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

Γ ⊢ 𝐵, Δ Γ ⊢ Δ Γ ⊢ 𝐵, Δ Γ ⊢ Δ Γ ⊢ 𝐵 ∧ 𝐶, Δ Linearity in logic Classical sequent calculus has weakening and contraction of formulas, which allows using any hypothesis any number of times: Γ, 𝐵, 𝐵 ⊢ Δ cL Γ ⊢ 𝐵, 𝐵, Δ cR wL wR Γ, 𝐵 ⊢ Δ Γ ⊢ 𝐵, Δ Γ, 𝐵 ⊢ Δ Γ ⊢ 𝐵, Δ These make the following rules equivalent: Γ ′ ⊢ 𝐶, Δ ′ Γ ⊢ 𝐶, Δ ∧ Ra ∧ 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: 𝐵 1 , …, 𝐵 𝑜 ⊢ 𝐶 “From hypotheses 𝐵 1 , …, 𝐵 𝑜 deduce 𝐶 .” A proof of this is interpreted as a way to make a proof of 𝐶 from proofs of the 𝐵 𝑗 a function from 𝐵 1 × ⋯ × 𝐵 𝑜 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: 𝐵 1 , …, 𝐵 𝑜 ⊢ 𝐶 1 , …, 𝐶 𝑞 “From hypotheses 𝐵 1 , …, 𝐵 𝑜 deduce 𝐶 1 or … or 𝐶 𝑞 .” Contraction and weakening are allowed on both sides. Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 15 / 84

& ⋯ 𝐶 𝑞 & Sequent calculi Sequents in linear logic: 𝐵 1 , …, 𝐵 𝑜 ⊢ 𝐶 1 , …, 𝐶 𝑞 “From hypotheses 𝐵 1 , …, 𝐵 𝑜 deduce 𝐶 1 or … or 𝐶 𝑞 linearly .” A proof of this is interpreted as a way to make a proof of 𝐶 from proofs of the 𝐵 𝑗 using each 𝐵 𝑗 exactly once a linear map from 𝐵 1 ⊗ ⋯ ⊗ 𝐵 𝑜 to 𝐶 1 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

We focus on MLL, the subsystem made only of multiplicative connectives and negation. Definition 𝐵 ⊸ 𝐶 is a notation for 𝐵 ⊥ 𝐶 . ! 𝐵, ?𝐵 𝐵, 𝐶 ∶= 𝛽 𝐵 ⊥ & 𝐵 ⊗ 𝐶, 𝐵 & 𝐶, 1, ⊥ 𝐵 & 𝐶, 𝐵 ⊕ 𝐶, ⊤, 0 Formulas and sequents In this talk we focus on the propositional structure: formulas propositional variable linear negation multiplicatives additives exponentials sequents with 𝑜, 𝑞 ≥ 0 Γ, Δ, Θ ∶= 𝐵 1 , …, 𝐵 𝑜 ⊢ 𝐶 1 , …, 𝐶 𝑞 Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 17 / 84

& 𝐵, 𝐶 ∶= 𝛽 𝐵 ⊥ 𝐵 ⊗ 𝐶, 𝐵 & 𝐶, 1, ⊥ 𝐵 & 𝐶, 𝐵 ⊕ 𝐶, ⊤, 0 ! 𝐵, ?𝐵 Formulas and sequents In this talk we focus on the propositional structure: formulas propositional variable linear negation multiplicatives additives exponentials sequents with 𝑜, 𝑞 ≥ 0 Γ, Δ, Θ ∶= 𝐵 1 , …, 𝐵 𝑜 ⊢ 𝐶 1 , …, 𝐶 𝑞 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

𝐵 ⊢ 𝐵 Γ ⊢ 𝐵, Δ Γ ′ , 𝐵 ⊢ Δ ′ MLL – the deductive structure The order of formulas is irrelevant: Γ, 𝐵, 𝐶, Δ ⊢ Θ exL Γ ⊢ Δ, 𝐵, 𝐶, Θ exR Γ, 𝐶, 𝐵, Δ ⊢ Θ Γ ⊢ Δ, 𝐶, 𝐵, Θ Axiom and cut rules: ax cut Γ, Γ ′ ⊢ Δ, Δ ′ Linear negation: Γ ⊢ 𝐵, Δ ⊥ L Γ, 𝐵 ⊢ Δ ⊥ R Γ, 𝐵 ⊥ ⊢ Δ Γ ⊢ 𝐵 ⊥ , Δ Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 18 / 84

Γ ⊢ Δ, 𝐵 Γ, 𝐵 ⊕ 𝐶 ⊢ Δ & Γ ⊢ Δ, 𝐶 Γ ⊢ Δ, 𝐵 ⊕ 𝐶 & Γ, 𝐵 ⊢ Δ 𝐶 ⊢ Δ, Δ ′ & Γ, Γ ′ , 𝐵 & Γ ⊢ Δ, 𝐵 ⊕ 𝐶 Γ ′ , 𝐶 ⊢ Δ ′ Γ, 𝐵 ⊢ Δ Γ, 𝐵 & 𝐶 ⊢ Δ Γ, 𝐶 ⊢ Δ Γ, 𝐵, 𝐶 ⊢ Δ Γ, 𝐵 & 𝐶 ⊢ Δ Γ ⊢ Δ, 𝐵 Γ ⊢ Δ, 𝐵 & 𝐶 𝐶 MLL – the connectives Multiplicatives: Γ ′ ⊢ Δ ′ , 𝐶 ⊗ R ⊗ L Γ ⊢ Δ, 𝐵 Γ, Γ ′ ⊢ Δ, Δ ′ , 𝐵 ⊗ 𝐶 Γ, 𝐵 ⊗ 𝐶 ⊢ Δ L R Γ, 𝐵 ⊢ Δ Γ ⊢ Δ, 𝐵, 𝐶 Γ ⊢ Δ, 𝐵 Additives: ⊕ R 1 ⊕ R 2 Γ, 𝐶 ⊢ Δ ⊕ L & L 1 & L 2 Γ ⊢ Δ, 𝐶 & R Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 19 / 84

𝐷 & & 𝐷) ⊢ (𝐵 ⊗ 𝐶) & 𝐵 ⊥ MLL – provability Example The following sequents are provable in MLL: multiplicative excluded middle: ⊢ 𝐵 semi-distributivity of tensor over par: 𝐵 ⊗ (𝐶 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

Theorem (De Morgan laws) For all formulas 𝐵 and 𝐶 , the following equivalences hold: 𝐵 ⊥ ⊗ 𝐶 ⊥ . Exercise: Prove this. Theorem A sequent 𝐵 1 , …, 𝐵 𝑜 ⊢ 𝐶 1 , …, 𝐶 𝑞 is provable if and only the sequent 𝑜 , 𝐶 1 , …, 𝐶 𝑞 is provable. 1 , …, 𝐵 ⊥ 𝐶) ⊥ & 𝐶 ⊥ , (𝐵 & 𝐵 ⊥ (𝐵 ⊗ 𝐶) ⊥ 𝐵 ⊥⊥ , 𝐵 ⊢ 𝐵 ⊥ Symmetries Let us see if we can simplify the system a bit. Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 21 / 84

Theorem A sequent 𝐵 1 , …, 𝐵 𝑜 ⊢ 𝐶 1 , …, 𝐶 𝑞 is provable if and only the sequent 𝑜 , 𝐶 1 , …, 𝐶 𝑞 is provable. 1 , …, 𝐵 ⊥ ⊢ 𝐵 ⊥ & 𝐶 ⊥ , (𝐵 & 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 ⊗ 𝐶) ⊥ ˛ 𝐵 ⊥ 𝐶) ⊥ ˛ 𝐵 ⊥ ⊗ 𝐶 ⊥ . 𝐵 ˛ 𝐵 ⊥⊥ , Exercise: Prove this. Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 21 / 84

& ⊢ 𝐵 ⊥ 𝐶 ⊥ , (𝐵 & 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 ⊗ 𝐶) ⊥ ˛ 𝐵 ⊥ 𝐶) ⊥ ˛ 𝐵 ⊥ ⊗ 𝐶 ⊥ . 𝐵 ˛ 𝐵 ⊥⊥ , Exercise: Prove this. Theorem A sequent 𝐵 1 , …, 𝐵 𝑜 ⊢ 𝐶 1 , …, 𝐶 𝑞 is provable if and only the sequent 𝑜 , 𝐶 1 , …, 𝐶 𝑞 is provable. 1 , …, 𝐵 ⊥ Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 21 / 84

& (𝐵 𝐶 ⊥ 𝐵, 𝐶 ∶= 𝛽 & 𝛽 ⊥ Γ, Δ, Θ ∶= ⊢ 𝐵 1 , …, 𝐵 𝑜 𝐵 ⊗ 𝐶, 𝐵 & 𝐶, 1, ⊥ ! 𝐵, ?𝐵 𝐵 & 𝐶, 𝐵 ⊕ 𝐶, ⊤, 0 One-sided presentation Redefine the language of formulas: formulas propositional variable negated variable multiplicatives additives exponentials sequents with 𝑜 ≥ 0 Definition Negation is the operation on formulas defined as (𝐵 ⊗ 𝐶) ⊥ ∶= 𝐵 ⊥ (𝐵 ⊕ 𝐶) ⊥ ∶= 𝐵 ⊥ & 𝐶 ⊥ (!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

⊢ Γ ⊥ ⊢ Γ, ⊤ ⊤ ⊢ 1 1 ⊢ Γ, ⊥ ⊢ Γ, 𝐵 ⊕ 𝐶 ⊕ 2 ⊢ Γ, 𝐶 ⊢ Γ, 𝐵 ⊕ 𝐶 ⊕ 1 ⊢ Γ, 𝐵 ⊢ Γ, 𝐵 & 𝐶 ⊢ Γ, 𝐵 𝐶 & ⊢ Γ, Δ ⊢ Γ, 𝐵 ⊢ Γ, 𝐵 & ⊢ Γ, 𝐵, 𝐶 ⊢ Δ, 𝐵 ⊥ One-sided sequent calculus Axiom and cut rules: ax cut ⊢ 𝐵 ⊥ , 𝐵 Multiplicatives: ⊢ Γ, 𝐵 ⊢ Δ, 𝐶 ⊗ ⊢ Γ, Δ, 𝐵 ⊗ 𝐶 Additives: ⊢ Γ, 𝐶 & Units: Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 23 / 84

⊢ 1 ⊢ Γ, 𝐵 1 ⊢ Γ, ⊤ ⊢ Γ ⊕ 2 ⊢ Γ, 𝐶 ⊥ ⊕ 1 ⊢ Γ, 𝐵 ⊢ Γ, ⊥ 𝐶 ⊤ & & ⊢ Γ, 𝐵, 𝐶 ⊢ Γ, Δ ⊢ Δ, 𝐵 ⊥ ⊢ Γ, 𝐵 One-sided sequent calculus Axiom and cut rules: ax cut ⊢ 𝐵 ⊥ , 𝐵 Multiplicatives: ⊢ Γ, 𝐵 ⊢ Δ, 𝐶 ⊗ ⊢ Γ, Δ, 𝐵 ⊗ 𝐶 Additives: ⊢ Γ, 𝐵 ⊢ Γ, 𝐶 & ⊢ Γ, 𝐵 & 𝐶 ⊢ Γ, 𝐵 ⊕ 𝐶 ⊢ Γ, 𝐵 ⊕ 𝐶 Units: Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 23 / 84

⊢ 1 & ⊤ ⊕ 2 ⊢ Γ, 𝐶 ⊢ Γ ⊕ 1 ⊢ Γ, 𝐵 ⊥ 𝐶 & 1 ⊢ Γ, 𝐵 ⊢ Γ, Δ ⊢ Γ, 𝐵, 𝐶 ⊢ Δ, 𝐵 ⊥ ⊢ Γ, 𝐵 One-sided sequent calculus Axiom and cut rules: ax cut ⊢ 𝐵 ⊥ , 𝐵 Multiplicatives: ⊢ Γ, 𝐵 ⊢ Δ, 𝐶 ⊗ ⊢ Γ, Δ, 𝐵 ⊗ 𝐶 Additives: ⊢ Γ, 𝐵 ⊢ Γ, 𝐶 & ⊢ Γ, 𝐵 & 𝐶 ⊢ Γ, 𝐵 ⊕ 𝐶 ⊢ Γ, 𝐵 ⊕ 𝐶 Units: ⊢ Γ, ⊤ ⊢ Γ, ⊥ Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 23 / 84

& & & ⊢ 𝐵 ⊥ , 𝐵 𝐷 ⊥ ) ⊸ 𝐵 ⊥ & ⊢ 𝐵 ⊥ , 𝐶 ⊥ , 𝐵 ⊗ 𝐶 ⊕ 1 ⊢ 𝐵 ⊥ , 𝐶 ⊥ , (𝐵 ⊗ 𝐶) ⊕ (𝐵 ⊗ 𝐷) & ⊢ 𝐵 ⊥ , 𝐵 ⊢ 𝐵 ⊥ 𝐶 ⊥ ) & (𝐵 ⊥ ⊢ 𝐵 ⊥ , 𝐷 ⊥ , 𝐵 ⊗ 𝐷 ⊕ 2 Additives vs multiplicatives Example: distributivity of ⊗ over ⊕ . ax ax ax ax ⊢ 𝐶 ⊥ , 𝐶 ⊗ ⊢ 𝐷 ⊥ , 𝐷 ⊗ ⊢ 𝐵 ⊥ , 𝐶 ⊥ , (𝐵 ⊗ 𝐶) ⊕ (𝐵 ⊗ 𝐷) & ⊢ 𝐵 ⊥ , 𝐶 ⊥ & 𝐷 ⊥ , (𝐵 ⊗ 𝐶) ⊕ (𝐵 ⊗ 𝐷) (𝐶 ⊥ & 𝐷 ⊥ ), (𝐵 ⊗ 𝐶) ⊕ (𝐵 ⊗ 𝐷) Hence 𝐵 ⊗ (𝐶 ⊕ 𝐷) ⊸ (𝐵 ⊗ 𝐶) ⊕ (𝐵 ⊗ 𝐷) , equivalently (𝐵 ⊥ (𝐶 ⊥ & 𝐷 ⊥ ) . Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 24 / 84

& & & ⊢ 𝐵 ⊥ 𝐶 ⊥ ) & (𝐵 ⊥ ⊢ 𝐵 ⊥ , 𝐷 ⊥ , 𝐵 ⊗ (𝐶 ⊕ 𝐷) & ⊕ 1 ⊢ 𝐷 ⊥ , 𝐷 𝐷 ⊥ ), 𝐵 ⊗ (𝐶 ⊕ 𝐷) ⊢ 𝐵 ⊥ , 𝐵 𝐶 ⊥ , 𝐵 ⊗ (𝐶 ⊕ 𝐷) ⊢ (𝐵 ⊥ & ⊢ 𝐵 ⊥ & ⊢ 𝐵 ⊥ , 𝐶 ⊥ , 𝐵 ⊗ (𝐶 ⊕ 𝐷) & ⊕ 1 ⊢ 𝐶 ⊥ , 𝐶 𝐶 ⊥ ) & (𝐵 ⊥ ⊢ 𝐵 ⊥ , 𝐵 & & Additives vs multiplicatives Example: distributivity of ⊗ over ⊕ . ax ax ax ax ⊢ 𝐶 ⊥ , 𝐶 ⊕ 𝐷 ⊗ ⊢ 𝐷 ⊥ , 𝐶 ⊕ 𝐷 ⊗ 𝐷 ⊥ , 𝐵 ⊗ (𝐶 ⊕ 𝐷) & Hence (𝐵 ⊗ 𝐶) ⊕ (𝐵 ⊗ 𝐷) ⊸ 𝐵 ⊗ (𝐶 ⊕ 𝐷) , equivalently 𝐵 ⊥ 𝐷 ⊥ ) . (𝐶 ⊥ & 𝐷 ⊥ ) ⊸ (𝐵 ⊥ Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 24 / 84

⊢ Γ Exponentials Contraction and weakening are crucial for logical expressiveness. Linear logic provides them through modalities . Allowed structural rules: ⊢ Γ, ?𝐵, ?𝐵 c w ⊢ Γ, 𝐵 ? ⊢ Γ, ?𝐵 ⊢ Γ, ?𝐵 ⊢ Γ, ?𝐵 Promotion: ⊢ ?𝐵 1 , …, ?𝐵 𝑜 , 𝐶 ! ⊢ ?𝐵 1 , …, ?𝐵 𝑜 , !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

?𝐵 = 𝑜 !𝐵 = (𝐵 ⊕ ⊥), 𝑜=0 & ∞ 𝑜=0 (𝐵 & 1). 𝐵. 𝑗=1 􏾀 𝑜=0 􏾀 & ∞ !𝐵 = 𝐵, 𝑗=1 & 𝑜 𝑜=0 􏾙 ∞ ?𝐵 = ∞ Exponentials – equivalences Wrong but not too much: A bit less wrong: 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

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. The question of consistency We have a definition of formulas, sequents and deduction rules. But how do we know if the system is consistent? Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 29 / 84

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

Use the argument sequent calculus was built for: Cut elimination. 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. 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

⊢ Γ, Δ, Θ ⊢ Θ, 𝐵 ⊥ ⊢ Γ, 𝐵 𝜌 1 ↘ 𝜌 3 ⊢ Θ, 𝐵 ⊥ , 𝐶 ⊥ 𝐶 ⊥ & & ⊢ Δ, 𝐶 ⊢ Θ, 𝐵 ⊥ , 𝐶 ⊥ 𝜌 3 ⊢ Γ, Δ, 𝐵 ⊗ 𝐶 ⊢ Δ, Θ, 𝐵 ⊥ 𝜌 2 ⊢ Γ, 𝐵 𝜌 1 ⊢ Γ, Δ, Θ 𝜌 2 Cut elimination Interaction rules Tensor versus par ⊢ Δ, 𝐶 ⊗ cut cut cut Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 32 / 84

⊢ Δ, 𝐵 ⊥ ⊢ Γ, Δ 𝜌 3 ⊢ Γ, 𝐵 𝜌 1 ↘ ⊢ Γ, Δ ⊕ 1 ⊢ Δ, 𝐵 ⊥ 𝜌 3 ⊢ Γ, 𝐵 & 𝐶 𝜌 2 ⊢ Γ, 𝐵 𝜌 1 Cut elimination Interaction rules With versus plus ⊢ Γ, 𝐶 & ⊢ Δ, 𝐵 ⊥ ⊕ 𝐶 ⊥ cut cut Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 33 / 84

⊢ ?Γ, !𝐵 𝜌 1 ⊢ ?Γ, Δ 𝜌 1 ⊢ ?Γ, ?Γ, Δ ⊢ ?Γ, !𝐵 𝜌 2 ⊢ Δ, ?𝐵 ⊥ , ?𝐵 ⊥ ⊢ ?Γ, Δ, ?𝐵 ⊥ ⊢ Δ, ?𝐵 ⊥ ⊢ Δ, ?𝐵 ⊥ , ?𝐵 ⊥ ⊢ ?Γ, Δ ↘ 𝜌 1 𝜌 2 ⊢ ?Γ, !𝐵 Cut elimination Interaction rules Promotion versus contraction c ⊢ ?Γ, 𝐵 ! cut ⊢ ?Γ, 𝐵 ! cut ⊢ ?Γ, 𝐵 ! cut c Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 34 / 84

! ! ? ! & ⊕ 2 & ⊕ 1 ⊥ 1 & ⊗ Cut elimination Interaction rules … plus a few other cancellation rules … left right action propagate the cuts to sub-formulas drop the proof of 1 keep only the left proof in the & rule 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

𝜌 1 𝜌 2 ⊢ Γ, Δ, 𝐵 ⊗ 𝐶, 𝐷 𝜌 1 ⊢ Γ, 𝐵 𝜌 2 ⊢ Θ, 𝐷 ⊥ ⊢ Γ, Δ, 𝐵 ⊗ 𝐶, 𝐷 𝜌 3 ⊢ Θ, 𝐷 ⊥ 𝜌 3 ⊢ Γ, Δ, Θ, 𝐵 ⊗ 𝐶 ↘ ⊢ Δ, 𝐶, 𝐷 ⊢ Γ, 𝐵 Cut elimination Commutation rules Commutation with tensor ⊢ Δ, 𝐶, 𝐷 ⊗ cut cut ⊢ Δ, Θ, 𝐶 ⊗ Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 36 / 84

𝜌 3 𝜌 3 ⊢ Δ, 𝐷 ⊥ ⊢ Γ, 𝐶, 𝐷 ⊢ Γ, 𝐵, 𝐷 𝜌 1 ↘ ⊢ Γ, Δ, 𝐵 & 𝐶 ⊢ Δ, 𝐷 ⊥ ⊢ Γ, Δ, 𝐵 𝜌 3 ⊢ Γ, 𝐵 & 𝐶, 𝐷 ⊢ Δ, 𝐷 ⊥ 𝜌 2 ⊢ Γ, 𝐵, 𝐷 𝜌 1 ⊢ Γ, Δ, 𝐵 & 𝐶 𝜌 2 Cut elimination Commutation rules Commutation with “with” ⊢ Γ, 𝐶, 𝐷 & cut cut cut ⊢ Γ, Δ, 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

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 Normalization With the right set of rules, clearly irreducible proofs are cut-free. How to prove that reduction always terminates? Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 39 / 84

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

Indirectly through more tractable systems polarized systems … more on this in a minute proof nets … more on this later 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. 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

We need a structure that plays in LL the role of Boolean algebras in LK. 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? 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

𝐵 ⊸ 𝐶 = 𝐵 ⊥ ⊥⊥ 𝐵 & 𝐵 ⊕ 𝐶 ∶= (𝐵 ∪ 𝐶) ⊥⊥ 𝐵 & 𝐶 ∶= 𝐵 ∩ 𝐶 0 ∶= ∅ ⊥⊥ ⊤ ∶= 𝑁 !𝐵 ∶= (𝐵 ∩ 𝐽) ⊥⊥ 1 ∶= {1} ⊥⊥ & Phase spaces Connectives Given (𝑁, ⊥) , for subsets 𝐵, 𝐶 ⊆ 𝑁 define 𝐶 ∶= (𝐵 ⊥ ⊗ 𝐶 ⊥ ) ⊥ 𝐵 ⊗ 𝐶 ∶= 􏿻𝑞𝑟 􏿗 𝑞 ∈ 𝐵, 𝑟 ∈ 𝐶􏿾 ?𝐵 ∶= (𝐵 ⊥ ∩ 𝐽) ⊥ where 𝐽 is the set of idempotents belonging to 1 . If propositional variables are interpreted as facts, then for any formula 𝐵 the interpretation � 𝐵 � 𝑁 is a fact. 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 ∈ � 𝐵 � 𝑁 in any phase space 𝑁 . Exercise: Check it by induction over proofs. Theorem (Completeness) If 1 ∈ � 𝐵 � 𝑁 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 � 𝐵 � 𝑁 = 􏿻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

& |𝐵 ⊗ 𝐶| = |𝐵 & 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 ′ . 𝐶| = |𝐵| × |𝐶| and (𝑦, 𝑧) ¨ 𝐵⊗𝐶 (𝑦 ′ , 𝑧 ′ ) if 𝑦 ¨ 𝐵 𝑦 ′ and 𝑧 ¨ 𝐶 𝑧 ′ , 𝐶 (𝑦 ′ , 𝑧 ′ ) if 𝑦 ˝ 𝐵 𝑦 ′ or 𝑧 ˝ 𝐶 𝑧 ′ . (𝑦, 𝑧) ˝ 𝐵 |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

Exponentials w c ⊢ 𝛿 ∶ Γ, 𝑏 ∪ 𝑏 ′ ∶ ?𝐵 􏿻⊢ 𝑏 1,𝑗 ∶ ?𝐵 1 , …𝑏 𝑜,𝑗 ∶ ?𝐵 𝑜 , 𝑐 𝑗 ∶ 𝐶􏿾 ⊢ ⋃ 𝑗∈𝐽 𝑏 1,𝑗 ∶ ?𝐵 1 , …⋃ 𝑗∈𝐽 𝑏 𝑜 𝑗 ∶ ?𝐵 𝑜 , {𝑐 𝑗 | 𝑗 ∈ 𝐽} ∶ !𝐶 ⊢ 𝛿 ∶ Γ, (𝛽, 𝛾) ∶ 𝐵 ⊢ 𝛿 ∶ Γ, {𝛽} ∶ ?𝐵 ? ⊢ 𝛿 ∶ Γ, 𝛽 ∶ 𝐵 ⊢ 𝛿 ∶ Γ, ∅ ∶ ?𝐵 𝐶 & ⊗ & ⊢ 𝛿 ∶ Γ, 𝛽 ∶ 𝐵, 𝛾 ∶ 𝐶 ⊢ 𝛿 ∶ Γ, 𝑏 ∶ ?𝐵, 𝑏 ′ ?𝐵 ⊢ 𝛾 ∶ 𝐶, 𝜀 ∶ Δ 𝑗∈𝐽 ⊢ 𝛽 ∶ 𝐵 ⊥ , 𝜀 ∶ Δ ⊢ 𝛿 ∶ Γ, 𝛽 ∶ 𝐵 ! ⊢ 𝛿 ∶ Γ Coherence spaces: interpreting proofs Identity ax cut ⊢ 𝛽 ∶ 𝐵 ⊥ , 𝛽 ∶ 𝐵 ⊢ 𝛿 ∶ Γ, 𝜀 ∶ Δ Multiplicatives ⊢ 𝛿 ∶ Γ, 𝛽 ∶ 𝐵 ⊢ 𝛿 ∶ Γ, (𝛽, 𝛾) ∶ 𝐵 ⊗ 𝐶, 𝜀 ∶ Δ Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 45 / 84

? 𝑗∈𝐽 ! 𝐶 & ⊢ 𝛿 ∶ Γ, (𝛽, 𝛾) ∶ 𝐵 & ⊢ 𝛿 ∶ Γ, 𝛽 ∶ 𝐵, 𝛾 ∶ 𝐶 ⊗ ⊢ 𝛾 ∶ 𝐶, 𝜀 ∶ Δ ⊢ 𝛿 ∶ Γ, 𝛽 ∶ 𝐵 ⊢ 𝛽 ∶ 𝐵 ⊥ , 𝜀 ∶ Δ Coherence spaces: interpreting proofs Identity ax cut ⊢ 𝛽 ∶ 𝐵 ⊥ , 𝛽 ∶ 𝐵 ⊢ 𝛿 ∶ Γ, 𝜀 ∶ Δ Multiplicatives ⊢ 𝛿 ∶ Γ, 𝛽 ∶ 𝐵 ⊢ 𝛿 ∶ Γ, (𝛽, 𝛾) ∶ 𝐵 ⊗ 𝐶, 𝜀 ∶ Δ Exponentials ⊢ 𝛿 ∶ Γ, 𝛽 ∶ 𝐵 ⊢ 𝛿 ∶ Γ w ⊢ 𝛿 ∶ Γ, 𝑏 ∶ ?𝐵, 𝑏 ′ ?𝐵 c ⊢ 𝛿 ∶ Γ, 𝑏 ∪ 𝑏 ′ ∶ ?𝐵 ⊢ 𝛿 ∶ Γ, ∅ ∶ ?𝐵 ⊢ 𝛿 ∶ Γ, {𝛽} ∶ ?𝐵 􏿻⊢ 𝑏 1,𝑗 ∶ ?𝐵 1 , …𝑏 𝑜,𝑗 ∶ ?𝐵 𝑜 , 𝑐 𝑗 ∶ 𝐶􏿾 ⊢ ⋃ 𝑗∈𝐽 𝑏 1,𝑗 ∶ ?𝐵 1 , …⋃ 𝑗∈𝐽 𝑏 𝑜 𝑗 ∶ ?𝐵 𝑜 , {𝑐 𝑗 | 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

⊢ Δ ∗ , !𝐵 ⊗ (𝐶 ∗ ) ⊥ , 𝐶 ∗ ⊢ Γ ∗ , ?(𝐵 ∗ ) ⊥ ⊗ ⊢ (𝐶 ∗ ) ⊥ , 𝐶 ∗ ⊢ Δ ∗ , !𝐵 ∗ ! ⊢ Δ ∗ , 𝐵 ∗ 𝐶 ∗ & ⊢ Γ ∗ , ?(𝐵 ∗ ) ⊥ ↝ 𝐶 ∗ ⊢ Γ ∗ , Δ ∗ , 𝐶 ∗ & & 𝐶 ↝ 𝐵 ⇒ 𝐶 = !𝐵 ⊸ 𝐶 = ?𝐵 ⊥ & ⊢ Γ ∗ , ?(𝐵 ∗ ) ⊥ , 𝐶 ∗ LJ expressed in linear logic Linear logic arises from the decomposition Deduction rules can be translated accordingly: Γ, 𝐵 ⊢ 𝑀𝐾 𝐶 Γ ⊢ 𝑀𝐾 𝐵 ⇒ 𝐶 ax Γ ⊢ 𝑀𝐾 𝐵 ⇒ 𝐶 Δ ⊢ 𝑀𝐾 𝐵 Γ, Δ ⊢ 𝑀𝐾 𝐶 cut 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 𝐵 1 , …, 𝐵 𝑜 ⊢ 𝐶 1 , …, 𝐶 𝑞 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

⊢ 𝐶 ⊥ & 𝐷 ⊥ ), 𝐵 ⊕ (𝐶 ⊗ 𝐷) & ⊢ 𝐵 ⊥ , 𝐵 ⊕ 1 ⊢ 𝐵 ⊥ , 𝐵 ⊕ (𝐶 ⊗ 𝐷) & ⊢ 𝐶 ⊥ , 𝐶 ⊢ 𝐶 ⊥ ⊕ 2 ⊢ 𝐶 ⊥ , 𝐷 ⊥ , 𝐶 ⊗ 𝐷 𝐷 ⊥ , 𝐶 ⊗ 𝐷 Type isomorphisms Technical aside: 𝜃 -equivalence Consider possible cut-free proofs of 𝐵 ⊕ (𝐶 ⊗ 𝐷) ⊸ 𝐵 ⊕ (𝐶 ⊗ 𝐷) . ax ax ⊢ 𝐷 ⊥ , 𝐷 ⊗ ax ax 𝐷 ⊥ , 𝐵 ⊕ (𝐶 ⊗ 𝐷) ax ⊢ 𝐵 ⊥ & (𝐶 ⊥ Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 52 / 84

⊢ 𝐶 ⊥ & 𝐷 ⊥ ), 𝐵 ⊕ (𝐶 ⊗ 𝐷) & ⊢ 𝐵 ⊥ , 𝐵 ⊕ 1 ⊢ 𝐵 ⊥ , 𝐵 ⊕ (𝐶 ⊗ 𝐷) & ⊢ 𝐶 ⊥ , 𝐶 ⊢ 𝐶 ⊥ ⊕ 2 ⊢ 𝐶 ⊥ , 𝐷 ⊥ , 𝐶 ⊗ 𝐷 𝐷 ⊥ , 𝐶 ⊗ 𝐷 Type isomorphisms Technical aside: 𝜃 -equivalence Consider possible cut-free proofs of 𝐵 ⊕ (𝐶 ⊗ 𝐷) ⊸ 𝐵 ⊕ (𝐶 ⊗ 𝐷) . ax ax ⊢ 𝐷 ⊥ , 𝐷 ⊗ ax ax 𝐷 ⊥ , 𝐵 ⊕ (𝐶 ⊗ 𝐷) & ⊢ 𝐵 ⊥ & (𝐶 ⊥ Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 52 / 84

⊢ 𝐶 ⊥ & 𝐷 ⊥ ), 𝐵 ⊕ (𝐶 ⊗ 𝐷) & ⊢ 𝐵 ⊥ , 𝐵 ⊕ 1 ⊢ 𝐵 ⊥ , 𝐵 ⊕ (𝐶 ⊗ 𝐷) & ⊢ 𝐶 ⊥ , 𝐶 ⊢ 𝐶 ⊥ ⊕ 2 ⊢ 𝐶 ⊥ , 𝐷 ⊥ , 𝐶 ⊗ 𝐷 & 𝐷 ⊥ , 𝐶 ⊗ 𝐷 Type isomorphisms Technical aside: 𝜃 -equivalence Consider possible cut-free proofs of 𝐵 ⊕ (𝐶 ⊗ 𝐷) ⊸ 𝐵 ⊕ (𝐶 ⊗ 𝐷) . ax ax ⊢ 𝐷 ⊥ , 𝐷 ⊗ ax 𝐷 ⊥ , 𝐵 ⊕ (𝐶 ⊗ 𝐷) & ⊢ 𝐵 ⊥ & (𝐶 ⊥ Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 52 / 84

& 𝐷 ⊥ , 𝐶 ⊗ 𝐷 𝐷 ⊥ ), 𝐵 ⊕ (𝐶 ⊗ 𝐷) & ⊢ 𝐵 ⊥ , 𝐵 ⊕ 1 ⊢ 𝐵 ⊥ , 𝐵 ⊕ (𝐶 ⊗ 𝐷) & ⊢ 𝐶 ⊥ , 𝐶 ⊢ 𝐶 ⊥ ⊕ 2 ⊢ 𝐶 ⊥ , 𝐷 ⊥ , 𝐶 ⊗ 𝐷 & ⊢ 𝐶 ⊥ Type isomorphisms Technical aside: 𝜃 -equivalence Consider possible cut-free proofs of 𝐵 ⊕ (𝐶 ⊗ 𝐷) ⊸ 𝐵 ⊕ (𝐶 ⊗ 𝐷) . ax ax ⊢ 𝐷 ⊥ , 𝐷 ⊗ ax 𝐷 ⊥ , 𝐵 ⊕ (𝐶 ⊗ 𝐷) & ⊢ 𝐵 ⊥ & (𝐶 ⊥ 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

𝐵 ⊗ 𝐶 ≃ 𝐶 ⊗ 𝐵 𝐵 ⊗ (𝐶 ⊕ 𝐷) ≃ (𝐵 ⊗ 𝐶) ⊕ (𝐵 ⊗ 𝐷) !(𝐵 & 𝐶) ≃ !𝐵 ⊗ !𝐶 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: 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

!(𝐵 & 𝐶) ≃ !𝐵 ⊗ !𝐶 (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 𝐵 ⊗ (𝐶 ⊕ 𝐷) ≃ (𝐵 ⊗ 𝐶) ⊕ (𝐵 ⊗ 𝐷) 𝐵 ⊗ 0 ≃ 0 !⊤ ≃ 1 Standard isomorphisms Remark that 𝐵 ≃ 𝐶 iff 𝐵 ⊥ ≃ 𝐶 ⊥ . Associativity and commutativity Distributivity Exponentiation Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 54 / 84

& & ⊢ Γ, 𝐵 & & Reversibility The rules for and & are reversible, i.e. 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 or 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

⋯ 𝜌 1 ⊢ Γ 1 , …, Γ 𝑙 , 𝑄 𝑗 𝜌 𝑙 & ⊢ Γ 1 , 𝑂 1 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 = 𝑄 1 , …, 𝑄 𝑜 be a provable sequent consisting of positive formulas only. Then there is a formula 𝑄 𝑗 and proof of ⊢ Γ of the form ⊢ Γ 𝑙 , 𝑂 𝑙 𝑆 where the 𝑂 𝑘 are the maximal negative subformulas of 𝑄 𝑗 and the last set of rules 𝑆 builds 𝑄 𝑗 from the 𝑂 𝑘 . Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 56 / 84

𝑌 𝑗 & ⊢ Γ, Φ ∗ (𝐵 1 , …, 𝐵 𝑜 ) ⊢ (Γ 𝑗 ) 𝑗∈𝐽 , Φ(𝐵 1 , …, 𝐵 𝑜 ) Φ 𝐽 𝑗∈𝐽 𝐽∈𝓙 􏾀 𝑗∈𝐽 𝑌 𝑗 Φ ∗ (𝑌 1 , …, 𝑌 𝑜 ) ≃ & 𝐽∈𝓙 Synthetic connectives Let Φ(𝑌 1 , …, 𝑌 𝑜 ) be a formula made of positive connectives from the variables 𝑌 1 , …, 𝑌 𝑜 . Call Φ ∗ the dual of Φ . Up to associativity/commutativity/neutrality, for some set 𝓙 ⊆ 𝒬({1, …, 𝑜}) one has Φ(𝑌 1 , …, 𝑌 𝑜 ) ≃ 􏾙 There is one family of rules (⊢ Γ 𝑗 , 𝐵 𝑗 ) 𝑗∈𝐽 (⊢ Γ, (𝐵 𝑗 ) 𝑗∈𝐽 ) 𝐽∈𝓙 Φ ∗ 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

⊢ Γ, 𝑂 ⊢ Γ ⊢ 𝑂 1 , …, 𝑂 𝑜 , !𝑂 𝑄, 𝑅 ∶= 𝛽, 𝑄 ⊗ 𝑅, 𝑄 ⊕ 𝑅, 1, 0, !𝑂 𝑁, 𝑂 ∶= 𝛽 ⊥ , 𝑁 & 𝑂, 𝑁 & 𝑂, ⊥, ⊤, ?𝑄 ⊢ Γ, 𝑂 Polarized linear logic Since connectives of the same polarity behave well, let us restrict to a system where polarities are never mixed: If 𝑄 is a positive formula where variables only appear under modalities, then 𝑄 ⊸ !𝑄 is provable. Hence the following rules are derivable: ⊢ Γ, 𝑂, 𝑂 C W ⊢ 𝑂 1 , …, 𝑂 𝑜 , 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

Definition (Formulas of MILL) propositional variable linear implication multiplicative conjunction Definition (Proof terms for MILL) variable — axiom linear abstraction — introduction of ⊸ linear application — elimination of ⊸ pair — introduction of ⊗ matching — elimination of ⊗ 𝐵 ⊸ 𝐶 𝐵, 𝐶 ∶= 𝛽 𝐵 ⊗ 𝐶 𝑢, 𝑣 ∶= 𝑦 𝑢(𝑦,𝑧∶=𝑣) 𝜇𝑦.𝑢 (𝑢)𝑣 (𝑢, 𝑣) Intuitionistic LL The 𝜇 -calculus is simpler because it is asymmetric. What if we made LL asymmetric too? Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 61 / 84

Definition (Proof terms for MILL) variable — axiom linear abstraction — introduction of ⊸ linear application — elimination of ⊸ pair — introduction of ⊗ matching — elimination of ⊗ 𝑢(𝑦,𝑧∶=𝑣) 𝐵, 𝐶 ∶= 𝛽 𝐵 ⊸ 𝐶 (𝑢, 𝑣) 𝐵 ⊗ 𝐶 𝑢, 𝑣 ∶= 𝑦 (𝑢)𝑣 𝜇𝑦.𝑢 Intuitionistic LL The 𝜇 -calculus is simpler because it is asymmetric. What if we made LL asymmetric too? Definition (Formulas of MILL) propositional variable linear implication multiplicative conjunction Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 61 / 84

(𝑢, 𝑣) 𝐵, 𝐶 ∶= 𝛽 𝐵 ⊸ 𝐶 𝐵 ⊗ 𝐶 𝑢, 𝑣 ∶= 𝑦 𝜇𝑦.𝑢 𝑢(𝑦,𝑧∶=𝑣) (𝑢)𝑣 Intuitionistic LL The 𝜇 -calculus is simpler because it is asymmetric. What if we made LL asymmetric too? Definition (Formulas of MILL) propositional variable linear implication multiplicative conjunction Definition (Proof terms for MILL) variable — axiom linear abstraction — introduction of ⊸ linear application — elimination of ⊸ pair — introduction of ⊗ matching — elimination of ⊗ Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 61 / 84

Δ ⊢ 𝑣 ∶ 𝐵 ⊗ 𝐶 𝑦 ∶ 𝐵 ⊢ 𝑦 ∶ 𝐵 Γ ⊢ 𝑢 ∶ 𝐵 ⊸ 𝐶 Γ, Δ ⊢ (𝑢)𝑣 ∶ 𝐶 Γ, 𝑦 ∶ 𝐵, 𝑧 ∶ 𝐶 ⊢ 𝑢 ∶ 𝐷 MILL – typing rules Identity ax Implication Δ ⊢ 𝑣 ∶ 𝐵 ⊸ E ⊸ R Γ, 𝑦 ∶ 𝐵 ⊢ 𝑢 ∶ 𝐶 Γ ⊢ 𝜇𝑦.𝑢 ∶ 𝐵 ⊸ 𝐶 Tensor Δ ⊢ 𝑣 ∶ 𝐶 ⊗ R ⊗ E Γ ⊢ 𝑢 ∶ 𝐵 Γ, Δ ⊢ (𝑢, 𝑣) ∶ 𝐵 ⊗ 𝐶 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

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. (𝜇𝑦.𝑢)𝑣 ⇝ 𝑢[𝑣/𝑦] 𝑢(𝑦,𝑧∶=(𝑣, 𝑤)) ⇝ 𝑢[𝑣/𝑦][𝑤/𝑧] MILL – reduction Definition Cut elimination for MILL is generated by the following rules: Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 63 / 84

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

𝜇𝑦.𝑢 𝑦 ⊸ Γ 𝑢 𝑢 Γ Δ 𝑣 𝐵 ⊸ 𝐶 𝐵 ⊸ 𝐶 𝐵 𝑦 ∶ 𝐵 (𝑢)𝑣 𝑦 ∶ 𝐵 MILL – a graphical notation Axiom and linear implication 𝐵 ⊸ 𝐶 . . . . . . . . . . . . . . . . . . . ax Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 64 / 84

⊗ ⊗ 𝑣 𝐵 ⊗ 𝐶 Δ 𝑢 Γ 𝑢(𝑦,𝑧∶=𝑣) (𝑢, 𝑣) 𝐷 𝐵 ⊗ 𝐶 𝑣 Δ 𝑢 Γ MILL – a graphical notation Tensor . . . . . . . . . . . . . . . Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 65 / 84

The cut rule is admissible. Graphically: . . . . . . Δ Γ, 𝑦 ∶ 𝐵 ⊢ 𝑢 ∶ 𝐶 Δ ⊢ 𝑣 ∶ 𝐵 Γ, Δ ⊢ 𝑢[𝑣/𝑦] ∶ 𝐶 𝑣 Γ 𝑢 𝐶 The substitution lemma Lemma if Γ and Δ have disjoint domains. Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 66 / 84

Graphically: . . . . . . Γ, 𝑦 ∶ 𝐵 ⊢ 𝑢 ∶ 𝐶 Δ ⊢ 𝑣 ∶ 𝐵 Γ, Δ ⊢ 𝑢[𝑣/𝑦] ∶ 𝐶 𝑣 Γ Δ 𝑢 𝐶 The substitution lemma Lemma if Γ and Δ have disjoint domains. The cut rule is admissible. Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 66 / 84

𝐶 𝐵 𝑢 𝑣 𝑦 ∶ 𝐵 Δ Γ Γ, Δ ⊢ 𝑢[𝑣/𝑦] ∶ 𝐶 Δ ⊢ 𝑣 ∶ 𝐵 Γ, 𝑦 ∶ 𝐵 ⊢ 𝑢 ∶ 𝐶 The substitution lemma Lemma if Γ and Δ have disjoint domains. The cut rule is admissible. Graphically: ax . . . . . . . . Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 66 / 84

Δ Γ, 𝑦 ∶ 𝐵 ⊢ 𝑢 ∶ 𝐶 Δ ⊢ 𝑣 ∶ 𝐵 Γ, Δ ⊢ 𝑢[𝑣/𝑦] ∶ 𝐶 𝐵 Γ 𝑢 𝐶 𝑣 The substitution lemma Lemma if Γ and Δ have disjoint domains. The cut rule is admissible. Graphically: . . . . . . Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 66 / 84

𝐶 Γ, 𝑦 ∶ 𝐵 ⊢ 𝑢 ∶ 𝐶 Δ ⊢ 𝑣 ∶ 𝐵 Γ, Δ ⊢ 𝑢[𝑣/𝑦] ∶ 𝐶 𝑣 Γ 𝑢 Δ The substitution lemma Lemma if Γ and Δ have disjoint domains. The cut rule is admissible. Graphically: . . . . . . Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 66 / 84

𝑣 𝑢 𝐶 Γ ⊸ Δ ⊸ MILL – graphical cut elimination Linear implication ax . . . . . . . . Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 67 / 84

Γ Δ 𝑢 𝑣 𝐶 MILL – graphical cut elimination Linear implication . . . . . Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 67 / 84

𝑣 𝑢 𝐷 ⊗ Γ ⊗ 𝑤 Θ Δ MILL – graphical cut elimination Tensor ax ax . . . . . . . . . . . . Emmanuel Beffara (IML, Marseille) Introduction to linear logic Torino – 27/8/2013 68 / 84