, 𝐵 ax ⊢ 𝐶⊥, 𝐶 ⊕1 ⊢ 𝐶⊥, 𝐶 ⊕ 𝐷 ⊗ ⊢ 𝐵⊥, 𝐶⊥, 𝐵 ⊗ (𝐶 ⊕ 𝐷) & ⊢ 𝐵⊥ & 𝐶⊥, 𝐵 ⊗ (𝐶 ⊕ 𝐷) ax ⊢ 𝐵⊥, 𝐵 ax ⊢ 𝐷⊥, 𝐷 ⊕1 ⊢ 𝐷⊥, 𝐶 ⊕ 𝐷 ⊗ ⊢ 𝐵⊥, 𝐷⊥, 𝐵 ⊗ (𝐶 ⊕ 𝐷) & ⊢ 𝐵⊥ & 𝐷⊥, 𝐵 ⊗ (𝐶 ⊕ 𝐷) & ⊢ (𝐵⊥ & 𝐶⊥) & (𝐵⊥ & 𝐷⊥), 𝐵 ⊗ (𝐶 ⊕ 𝐷) Hence (𝐵 ⊗ 𝐶) ⊕ (𝐵 ⊗ 𝐷) ⊸ 𝐵 ⊗ (𝐶 ⊕ 𝐷), equivalently 𝐵⊥ & (𝐶⊥ & 𝐷⊥) ⊸ (𝐵⊥ & 𝐶⊥) & (𝐵⊥ & 𝐷⊥).

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 24 / 84

Exponentials

Contraction and weakening are crucial for logical expressiveness. Linear logic provides them through modalities. Allowed structural rules: ⊢ Γ, 𝐵 ? ⊢ Γ, ?𝐵 ⊢ Γ w ⊢ Γ, ?𝐵 ⊢ Γ, ?𝐵, ?𝐵 c ⊢ Γ, ?𝐵 Promotion: ⊢ ?𝐵1, …, ?𝐵𝑜, 𝐶 ! ⊢ ?𝐵1, …, ?𝐵𝑜, !𝐶 Idea: ?𝐵 means “𝐵 some number of times” !𝐵 means “as many 𝐵 as necessary”

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 25 / 84

Exponentials – equivalences

Wrong but not too much: ?𝐵 =

∞

􏾙

𝑜=0 𝑜

&

𝑗=1

𝐵, !𝐵 =

∞

&

𝑜=0 𝑜

􏾀

𝑗=1

𝐵. A bit less wrong: ?𝐵 =

∞

&

𝑜=0

(𝐵 ⊕ ⊥), !𝐵 =

∞

􏾀

𝑜=0

(𝐵 & 1). Actually true: !(𝐵 & 𝐶) ˛ !𝐵 ⊗ !𝐶 !𝐵 ⊗ !𝐵 ˛ !𝐵 !!𝐵 ˛ !𝐵 !?!?𝐵 ˛ !?𝐵

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 26 / 84

Fragments

Many fragments are interesting: (possibly) restrict the set of formulas restrict the rules to allowed formulas (possibly) further restrict the set of rules For instance: MLL = multiplicative = keep only ⊗ and & MELL = multiplicative-exponential = remove additives MALL = multiplicative-additive = remove exponentials ILL = “intuitionistic” = two-sided, one formula on the right focalized = more on this later polarized = more on this later LJ, LK = more on this later

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 27 / 84

Plan

The proof-program correspondence Linear sequent calculus A bit of semantics Cut elimination and consistency Provability semantics Proof semantics in coherence spaces A bit of proof theory Proof nets

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 28 / 84

The question of consistency

We have a definition of formulas, sequents and deduction rules. But how do we know if the system is consistent? Provability in LK is preserved through translations. This is a good hint but it doesn’t say much of LL! LL has a model in coherent spaces, of course. But this does not inform us on the possibilities of the system. Use the argument sequent calculus was built for: Cut elimination.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 29 / 84

The question of consistency

We have a definition of formulas, sequents and deduction rules. But how do we know if the system is consistent? Provability in LK is preserved through translations. This is a good hint but it doesn’t say much of LL! LL has a model in coherent spaces, of course. But this does not inform us on the possibilities of the system. Use the argument sequent calculus was built for: Cut elimination.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 29 / 84

The question of consistency

We have a definition of formulas, sequents and deduction rules. But how do we know if the system is consistent? Provability in LK is preserved through translations. This is a good hint but it doesn’t say much of LL! LL has a model in coherent spaces, of course. But this does not inform us on the possibilities of the system. Use the argument sequent calculus was built for: Cut elimination.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 29 / 84

The question of consistency

We have a definition of formulas, sequents and deduction rules. But how do we know if the system is consistent? Provability in LK is preserved through translations. This is a good hint but it doesn’t say much of LL! LL has a model in coherent spaces, of course. But this does not inform us on the possibilities of the system. Use the argument sequent calculus was built for: Cut elimination.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 29 / 84

Consistency by cut elimination

Theorem (Admissibility of cut)

A sequent is provable if and only if it is provable without the cut rule.

Corollary (Consistency)

The empty sequent ⊢ is not provable.

Proof.

All rules except cut have at least one formula in conclusion. Hence you cannot prove both 𝐵 and 𝐵⊥.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 30 / 84

Cut elimination

Define reduction rules over proofs that locally eliminate cuts. Prove well-foundedness of the reduction relation. Prove that irreducible proofs are cut-free. Conclude.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 31 / 84

Cut elimination

Interaction rules

Tensor versus par 𝜌1 ⊢ Γ, 𝐵 𝜌2 ⊢ Δ, 𝐶 ⊗ ⊢ Γ, Δ, 𝐵 ⊗ 𝐶 𝜌3 ⊢ Θ, 𝐵⊥, 𝐶⊥ & ⊢ Θ, 𝐵⊥ & 𝐶⊥ cut ⊢ Γ, Δ, Θ ↘ 𝜌1 ⊢ Γ, 𝐵 𝜌2 ⊢ Δ, 𝐶 𝜌3 ⊢ Θ, 𝐵⊥, 𝐶⊥ cut ⊢ Δ, Θ, 𝐵⊥ cut ⊢ Γ, Δ, Θ

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 32 / 84

Cut elimination

Interaction rules

With versus plus 𝜌1 ⊢ Γ, 𝐵 𝜌2 ⊢ Γ, 𝐶 & ⊢ Γ, 𝐵 & 𝐶 𝜌3 ⊢ Δ, 𝐵⊥ ⊕1 ⊢ Δ, 𝐵⊥ ⊕ 𝐶⊥ cut ⊢ Γ, Δ ↘ 𝜌1 ⊢ Γ, 𝐵 𝜌3 ⊢ Δ, 𝐵⊥ cut ⊢ Γ, Δ

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 33 / 84

Cut elimination

Interaction rules

Promotion versus contraction 𝜌1 ⊢ ?Γ, 𝐵 ! ⊢ ?Γ, !𝐵 𝜌2 ⊢ Δ, ?𝐵⊥, ?𝐵⊥ c ⊢ Δ, ?𝐵⊥ cut ⊢ ?Γ, Δ ↘ 𝜌1 ⊢ ?Γ, 𝐵 ! ⊢ ?Γ, !𝐵 𝜌1 ⊢ ?Γ, 𝐵 ! ⊢ ?Γ, !𝐵 𝜌2 ⊢ Δ, ?𝐵⊥, ?𝐵⊥ cut ⊢ ?Γ, Δ, ?𝐵⊥ cut ⊢ ?Γ, ?Γ, Δ c ⊢ ?Γ, Δ

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 34 / 84

Cut elimination

Interaction rules

… plus a few other cancellation rules … left right action ⊗ & propagate the cuts to sub-formulas 1 ⊥ drop the proof of 1 ⊕1 & keep only the left proof in the & rule ⊕2 & keep only the right proof in the & rule ! ? propagate the cut to the sub-formula ! w drop the proof from the promotion ! c duplicate the proof from the promotion ax anything drop the axiom

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 35 / 84

Cut elimination

Commutation rules

Commutation with tensor 𝜌1 ⊢ Γ, 𝐵 𝜌2 ⊢ Δ, 𝐶, 𝐷 ⊗ ⊢ Γ, Δ, 𝐵 ⊗ 𝐶, 𝐷 𝜌3 ⊢ Θ, 𝐷⊥ cut ⊢ Γ, Δ, Θ, 𝐵 ⊗ 𝐶 ↘ 𝜌1 ⊢ Γ, 𝐵 𝜌2 ⊢ Δ, 𝐶, 𝐷 𝜌3 ⊢ Θ, 𝐷⊥ cut ⊢ Δ, Θ, 𝐶 ⊗ ⊢ Γ, Δ, 𝐵 ⊗ 𝐶, 𝐷

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 36 / 84

Cut elimination

Commutation rules

Commutation with “with” 𝜌1 ⊢ Γ, 𝐵, 𝐷 𝜌2 ⊢ Γ, 𝐶, 𝐷 & ⊢ Γ, 𝐵 & 𝐶, 𝐷 𝜌3 ⊢ Δ, 𝐷⊥ cut ⊢ Γ, Δ, 𝐵 & 𝐶 ↘ 𝜌1 ⊢ Γ, 𝐵, 𝐷 𝜌3 ⊢ Δ, 𝐷⊥ cut ⊢ Γ, Δ, 𝐵 𝜌2 ⊢ Γ, 𝐶, 𝐷 𝜌3 ⊢ Δ, 𝐷⊥ cut ⊢ Γ, Δ, 𝐶 & ⊢ Γ, Δ, 𝐵 & 𝐶

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 37 / 84

Cut elimination

Commutation rules

… plus a lot more commutation rules …

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 38 / 84

Normalization

With the right set of rules, clearly irreducible proofs are cut-free. How to prove that reduction always terminates? Using a clever induction on formulas and proofs. Works only in the absence of second-order quantification. Using reducibility candidates, like in system F. Lots of technical points to cope with, but it works. Indirectly through more tractable systems

polarized systems … more on this in a minute proof nets … more on this later

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 39 / 84

Normalization

With the right set of rules, clearly irreducible proofs are cut-free. How to prove that reduction always terminates? Using a clever induction on formulas and proofs. Works only in the absence of second-order quantification. Using reducibility candidates, like in system F. Lots of technical points to cope with, but it works. Indirectly through more tractable systems

polarized systems … more on this in a minute proof nets … more on this later

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 39 / 84

Normalization

With the right set of rules, clearly irreducible proofs are cut-free. How to prove that reduction always terminates? Using a clever induction on formulas and proofs. Works only in the absence of second-order quantification. Using reducibility candidates, like in system F. Lots of technical points to cope with, but it works. Indirectly through more tractable systems

polarized systems … more on this in a minute proof nets … more on this later

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 39 / 84

Normalization

With the right set of rules, clearly irreducible proofs are cut-free. How to prove that reduction always terminates? Using a clever induction on formulas and proofs. Works only in the absence of second-order quantification. Using reducibility candidates, like in system F. Lots of technical points to cope with, but it works. Indirectly through more tractable systems

polarized systems … more on this in a minute proof nets … more on this later

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 39 / 84

The question of completeness

How do we know we are not missing some rules?

Theorem (Completeness)

If a formula 𝐵 is satisfied in every interpretation, then ⊢ 𝐵 is provable in LL. But what is an interpretation? We need a structure that plays in LL the role of Boolean algebras in LK.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 40 / 84

The question of completeness

How do we know we are not missing some rules?

Theorem (Completeness)

If a formula 𝐵 is satisfied in every interpretation, then ⊢ 𝐵 is provable in LL. But what is an interpretation? We need a structure that plays in LL the role of Boolean algebras in LK.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 40 / 84

Phase spaces

Definition

A phase space is a pair (𝑁, ⊥) where 𝑁 is a commutative monoid and ⊥ is a subset of 𝑁. points of 𝑁 are tests/interactions/processes… elements of ⊥ are successful tests, valid interactions… ⊥ is the rule of the game

Definition

Two points 𝑦, 𝑧 ∈ 𝑁 are orthogonal if 𝑦𝑧 ∈ ⊥. For 𝐵 ⊆ 𝑁, let 𝐵⊥ ∶= 􏿻𝑧 ∈ 𝑁 􏿗 ∀𝑦 ∈ 𝐵, 𝑦𝑧 ∈ ⊥􏿾. A fact is a set of the form 𝐵⊥. Exercise: Prove that 𝐵 ⊆ 𝐶 implies 𝐶⊥ ⊆ 𝐵⊥ and that 𝐵 ⊆ 𝐵⊥⊥ and 𝐵⊥⊥⊥ = 𝐵⊥. Facts play the role of truth values.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 41 / 84

Phase spaces

Connectives

Given (𝑁, ⊥), for subsets 𝐵, 𝐶 ⊆ 𝑁 define 𝐵 ⊗ 𝐶 ∶= 􏿻𝑞𝑟 􏿗 𝑞 ∈ 𝐵, 𝑟 ∈ 𝐶􏿾

⊥⊥

𝐵 & 𝐶 ∶= (𝐵⊥ ⊗ 𝐶⊥)⊥ 𝐵 ⊕ 𝐶 ∶= (𝐵 ∪ 𝐶)⊥⊥ 𝐵 & 𝐶 ∶= 𝐵 ∩ 𝐶 0 ∶= ∅⊥⊥ ⊤ ∶= 𝑁 !𝐵 ∶= (𝐵 ∩ 𝐽)⊥⊥ ?𝐵 ∶= (𝐵⊥ ∩ 𝐽)⊥ 1 ∶= {1}⊥⊥ where 𝐽 is the set of idempotents belonging to 1. If propositional variables are interpreted as facts, then for any formula 𝐵 the interpretation 𝐵𝑁 is a fact. 𝐵 ⊸ 𝐶 = 𝐵⊥ & 𝐶 = 􏿻𝑦 ∈ 𝑁 􏿗 ∀𝑧 ∈ 𝐵, 𝑦𝑧 ∈ 𝐶􏿾 If ⊥ = ∅ then we get the elementary Boolean algebra {∅, ⊤}.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 42 / 84

Phase spaces

Soundness and completeness

Theorem (Soundness)

If ⊢ 𝐵 is provable, then 1 ∈ 𝐵𝑁 in any phase space 𝑁. Exercise: Check it by induction over proofs.

Theorem (Completeness)

If 1 ∈ 𝐵𝑁 in any phase space 𝑁, then ⊢ 𝐵 is provable.

Proof.

Take for 𝑁 the sequents (up to duplication of ? formulas) and for ⊥ the provable ones. Check that 𝐵𝑁 = 􏿻Γ 􏿗 ⊢ Γ, 𝐵 is provable􏿾. The neutral element is the empty sequent so ⊢ 𝐵 is provable.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 43 / 84

Coherence spaces: interpreting formulas

Linear logic was extracted from the notion of linearity observed when interpreting the 𝜇-calculus in coherence spaces. It can itself be interpreted in coherence spaces:

Definition

􏿗𝐵⊥􏿗 = |𝐵| and 𝑦 ¨𝐵⊥ 𝑦′ unless 𝑦 ˝𝐵 𝑦′. |𝐵 ⊗ 𝐶| = |𝐵 & 𝐶| = |𝐵| × |𝐶| and

(𝑦, 𝑧) ¨𝐵⊗𝐶 (𝑦′, 𝑧′) if 𝑦 ¨𝐵 𝑦′ and 𝑧 ¨𝐶 𝑧′, (𝑦, 𝑧) ˝𝐵

& 𝐶 (𝑦′, 𝑧′) if 𝑦 ˝𝐵 𝑦′ or 𝑧 ˝𝐶 𝑧′.

|𝐵 ⊕ 𝐶| = |𝐵 & 𝐶| = ({1} × |𝐵|) ∪ ({2} × |𝐶|) and

(𝑗, 𝑦) ¨𝐵⊕𝐶 (𝑘, 𝑦′) if 𝑗 = 𝑘 and 𝑦 ¨ 𝑦′. (𝑗, 𝑦) ¨𝐵&𝐶 (𝑘, 𝑦′) if 𝑗 ≠ 𝑘 or 𝑦 ¨ 𝑦′.

|!𝐵| is the set of finite cliques of 𝐵, 𝑦 ¨!𝐵 𝑦′ if 𝑦 ∪ 𝑦′ is a clique in 𝐵. where 𝑦 ˝ 𝑦′ means 𝑦 ¨ 𝑦′ and 𝑦 ≠ 𝑦′.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 44 / 84

Coherence spaces: interpreting proofs

Identity ax ⊢ 𝛽 ∶ 𝐵⊥, 𝛽 ∶ 𝐵 ⊢ 𝛿 ∶ Γ, 𝛽 ∶ 𝐵 ⊢ 𝛽 ∶ 𝐵⊥, 𝜀 ∶ Δ cut ⊢ 𝛿 ∶ Γ, 𝜀 ∶ Δ Multiplicatives ⊢ 𝛿 ∶ Γ, 𝛽 ∶ 𝐵 ⊢ 𝛾 ∶ 𝐶, 𝜀 ∶ Δ ⊗ ⊢ 𝛿 ∶ Γ, (𝛽, 𝛾) ∶ 𝐵 ⊗ 𝐶, 𝜀 ∶ Δ ⊢ 𝛿 ∶ Γ, 𝛽 ∶ 𝐵, 𝛾 ∶ 𝐶 & ⊢ 𝛿 ∶ Γ, (𝛽, 𝛾) ∶ 𝐵 & 𝐶 Exponentials ⊢ 𝛿 ∶ Γ, 𝛽 ∶ 𝐵 ? ⊢ 𝛿 ∶ Γ, {𝛽} ∶ ?𝐵 ⊢ 𝛿 ∶ Γ w ⊢ 𝛿 ∶ Γ, ∅ ∶ ?𝐵 ⊢ 𝛿 ∶ Γ, 𝑏 ∶ ?𝐵, 𝑏′?𝐵 c ⊢ 𝛿 ∶ Γ, 𝑏 ∪ 𝑏′ ∶ ?𝐵 􏿻⊢ 𝑏1,𝑗 ∶ ?𝐵1, …𝑏𝑜,𝑗 ∶ ?𝐵𝑜, 𝑐𝑗 ∶ 𝐶􏿾

𝑗∈𝐽

! ⊢ ⋃𝑗∈𝐽 𝑏1,𝑗 ∶ ?𝐵1, …⋃𝑗∈𝐽 𝑏𝑜𝑗 ∶ ?𝐵𝑜, {𝑐𝑗 | 𝑗 ∈ 𝐽} ∶ !𝐶

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 45 / 84

Coherence spaces: interpreting proofs

Identity ax ⊢ 𝛽 ∶ 𝐵⊥, 𝛽 ∶ 𝐵 ⊢ 𝛿 ∶ Γ, 𝛽 ∶ 𝐵 ⊢ 𝛽 ∶ 𝐵⊥, 𝜀 ∶ Δ cut ⊢ 𝛿 ∶ Γ, 𝜀 ∶ Δ Multiplicatives ⊢ 𝛿 ∶ Γ, 𝛽 ∶ 𝐵 ⊢ 𝛾 ∶ 𝐶, 𝜀 ∶ Δ ⊗ ⊢ 𝛿 ∶ Γ, (𝛽, 𝛾) ∶ 𝐵 ⊗ 𝐶, 𝜀 ∶ Δ ⊢ 𝛿 ∶ Γ, 𝛽 ∶ 𝐵, 𝛾 ∶ 𝐶 & ⊢ 𝛿 ∶ Γ, (𝛽, 𝛾) ∶ 𝐵 & 𝐶 Exponentials ⊢ 𝛿 ∶ Γ, 𝛽 ∶ 𝐵 ? ⊢ 𝛿 ∶ Γ, {𝛽} ∶ ?𝐵 ⊢ 𝛿 ∶ Γ w ⊢ 𝛿 ∶ Γ, ∅ ∶ ?𝐵 ⊢ 𝛿 ∶ Γ, 𝑏 ∶ ?𝐵, 𝑏′?𝐵 c ⊢ 𝛿 ∶ Γ, 𝑏 ∪ 𝑏′ ∶ ?𝐵 􏿻⊢ 𝑏1,𝑗 ∶ ?𝐵1, …𝑏𝑜,𝑗 ∶ ?𝐵𝑜, 𝑐𝑗 ∶ 𝐶􏿾

𝑗∈𝐽

! ⊢ ⋃𝑗∈𝐽 𝑏1,𝑗 ∶ ?𝐵1, …⋃𝑗∈𝐽 𝑏𝑜𝑗 ∶ ?𝐵𝑜, {𝑐𝑗 | 𝑗 ∈ 𝐽} ∶ !𝐶

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 45 / 84

Coherence spaces: sanity check

Theorem

The set of tuples in the interpretation of a proof is always a clique.

Proof.

By a simple induction of proofs.

Theorem

The interpretation of proofs in coherence spaces is invariant by cut elimination.

Proof.

By case analysis on the various cases of cut elimination.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 46 / 84

Plan

The proof-program correspondence Linear sequent calculus A bit of semantics A bit of proof theory Intuitionistic and classical logics as fragments Cut elimination and proof equivalence Reversibility and focalization Proof nets

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 47 / 84

LJ expressed in linear logic

Linear logic arises from the decomposition 𝐵 ⇒ 𝐶 = !𝐵 ⊸ 𝐶 = ?𝐵⊥ & 𝐶 Deduction rules can be translated accordingly:

Γ, 𝐵 ⊢𝑀𝐾 𝐶 Γ ⊢𝑀𝐾 𝐵 ⇒ 𝐶 ↝ ⊢ Γ∗, ?(𝐵∗)⊥, 𝐶∗ & ⊢ Γ∗, ?(𝐵∗)⊥ & 𝐶∗ Γ ⊢𝑀𝐾 𝐵 ⇒ 𝐶 Δ ⊢𝑀𝐾 𝐵 Γ, Δ ⊢𝑀𝐾 𝐶 ↝ ⊢ Γ∗, ?(𝐵∗)⊥ & 𝐶∗ ⊢ Δ∗, 𝐵∗ ! ⊢ Δ∗, !𝐵∗ ax ⊢ (𝐶∗)⊥, 𝐶∗ ⊗ ⊢ Δ∗, !𝐵 ⊗ (𝐶∗)⊥, 𝐶∗ cut ⊢ Γ∗, Δ∗, 𝐶∗

The other connectives have adequate translations.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 48 / 84

LK expressed in linear logic

Classical sequents have the shape 𝐵1, …, 𝐵𝑜 ⊢ 𝐶1, …, 𝐶𝑞 with contraction and weakening allowed on both sides. This suggests translating 𝐵 ⇒ 𝐶 into something like !𝐵 ⊸ ?𝐶. This does not work, but !𝐵 ⊸ ?!𝐶 and !?𝐵 ⊸ ?𝐶 do work.

Theorem

A sequent is provable in classical sequent calculus if and only if its translation in linear logic, by any of the above translations, is provable. LK proofs are translated into LL proofs, mapping linear connectives to classical ones is the reverse translation. Exercise: Prove that in more detail.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 49 / 84

LK as two fragments?

There are two families of translations: “left-handed”: !?𝐵 ⊸ ?𝐶 the associated reduction for 𝜇-calculus is call by name “right-handed”: !𝐵 ⊸ ?!𝐶 the associated reduction for 𝜇-calculus is call by value More precise study of control operators is possible along these lines.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 50 / 84

Cut-elimination as computation

Let us look again at cut elimination. It is a computational process for turning arbitrary proofs into cut-free canonical proofs: cut-free proofs are like values, a proof of 𝐵 ⊸ 𝐶 maps values of 𝐵 to values of 𝐶, equivalence modulo cut-elimination implies semantic equality. Incidentally, it decomposes the reduction of the 𝜇-calculus. It turns arbitrary proofs into explicit, direct proofs: subformula property, mechanical proof search is possible.

In the absence of second-order quantification.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 51 / 84

Type isomorphisms

Technical aside: 𝜃-equivalence

Consider possible cut-free proofs of 𝐵 ⊕ (𝐶 ⊗ 𝐷) ⊸ 𝐵 ⊕ (𝐶 ⊗ 𝐷). ax ⊢ 𝐵⊥, 𝐵 ⊕1 ⊢ 𝐵⊥, 𝐵 ⊕ (𝐶 ⊗ 𝐷) ax ⊢ 𝐶⊥, 𝐶 ax ⊢ 𝐷⊥, 𝐷 ⊗ ⊢ 𝐶⊥, 𝐷⊥, 𝐶 ⊗ 𝐷 ax ⊢ 𝐶⊥ & 𝐷⊥, 𝐶 ⊗ 𝐷 ⊕2 ⊢ 𝐶⊥ & 𝐷⊥, 𝐵 ⊕ (𝐶 ⊗ 𝐷) ax ⊢ 𝐵⊥ & (𝐶⊥ & 𝐷⊥), 𝐵 ⊕ (𝐶 ⊗ 𝐷)

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 52 / 84

Type isomorphisms

Technical aside: 𝜃-equivalence

Consider possible cut-free proofs of 𝐵 ⊕ (𝐶 ⊗ 𝐷) ⊸ 𝐵 ⊕ (𝐶 ⊗ 𝐷). ax ⊢ 𝐵⊥, 𝐵 ⊕1 ⊢ 𝐵⊥, 𝐵 ⊕ (𝐶 ⊗ 𝐷) ax ⊢ 𝐶⊥, 𝐶 ax ⊢ 𝐷⊥, 𝐷 ⊗ ⊢ 𝐶⊥, 𝐷⊥, 𝐶 ⊗ 𝐷 ax ⊢ 𝐶⊥ & 𝐷⊥, 𝐶 ⊗ 𝐷 ⊕2 ⊢ 𝐶⊥ & 𝐷⊥, 𝐵 ⊕ (𝐶 ⊗ 𝐷) & ⊢ 𝐵⊥ & (𝐶⊥ & 𝐷⊥), 𝐵 ⊕ (𝐶 ⊗ 𝐷)

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 52 / 84

Type isomorphisms

Technical aside: 𝜃-equivalence

Consider possible cut-free proofs of 𝐵 ⊕ (𝐶 ⊗ 𝐷) ⊸ 𝐵 ⊕ (𝐶 ⊗ 𝐷). ax ⊢ 𝐵⊥, 𝐵 ⊕1 ⊢ 𝐵⊥, 𝐵 ⊕ (𝐶 ⊗ 𝐷) ax ⊢ 𝐶⊥, 𝐶 ax ⊢ 𝐷⊥, 𝐷 ⊗ ⊢ 𝐶⊥, 𝐷⊥, 𝐶 ⊗ 𝐷 & ⊢ 𝐶⊥ & 𝐷⊥, 𝐶 ⊗ 𝐷 ⊕2 ⊢ 𝐶⊥ & 𝐷⊥, 𝐵 ⊕ (𝐶 ⊗ 𝐷) & ⊢ 𝐵⊥ & (𝐶⊥ & 𝐷⊥), 𝐵 ⊕ (𝐶 ⊗ 𝐷)

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 52 / 84

Type isomorphisms

Technical aside: 𝜃-equivalence

Consider possible cut-free proofs of 𝐵 ⊕ (𝐶 ⊗ 𝐷) ⊸ 𝐵 ⊕ (𝐶 ⊗ 𝐷). ax ⊢ 𝐵⊥, 𝐵 ⊕1 ⊢ 𝐵⊥, 𝐵 ⊕ (𝐶 ⊗ 𝐷) ax ⊢ 𝐶⊥, 𝐶 ax ⊢ 𝐷⊥, 𝐷 ⊗ ⊢ 𝐶⊥, 𝐷⊥, 𝐶 ⊗ 𝐷 & ⊢ 𝐶⊥ & 𝐷⊥, 𝐶 ⊗ 𝐷 ⊕2 ⊢ 𝐶⊥ & 𝐷⊥, 𝐵 ⊕ (𝐶 ⊗ 𝐷) & ⊢ 𝐵⊥ & (𝐶⊥ & 𝐷⊥), 𝐵 ⊕ (𝐶 ⊗ 𝐷) We will consider these proofs as equivalent. This is the LL version of 𝜃-equivalence in the 𝜇-calculus: 𝑢 ≃𝜃 𝜇𝑦.(𝑢)𝑦.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 52 / 84

Type isomorphisms

Definition

Two formulas 𝐵 and 𝐶 are isomorphic if there are proofs 𝜌 ⊢ 𝐵⊥, 𝐶 and 𝜍 ⊢ 𝐶⊥, 𝐵 𝜌 cut with 𝜍 on 𝐵 is equivalent to the axiom on 𝐶 𝜌 cut with 𝜍 on 𝐶 is equivalent to the axiom on 𝐵 This implies isomorphism in any model. These equivalences are isomorphisms: 𝐵 ⊗ 𝐶 ≃ 𝐶 ⊗ 𝐵 𝐵 ⊗ (𝐶 ⊕ 𝐷) ≃ (𝐵 ⊗ 𝐶) ⊕ (𝐵 ⊗ 𝐷) !(𝐵 & 𝐶) ≃ !𝐵 ⊗ !𝐶 Exercise: Prove it! These are not: 𝐵 ⊕ 𝐵 ˛ 𝐵 !𝐵 ⊗ !𝐵 ˛ !𝐵 !!𝐵 ˛ !𝐵 !?!?𝐵 ˛ !?𝐵 Exercise: Explain why!

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 53 / 84

Standard isomorphisms

Remark that 𝐵 ≃ 𝐶 iff 𝐵⊥ ≃ 𝐶⊥. Associativity and commutativity (𝐵 ⊕ 𝐶) ⊕ 𝐷 ≃ 𝐵 ⊕ (𝐶 ⊕ 𝐷) (𝐵 ⊗ 𝐶) ⊗ 𝐷 ≃ 𝐵 ⊗ (𝐶 ⊗ 𝐷) 𝐵 ⊕ 𝐶 ≃ 𝐶 ⊕ 𝐵 𝐵 ⊕ 𝐶 ≃ 𝐶 ⊕ 𝐵 𝐵 ⊕ 0 ≃ 𝐵 𝐵 ⊗ 1 ≃ 𝐵 Distributivity 𝐵 ⊗ (𝐶 ⊕ 𝐷) ≃ (𝐵 ⊗ 𝐶) ⊕ (𝐵 ⊗ 𝐷) 𝐵 ⊗ 0 ≃ 0 Exponentiation !(𝐵 & 𝐶) ≃ !𝐵 ⊗ !𝐶 !⊤ ≃ 1

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 54 / 84

Reversibility

The rules for & and & are reversible, i.e. ⊢ Γ, 𝐵 & 𝐶 is provable iff ⊢ Γ, 𝐵, 𝐶 is provable, ⊢ Γ, 𝐵 & 𝐶 is provable iff ⊢ Γ, 𝐵 and ⊢ Γ, 𝐶 are provable, i.e. one can always assume that the introduction rule for a &

• r for a &

comes last. Moreover: this can be proved directly using only permutations of rules moving these rules down does not change the behaviour of the proofs w.r.t. cut-elimination & , &, ⊥, ⊤ are called negative.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 55 / 84

Focalization

Definition

A formula is positive if its main connective is ⊗, ⊕, 1, 0 or !. It is negative if its main connective is & , &, ⊥, ⊤ or ?. Let Γ = 𝑄1, …, 𝑄𝑜 be a provable sequent consisting of positive formulas

• nly. Then there is a formula 𝑄𝑗 and proof of ⊢ Γ of the form

𝜌1 ⊢ Γ1, 𝑂1 ⋯ 𝜌𝑙 ⊢ Γ𝑙, 𝑂𝑙 𝑆 ⊢ Γ1, …, Γ𝑙, 𝑄𝑗 where the 𝑂𝑘 are the maximal negative subformulas of 𝑄𝑗 and the last set of rules 𝑆 builds 𝑄𝑗 from the 𝑂𝑘.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 56 / 84

Synthetic connectives

Let Φ(𝑌1, …, 𝑌𝑜) be a formula made of positive connectives from the variables 𝑌1, …, 𝑌𝑜. Call Φ∗ the dual of Φ. Up to associativity/commutativity/neutrality, for some set 𝓙 ⊆ 𝒬({1, …, 𝑜}) one has Φ(𝑌1, …, 𝑌𝑜) ≃ 􏾙

𝐽∈𝓙

􏾀

𝑗∈𝐽

𝑌𝑗 Φ∗(𝑌1, …, 𝑌𝑜) ≃&

𝐽∈𝓙

&

𝑗∈𝐽

𝑌𝑗 There is one family of rules (⊢ Γ𝑗, 𝐵𝑗 )𝑗∈𝐽 Φ𝐽 ⊢ (Γ𝑗)𝑗∈𝐽, Φ(𝐵1, …, 𝐵𝑜) (⊢ Γ, (𝐵𝑗)𝑗∈𝐽 )𝐽∈𝓙 Φ∗ ⊢ Γ, Φ∗(𝐵1, …, 𝐵𝑜) Any provable sequent using Φ and Φ∗ can be proved with these rules without decomposing Φ and Φ∗. Push this further and you get ludics…

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 57 / 84

Polarized linear logic

Since connectives of the same polarity behave well, let us restrict to a system where polarities are never mixed: 𝑄, 𝑅 ∶= 𝛽, 𝑄 ⊗ 𝑅, 𝑄 ⊕ 𝑅, 1, 0, !𝑂 𝑁, 𝑂 ∶= 𝛽⊥, 𝑁 & 𝑂, 𝑁 & 𝑂, ⊥, ⊤, ?𝑄 If 𝑄 is a positive formula where variables only appear under modalities, then 𝑄 ⊸ !𝑄 is provable. Hence the following rules are derivable: ⊢ Γ W ⊢ Γ, 𝑂 ⊢ Γ, 𝑂, 𝑂 C ⊢ Γ, 𝑂 ⊢ 𝑂1, …, 𝑂𝑜, 𝑂 ! ⊢ 𝑂1, …, 𝑂𝑜, !𝑂 Any provable polarized sequent has at most one positive formula (assuming the ⊤ rule respects this as a constraint). Push this further and you get LLP…

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 58 / 84

Plan

The proof-program correspondence Linear sequent calculus A bit of semantics A bit of proof theory Proof nets Intuitionistic LL and natural deduction Proof structures Correctness criteria

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 59 / 84

Proof nets

Why would we need another formalism for proofs? Cut elimination in LL requires a lot of commutation rules as in other sequent calculi, Proofs that differ only by commutation are equivalent w.r.t. cut elimination. On the other hand: Normalization in the 𝜇-calculus only has one rule unless we use explicit substitutions, There are separation results. We would like a natural deduction for LL.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 60 / 84

Intuitionistic LL

The 𝜇-calculus is simpler because it is asymmetric. What if we made LL asymmetric too?

Definition (Formulas of MILL)

𝐵, 𝐶 ∶= 𝛽 propositional variable 𝐵 ⊸ 𝐶 linear implication 𝐵 ⊗ 𝐶 multiplicative conjunction

Definition (Proof terms for MILL)

𝑢, 𝑣 ∶= 𝑦 variable — axiom 𝜇𝑦.𝑢 linear abstraction — introduction of ⊸ (𝑢)𝑣 linear application — elimination of ⊸ (𝑢, 𝑣) pair — introduction of ⊗ 𝑢(𝑦,𝑧∶=𝑣) matching — elimination of ⊗

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 61 / 84

Intuitionistic LL

The 𝜇-calculus is simpler because it is asymmetric. What if we made LL asymmetric too?

Definition (Formulas of MILL)

𝐵, 𝐶 ∶= 𝛽 propositional variable 𝐵 ⊸ 𝐶 linear implication 𝐵 ⊗ 𝐶 multiplicative conjunction

Definition (Proof terms for MILL)

𝑢, 𝑣 ∶= 𝑦 variable — axiom 𝜇𝑦.𝑢 linear abstraction — introduction of ⊸ (𝑢)𝑣 linear application — elimination of ⊸ (𝑢, 𝑣) pair — introduction of ⊗ 𝑢(𝑦,𝑧∶=𝑣) matching — elimination of ⊗

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 61 / 84

Intuitionistic LL

The 𝜇-calculus is simpler because it is asymmetric. What if we made LL asymmetric too?

Definition (Formulas of MILL)

𝐵, 𝐶 ∶= 𝛽 propositional variable 𝐵 ⊸ 𝐶 linear implication 𝐵 ⊗ 𝐶 multiplicative conjunction

Definition (Proof terms for MILL)

𝑢, 𝑣 ∶= 𝑦 variable — axiom 𝜇𝑦.𝑢 linear abstraction — introduction of ⊸ (𝑢)𝑣 linear application — elimination of ⊸ (𝑢, 𝑣) pair — introduction of ⊗ 𝑢(𝑦,𝑧∶=𝑣) matching — elimination of ⊗

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 61 / 84

MILL – typing rules

Identity ax 𝑦 ∶ 𝐵 ⊢ 𝑦 ∶ 𝐵 Implication Γ, 𝑦 ∶ 𝐵 ⊢ 𝑢 ∶ 𝐶 ⊸R Γ ⊢ 𝜇𝑦.𝑢 ∶ 𝐵 ⊸ 𝐶 Γ ⊢ 𝑢 ∶ 𝐵 ⊸ 𝐶 Δ ⊢ 𝑣 ∶ 𝐵 ⊸E Γ, Δ ⊢ (𝑢)𝑣 ∶ 𝐶 Tensor Γ ⊢ 𝑢 ∶ 𝐵 Δ ⊢ 𝑣 ∶ 𝐶 ⊗R Γ, Δ ⊢ (𝑢, 𝑣) ∶ 𝐵 ⊗ 𝐶 Γ, 𝑦 ∶ 𝐵, 𝑧 ∶ 𝐶 ⊢ 𝑢 ∶ 𝐷 Δ ⊢ 𝑣 ∶ 𝐵 ⊗ 𝐶 ⊗E Γ, Δ ⊢ 𝑢(𝑦,𝑧∶=𝑣) ∶ 𝐷 No contraction or weakening, of course.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 62 / 84

MILL – reduction

Definition

Cut elimination for MILL is generated by the following rules: (𝜇𝑦.𝑢)𝑣 ⇝ 𝑢[𝑣/𝑦] 𝑢(𝑦,𝑧∶=(𝑣, 𝑤)) ⇝ 𝑢[𝑣/𝑦][𝑤/𝑧]

Theorem

Cut elimination in MILL computes a unique normal form for every proof. Subject reduction: straightforward. Strong normalization: each step decreases the number of typing rules. Confluence: MILL is strongly confluent. Linearity makes things simpler than in the 𝜇-calculus.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 63 / 84

MILL – reduction

Definition

Cut elimination for MILL is generated by the following rules: (𝜇𝑦.𝑢)𝑣 ⇝ 𝑢[𝑣/𝑦] 𝑢(𝑦,𝑧∶=(𝑣, 𝑤)) ⇝ 𝑢[𝑣/𝑦][𝑤/𝑧]

Theorem

Cut elimination in MILL computes a unique normal form for every proof. Subject reduction: straightforward. Strong normalization: each step decreases the number of typing rules. Confluence: MILL is strongly confluent. Linearity makes things simpler than in the 𝜇-calculus.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 63 / 84

MILL – a graphical notation

Axiom and linear implication

. 𝑦 ∶ 𝐵 . ax . 𝐵 . 𝑦 . Γ . 𝑢 . ⊸ . 𝐵 ⊸ 𝐶 . 𝑦 ∶ 𝐵 . 𝜇𝑦.𝑢 . Γ . 𝑢 . Δ . 𝑣 . ⊸ . 𝐶 . 𝐵 ⊸ 𝐶 . 𝐵 . (𝑢)𝑣

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 64 / 84

MILL – a graphical notation

Tensor

. Γ . 𝑢 . Δ . 𝑣 . ⊗ . 𝐵 ⊗ 𝐶 . (𝑢, 𝑣) . Γ . 𝑢 . Δ . 𝑣 . ⊗ . 𝐷 . 𝐵 ⊗ 𝐶 . 𝑢(𝑦,𝑧∶=𝑣)

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 65 / 84

The substitution lemma

Lemma

Γ, 𝑦 ∶ 𝐵 ⊢ 𝑢 ∶ 𝐶 Δ ⊢ 𝑣 ∶ 𝐵 Γ, Δ ⊢ 𝑢[𝑣/𝑦] ∶ 𝐶 if Γ and Δ have disjoint domains. The cut rule is admissible. Graphically: . . Γ . 𝑢 . 𝐶 . Δ . 𝑣

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 66 / 84

The substitution lemma

Lemma

Γ, 𝑦 ∶ 𝐵 ⊢ 𝑢 ∶ 𝐶 Δ ⊢ 𝑣 ∶ 𝐵 Γ, Δ ⊢ 𝑢[𝑣/𝑦] ∶ 𝐶 if Γ and Δ have disjoint domains. The cut rule is admissible. Graphically: . . Γ . 𝑢 . 𝐶 . Δ . 𝑣

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 66 / 84

The substitution lemma

Lemma

Γ, 𝑦 ∶ 𝐵 ⊢ 𝑢 ∶ 𝐶 Δ ⊢ 𝑣 ∶ 𝐵 Γ, Δ ⊢ 𝑢[𝑣/𝑦] ∶ 𝐶 if Γ and Δ have disjoint domains. The cut rule is admissible. Graphically: . Γ . 𝑦 ∶ 𝐵 . ax . 𝑢 . 𝐶 . Δ . 𝑣 . 𝐵

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 66 / 84

The substitution lemma

Lemma

Γ, 𝑦 ∶ 𝐵 ⊢ 𝑢 ∶ 𝐶 Δ ⊢ 𝑣 ∶ 𝐵 Γ, Δ ⊢ 𝑢[𝑣/𝑦] ∶ 𝐶 if Γ and Δ have disjoint domains. The cut rule is admissible. Graphically: . Γ . 𝑢 . 𝐶 . Δ . 𝑣 . 𝐵

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 66 / 84

The substitution lemma

Lemma

Γ, 𝑦 ∶ 𝐵 ⊢ 𝑢 ∶ 𝐶 Δ ⊢ 𝑣 ∶ 𝐵 Γ, Δ ⊢ 𝑢[𝑣/𝑦] ∶ 𝐶 if Γ and Δ have disjoint domains. The cut rule is admissible. Graphically: . . Γ . 𝑢 . 𝐶 . Δ . 𝑣

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 66 / 84

MILL – graphical cut elimination

Linear implication

. Γ . Δ . ax . 𝑢 . 𝑣 . ⊸ . ⊸ . 𝐶

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 67 / 84

MILL – graphical cut elimination

Linear implication

. Γ . Δ . 𝑢 . 𝑣 . 𝐶

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 67 / 84

MILL – graphical cut elimination

Tensor

. . Γ . ax . ax . 𝑢 . Δ . 𝑣 . Θ . 𝑤 . ⊗ . ⊗ . 𝐷

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 68 / 84

MILL – graphical cut elimination

Tensor

. Γ . 𝑢 . Δ . 𝑣 . Θ . 𝑤 . 𝐷

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 68 / 84

Proof structures

We extend the graphical formalism to MLL sequent calculus.

1 Allow several formulas on the right hand side of sequents.

⇒ arbitrary number of outputs

2 Reintroduce negation

⇒ transform a hypothesis into a conclusion and vice versa

3 Hard-wire De Morgan duality

⇒ negation is again an operation on formulas and sequents

4 Forget about inputs.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 69 / 84

Proof structures

We extend the graphical formalism to MLL sequent calculus.

1 Allow several formulas on the right hand side of sequents.

⇒ arbitrary number of outputs

2 Reintroduce negation

⇒ transform a hypothesis into a conclusion and vice versa

3 Hard-wire De Morgan duality

⇒ negation is again an operation on formulas and sequents

4 Forget about inputs.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 69 / 84

Proof structures

We extend the graphical formalism to MLL sequent calculus.

1 Allow several formulas on the right hand side of sequents.

⇒ arbitrary number of outputs

2 Reintroduce negation

⇒ transform a hypothesis into a conclusion and vice versa

3 Hard-wire De Morgan duality

⇒ negation is again an operation on formulas and sequents

4 Forget about inputs.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 69 / 84

Proof structures

We extend the graphical formalism to MLL sequent calculus.

1 Allow several formulas on the right hand side of sequents.

⇒ arbitrary number of outputs

2 Reintroduce negation

⇒ transform a hypothesis into a conclusion and vice versa

3 Hard-wire De Morgan duality

⇒ negation is again an operation on formulas and sequents

4 Forget about inputs.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 69 / 84

Proof structures – MLL proofs

. . 𝜌 . 𝜍 . ⊗ . Γ . 𝐵 ⊗ 𝐶 . Δ . 𝐵 . 𝐶 . . 𝜌 . & . Γ . 𝐵 & 𝐶 . 𝐵 . 𝐶 . ax . 𝐵⊥ . 𝐵 . 𝜌 . 𝜍 . cut . Γ . Δ . 𝐵⊥ . 𝐵

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 70 / 84

Proof structures – MLL proofs

. . 𝜌 . 𝜍 . ⊗ . Γ . 𝐵 ⊗ 𝐶 . Δ . 𝐵 . 𝐶 . . 𝜌 . & . Γ . 𝐵 & 𝐶 . 𝐵 . 𝐶 . . ax . 𝐵⊥ . 𝐵 . 𝜌 . 𝜍 . cut . Γ . Δ . 𝐵⊥ . 𝐵

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 70 / 84

Proof structures – a definition

Definition

An MLL proof structure is a directed multigraph with edges labelled by MLL formulas and nodes labelled by rule names or the symbol “c”, with a total order on incoming and outgoing edges on each node, where nodes have one of these shapes: . . ax . 𝐵 . 𝐵⊥ . 𝐵 . 𝐵⊥ . cut . . 𝐵 . 𝐶 . ⊗ . 𝐵 ⊗ 𝐶 . 𝐵 . 𝐶 . & . 𝐵 & 𝐶 The nodes labeled “c” are called the conclusions of the structure.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 71 / 84

Proof structures – an example

Rule commutations are ignored

ax ⊢ 𝐵⊥, 𝐵 ax ⊢ 𝐷⊥, 𝐷 ax ⊢ 𝐶⊥, 𝐶 ⊗ ⊢ 𝐷⊥ ⊗ 𝐶⊥, 𝐶, 𝐷 ⊗ ⊢ 𝐷⊥ ⊗ 𝐶⊥, 𝐵⊥, 𝐵 ⊗ 𝐶, 𝐷 & ⊢ 𝐷⊥ ⊗ 𝐶⊥, 𝐵⊥, (𝐵 ⊗ 𝐶) & 𝐷 & ⊢ (𝐷⊥ ⊗ 𝐶⊥) & 𝐵⊥, (𝐵 ⊗ 𝐶) & 𝐷

. . ⊗ . & . (𝐷⊥ ⊗ 𝐶⊥) & 𝐵⊥ . ⊗ . & . (𝐵 ⊗ 𝐶) & 𝐷 . ax . ax . 𝐷 . 𝐶 . 𝐷⊥ . 𝐶⊥ . ax . 𝐵⊥ . 𝐵

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 72 / 84

Proof structures – an example

Rule commutations are ignored

ax ⊢ 𝐵⊥, 𝐵 ax ⊢ 𝐷⊥, 𝐷 ax ⊢ 𝐶⊥, 𝐶 ⊗ ⊢ 𝐷⊥ ⊗ 𝐶⊥, 𝐶, 𝐷 ⊗ ⊢ 𝐷⊥ ⊗ 𝐶⊥, 𝐵⊥, 𝐵 ⊗ 𝐶, 𝐷 & ⊢ (𝐷⊥ ⊗ 𝐶⊥) & 𝐵⊥, 𝐵 ⊗ 𝐶, 𝐷 & ⊢ (𝐷⊥ ⊗ 𝐶⊥) & 𝐵⊥, (𝐵 ⊗ 𝐶) & 𝐷

. . ⊗ . & . (𝐷⊥ ⊗ 𝐶⊥) & 𝐵⊥ . ⊗ . & . (𝐵 ⊗ 𝐶) & 𝐷 . ax . ax . 𝐷 . 𝐶 . 𝐷⊥ . 𝐶⊥ . ax . 𝐵⊥ . 𝐵

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 72 / 84

Proof structures – an example

Rule commutations are ignored

ax ⊢ 𝐷⊥, 𝐷 ax ⊢ 𝐵⊥, 𝐵 ax ⊢ 𝐶⊥, 𝐶 ⊗ ⊢ 𝐶⊥, 𝐵⊥, 𝐵 ⊗ 𝐶 ⊗ ⊢ 𝐷⊥ ⊗ 𝐶⊥, 𝐵⊥, 𝐵 ⊗ 𝐶, 𝐷 & ⊢ 𝐷⊥ ⊗ 𝐶⊥, 𝐵⊥, (𝐵 ⊗ 𝐶) & 𝐷 & ⊢ (𝐷⊥ ⊗ 𝐶⊥) & 𝐵⊥, (𝐵 ⊗ 𝐶) & 𝐷

. . ⊗ . & . (𝐷⊥ ⊗ 𝐶⊥) & 𝐵⊥ . ⊗ . & . (𝐵 ⊗ 𝐶) & 𝐷 . ax . ax . 𝐷 . 𝐶 . 𝐷⊥ . 𝐶⊥ . ax . 𝐵⊥ . 𝐵

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 72 / 84

Proof structures – an example

Rule commutations are ignored

ax ⊢ 𝐷⊥, 𝐷 ax ⊢ 𝐵⊥, 𝐵 ax ⊢ 𝐶⊥, 𝐶 ⊗ ⊢ 𝐶⊥, 𝐵⊥, 𝐵 ⊗ 𝐶 ⊗ ⊢ 𝐷⊥ ⊗ 𝐶⊥, 𝐵⊥, 𝐵 ⊗ 𝐶, 𝐷 & ⊢ (𝐷⊥ ⊗ 𝐶⊥) & 𝐵⊥, 𝐵 ⊗ 𝐶, 𝐷 & ⊢ (𝐷⊥ ⊗ 𝐶⊥) & 𝐵⊥, (𝐵 ⊗ 𝐶) & 𝐷

. . ⊗ . & . (𝐷⊥ ⊗ 𝐶⊥) & 𝐵⊥ . ⊗ . & . (𝐵 ⊗ 𝐶) & 𝐷 . ax . ax . 𝐷 . 𝐶 . 𝐷⊥ . 𝐶⊥ . ax . 𝐵⊥ . 𝐵

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 72 / 84

Proof structures – an example

Not all proofs are identified

ax ⊢ 𝐵⊥, 𝐵 ax ⊢ 𝐶⊥, 𝐶 ax ⊢ 𝐶⊥, 𝐶 ⊗ ⊢ 𝐶⊥ ⊗ 𝐶⊥, 𝐶, 𝐶 ⊗ ⊢ 𝐶⊥ ⊗ 𝐶⊥, 𝐵⊥, 𝐵 ⊗ 𝐶, 𝐶 & ⊢ 𝐶⊥ ⊗ 𝐶⊥, 𝐵⊥, (𝐵 ⊗ 𝐶) & 𝐶 & ⊢ (𝐶⊥ ⊗ 𝐶⊥) & 𝐵⊥, (𝐵 ⊗ 𝐶) & 𝐶

. . ⊗ . & . (𝐶⊥ ⊗ 𝐶⊥) & 𝐵⊥ . ⊗ . & . (𝐵 ⊗ 𝐶) & 𝐶 . ax . ax . 𝐶 . 𝐶 . 𝐶⊥ . 𝐶⊥ . ax . 𝐵⊥ . 𝐵

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 72 / 84

Proof structures – an example

Not all proofs are identified

ax ⊢ 𝐵⊥, 𝐵 ax ⊢ 𝐶⊥, 𝐶 ax ⊢ 𝐶⊥, 𝐶 ⊗,ex ⊢ 𝐶⊥ ⊗ 𝐶⊥, 𝐶, 𝐶 ⊗ ⊢ 𝐶⊥ ⊗ 𝐶⊥, 𝐵⊥, 𝐵 ⊗ 𝐶, 𝐶 & ⊢ 𝐶⊥ ⊗ 𝐶⊥, 𝐵⊥, (𝐵 ⊗ 𝐶) & 𝐶 & ⊢ (𝐶⊥ ⊗ 𝐶⊥) & 𝐵⊥, (𝐵 ⊗ 𝐶) & 𝐶

. . ⊗ . & . (𝐶⊥ ⊗ 𝐶⊥) & 𝐵⊥ . ⊗ . & . (𝐵 ⊗ 𝐶) & 𝐶 . ax . ax . 𝐶 . 𝐶 . 𝐶⊥ . 𝐶⊥ . ax . 𝐵⊥ . 𝐵

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 72 / 84

Correctness

Not all proof structures are translations of sequential proofs: . . & . ⊗ . (𝐷⊥ & 𝐶⊥) ⊗ 𝐵⊥ . & . ⊗ . (𝐵 & 𝐶) ⊗ 𝐷 . ax . 𝐷⊥ . 𝐷 . ax . 𝐶⊥ . 𝐶 . ax . 𝐵⊥ . 𝐵 Indeed, the conclusion is not provable.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 73 / 84

Proof nets

Definition

A proof net is a proof structure that is the translation of some sequential proof. Exercise: Enumerate all the cut-free proof structures with conclusions (𝐵⊥ ⊗ 𝐵⊥) & 𝐵⊥, (𝐵 ⊗ 𝐵) & 𝐵 and identify which ones are proof nets.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 74 / 84

Cut elimination in proof structures

Tensor versus par: . . 𝐵 . 𝐶 . ⊗ . 𝐵⊥ . 𝐶⊥ . & . cut ⇝ . . 𝐵 . 𝐶 . 𝐵⊥ . 𝐶⊥ . cut . cut Plus the same rules with the left and right premisses of the cut exchanged.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 75 / 84

Cut elimination in proof structures

Axiom: . . 𝐵 . ax . cut . 𝐵 ⇝ . . 𝐵 . 𝐵 This assumes that the right premiss of the cut node is not the left conclusion of the axiom node.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 76 / 84

Cut elimination in proof structures

Theorem (Strong normalization)

In any MLL proof structure, all maximal sequences of cut elimination steps are finite. Each step decreases the number of nodes.

Theorem (Strong confluence)

They all have the same length and they all reach the same irreducible proof structure (up to graph isomorphism). The only critical pairs are in these situations: . A . ax . cut . ax . A . . A . cut . ax . cut . A

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 77 / 84

Correctness

Theorem (Subject reduction)

Irreducible proof structures are cut free.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 78 / 84

Correctness

Problem

Not all irreducible proof structures are cut free. . . ax . cut

Related problem

How do we know that reducing a proof net gives a proof net?

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 78 / 84

Correctness criteria

A correctness criterion is characterization of correct proofs among proof structures. It should be reasonably easy to prove that correctness is preserved by cut elimination. The complexity of actually computing whether a structure satisfies the criterion is directly related to the complexity of the decision problem for the considered logic.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 79 / 84

Reversibility revisited

The & nodes in conclusion are irrelevant for correctness: . . ⊗ . ⊗ . & . (𝐷⊥ ⊗ 𝐶⊥) & 𝐵⊥ . & . (𝐵 ⊗ 𝐶) & 𝐷 . ax . ax . 𝐷 . 𝐶 . 𝐷⊥ . 𝐶⊥ . ax . 𝐵⊥ . 𝐵

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 80 / 84

Reversibility revisited

The & nodes in conclusion are irrelevant for correctness: . . ⊗ . ⊗ . 𝐷⊥ ⊗ 𝐶⊥ . 𝐵⊥ . 𝐵 ⊗ 𝐶 . 𝐷 . ax . ax . 𝐷 . 𝐶 . 𝐷⊥ . 𝐶⊥ . ax . 𝐵⊥ . 𝐵

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 80 / 84

Switchings

The reversibility property can be applied even inside proofs: . Δ . Φ(𝐵 & 𝐶) . 𝐵 . 𝐶 . 𝐵 & 𝐶 . &

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 81 / 84

Switchings

The reversibility property can be applied even inside proofs: . Δ . Φ(𝐵) . 𝐵 . 𝐶 . 𝐵 . 𝐶

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 81 / 84

Switchings

The reversibility property can be applied even inside proofs: . . Δ . Φ(𝐶) . 𝐵 . 𝐶 . 𝐶 . 𝐵

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 81 / 84

Switchings

The reversibility property can be applied even inside proofs: . . Δ . Φ(𝐶) . 𝐵 . 𝐶 . 𝐶 . 𝐵

Lemma

If 𝜌 is a correct cut-free proof structure, then all its &

• switchings are correct.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 81 / 84

Switchings

How can we recognize if a proof structure with only axioms and tensors is correct? . . 𝜌 . 𝜍 . ⊗ . Γ . 𝐵 ⊗ 𝐶 . Δ . 𝐵 . 𝐶 . . ax . 𝐵⊥ . 𝐵

Fact

The structures built using these rules are the acyclic and connected ones.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 82 / 84

Switchings

How can we recognize if a proof structure with only axioms and tensors is correct? . . 𝜌 . 𝜍 . ⊗ . Γ . 𝐵 ⊗ 𝐶 . Δ . 𝐵 . 𝐶 . . ax . 𝐵⊥ . 𝐵

Fact

The structures built using these rules are the acyclic and connected ones.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 82 / 84

Switchings

How can we recognize if a proof structure with only axioms and tensors is correct? . . 𝜌 . 𝜍 . ⊗ . Γ . 𝐵 ⊗ 𝐶 . Δ . 𝐵 . 𝐶 . . ax . 𝐵⊥ . 𝐵

Fact

The structures built using these rules are the acyclic and connected ones.

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 82 / 84

The DR criterion

Theorem (Danos-Regnier)

An MLL proof structure is sequentializable if and only if all its switchings are acyclic and connected. The “only if” part is essentially contained in the previous arguments. For the “if” part, the key point is to prove that the condition implies the existence of a splitting ⊗ node. More on this tomorrow…

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 83 / 84

Cut elimination preserves correctness

Take a tensor/par cut. . . 𝐵 . 𝐶 . 𝐵⊥ . 𝐶⊥ . ⊗ . cut . &

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 84 / 84

Cut elimination preserves correctness

Switch it. . . 𝜌1 . 𝜌2 . 𝜌3 . 𝜌4 . ⊗ . cut

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 84 / 84

Cut elimination preserves correctness

It is connected. . . 𝜌1 . 𝜌2 . 𝜌3 . 𝜌4 . ⊗ . cut

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 84 / 84

Cut elimination preserves correctness

It is connected and acyclic. . . 𝜌1 . 𝜌2 . 𝜌3 . 𝜌4 . ⊗ . cut

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 84 / 84

Cut elimination preserves correctness

It is connected and acyclic. . . 𝜌1 . 𝜌2 . 𝜌3 . 𝜌4 . ⊗ . cut

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 84 / 84

Cut elimination preserves correctness

It is connected and acyclic. . . 𝜌1 . 𝜌2 . 𝜌3 . 𝜌4 . ⊗ . cut

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 84 / 84

Cut elimination preserves correctness

It is connected and acyclic. . . 𝜌1 . 𝜌2 . 𝜌3 . 𝜌4 . ⊗ . cut

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 84 / 84

Cut elimination preserves correctness

Reduce the cut. . . 𝜌1 . 𝜌2 . 𝜌3 . 𝜌4 . cut . cut

Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 84 / 84