Introduction MILL MLL MALL Exponentials Conclusion Introduction to Linear Logic Shane Steinert-Threlkeld November 29, 2011
Introduction MILL MLL MALL Exponentials Conclusion What is Linear Logic? Structural Motivations Introduced by Jean-Yves Girard in 1987 [Gir87]. Linear logic is: Sequent calculus without weakening and contraction. As (or more) constructive than intuitionistic logic, while maintaining desirable features of classical logic. Finding more and more applications in theoretical computer science.
Introduction MILL MLL MALL Exponentials Conclusion What is Linear Logic? High-Level Motivations Linear logic is: a logic of actions [Gir89].
Introduction MILL MLL MALL Exponentials Conclusion What is Linear Logic? High-Level Motivations Linear logic is: a logic of actions [Gir89]. In all traditional logics, consider modus ponens: A → B A B
Introduction MILL MLL MALL Exponentials Conclusion What is Linear Logic? High-Level Motivations Linear logic is: a logic of actions [Gir89]. In all traditional logics, consider modus ponens: A → B A B In the conclusion, A still holds. This is perfectly well-suited to mathematics, which deals with stable truths.
Introduction MILL MLL MALL Exponentials Conclusion What is Linear Logic? High-Level Motivations Linear logic is: a logic of actions [Gir89]. In all traditional logics, consider modus ponens: A → B A B In the conclusion, A still holds. This is perfectly well-suited to mathematics, which deals with stable truths. “But wrong in real life, since real implication is causal .” For beautiful connections with physics, see Baez and Stay 2011 “Physics, Topology, Logic, Computation: a Rosetta Stone” [BS11].
Introduction MILL MLL MALL Exponentials Conclusion What is Linear Logic? High-Level Motivations In linear logic, we do not have A ⊸ A ⊗ A By eliminating weakening and contraction, we eliminate free duplication and elimination of formulas. (We will develop tools to restore these in a controlled manner.)
Introduction MILL MLL MALL Exponentials Conclusion What is Linear Logic? High-Level Motivations In linear logic, we do not have A ⊸ A ⊗ A By eliminating weakening and contraction, we eliminate free duplication and elimination of formulas. (We will develop tools to restore these in a controlled manner.) This motivates thinking of formulas in linear logic as resources as opposed to eternally true/false propositions. For instance [Gir89, p. 74]: state of a Turing machine state of a chess game chemical solution before/after reaction
Introduction MILL MLL MALL Exponentials Conclusion Additive vs. Multiplicative Connectives Two Sequent Calculi Consider a standard sequent calculus. Call these “M”-rules: Γ ′ , B ⊢ ∆ ′ Γ , A , B ⊢ ∆ Γ , A ⊢ ∆ (LM ∧ ) Γ , A ∧ B ⊢ ∆ (LM ∨ ) Γ , Γ ′ , A ∨ B ⊢ ∆ , ∆ ′ Γ ′ ⊢ ∆ ′ , B Γ ⊢ ∆ , A Γ ⊢ A , B , ∆ (RM ∨ ) Γ ⊢ A ∨ B , ∆ (RM ∧ ) Γ , Γ ′ ⊢ ∆ , ∆ ′ , A ∧ B Table: “M”-rules for sequent calculus.
Introduction MILL MLL MALL Exponentials Conclusion Additive vs. Multiplicative Connectives Two Sequent Calculi Consider a standard sequent calculus. Call these “A”-rules: Γ , A ⊢ ∆ Γ , A ⊢ ∆ Γ , B ⊢ ∆ (LA ∧ -1) Γ , A ∧ B ⊢ ∆ (LA ∨ ) Γ , A ∨ B ⊢ ∆ Γ , B ⊢ ∆ Γ ⊢ ∆ , A (LA ∧ -2) Γ , A ∧ B ⊢ ∆ (RA ∨ -1) Γ ⊢ ∆ , A ∨ B Γ ⊢ ∆ , A Γ ⊢ ∆ , B Γ ⊢ ∆ , B (RA ∨ -2) Γ ⊢ ∆ , A ∨ B (RA ∧ ) Γ ⊢ ∆ , A ∧ B Table: “A”-rules for sequent calculus.
Introduction MILL MLL MALL Exponentials Conclusion Additive vs. Multiplicative Connectives Interderivability of “M” and “A” Rules In both intuitionistic and classical logic, the two formulations are equivalent. Here we derive the “M” rules for ∧ using the “A” rules: Γ , A , B ⊢ ∆ Γ ′ ⊢ ∆ ′ , B Γ ⊢ ∆ , A (LA ∧ -1) Γ , A ∧ B , B ⊢ ∆ Γ , Γ ′ ⊢ ∆ , ∆ ′ , A Γ , Γ ′ ⊢ ∆ , ∆ ′ , B (LA ∧ -2) Γ , A ∧ B , A ∧ B ⊢ ∆ (RA ∧ ) Γ , Γ ′ ⊢ ∆ , ∆ ′ , A ∧ B Γ , A ∧ B ⊢ ∆ Table: “M” rules derived in “A” system.
Introduction MILL MLL MALL Exponentials Conclusion Additive vs. Multiplicative Connectives Interderivability of “M” and “A” Rules In both intuitionistic and classical logic, the two formulations are equivalent. Here we derive the “A” rules for ∧ using the “M” rules: Γ ⊢ ∆ , A Γ ⊢ ∆ , B Γ , A ⊢ ∆ (RM ∧ ) Γ , Γ ⊢ ∆ , ∆ , A ∧ B Γ , A , B ⊢ ∆ (LM ∧ ) Γ , A ∧ B ⊢ ∆ Γ ⊢ ∆ , A ∧ B Table: “A” rules derived in “M” system. Exercise. Carry out the same procedure for the ∨ rules.
Introduction MILL MLL MALL Exponentials Conclusion Additive vs. Multiplicative Connectives Interderivability of “M” and “A” Rules Notice anything?
Introduction MILL MLL MALL Exponentials Conclusion Additive vs. Multiplicative Connectives Interderivability of “M” and “A” Rules Notice anything? Every one of those proofs used contraction and/or weakening. In linear logic, we will have both multiplicative and additive connectives corresponding to these two sets of rules which are no longer equivalent.
Introduction MILL MLL MALL Exponentials Conclusion The Plan The Full Language of (Propositional) Classical Linear Logic Propositional variables: A , B , C , · · · , P , Q , R , · · · Constants: Multiplicative: 1 , ⊥ (units, resp. of ⊗ , ` ) Additive: ⊤ , 0 (units, resp. of & , ⊕ ) Connectives: Multiplicative: ⊗ , ` , ⊸ Additive: & , ⊕ Exponential modalities: ! , ? Linear negation: ( · ) ⊥
Introduction MILL MLL MALL Exponentials Conclusion The Plan Outline I Introduction 1 What is Linear Logic? Additive vs. Multiplicative Connectives The Plan MILL 2 Syntax and Sequent Calculus Natural Deduction and Term Calculus Categorical Semantics MLL 3 Sequent Calculus Proof Nets MALL 4 Additives Proof Nets Phase Semantics
Introduction MILL MLL MALL Exponentials Conclusion The Plan Outline II Exponentials 5 Exponential Modalities Translation of Intuitionistic Logic Extension of Phase Semantics Conclusion 6 Other Topics References
Introduction MILL MLL MALL Exponentials Conclusion Syntax and Sequent Calculus Sequent Calculus We now consider the ( ⊗ , ⊸ , 1 )-fragment, multiplicative intuitionistic linear logic . Γ , P , Q , ∆ ⊢ C (Ax) P ⊢ P (Ex) Γ , Q , P , ∆ ⊢ C Γ ⊢ P P , ∆ ⊢ Q (Cut) Γ , ∆ ⊢ Q ( 1 -R) ⊢ 1 Γ ⊢ P ( 1 -L) Γ , 1 ⊢ P Γ ⊢ P ∆ ⊢ Q Γ , P , Q ⊢ R ( ⊗ -L) Γ , P ⊗ Q ⊢ R ( ⊗ -R) Γ , ∆ ⊢ P ⊗ Q Γ , P ⊢ Q Γ ⊢ P Q , ∆ ⊢ R ( ⊸ -R) Γ ⊢ P ⊸ Q ( ⊸ -L) Γ , P ⊸ Q , ∆ ⊢ R Table: Sequent Calculus for MILL
Introduction MILL MLL MALL Exponentials Conclusion Syntax and Sequent Calculus Consequences Theorem MILL satisfies cut-elimination. Proof. Requires defining new commuting conversions, but otherwise is similar to regular intuitionistic case. See [BBPH93] for a proof (also with !).
Introduction MILL MLL MALL Exponentials Conclusion Natural Deduction and Term Calculus Natural Deduction of MILL P ⊢ P Γ , P ⊢ Q Γ ⊢ P ⊸ Q ∆ ⊢ Q ( ⊸ I ) Γ ⊢ P ⊸ Q ( ⊸ E ) Γ , ∆ ⊢ Q Γ ⊢ P ∆ ⊢ I ( I E ) ⊢ I Γ , ∆ ⊢ P Γ ⊢ P ⊗ Q ∆ , P , Q ⊢ R Γ ⊢ P ∆ ⊢ Q ( ⊗ I ) ( ⊗ E ) Γ , ∆ ⊢ P ⊗ Q Γ , ∆ ⊢ R Table: Natural Deduction (Sequent Style) for MILL
Introduction MILL MLL MALL Exponentials Conclusion Natural Deduction and Term Calculus Term Assignment x : P ⊢ x : P Γ , x : P ⊢ f : Q Γ ⊢ f : P ⊸ Q ∆ ⊢ g : Q ( ⊸ I ) Γ ⊢ λ x . f : P ⊸ Q ( ⊸ E ) Γ , ∆ ⊢ fg : Q Γ ⊢ f : P ∆ ⊢ g : I ( I E ) Γ , ∆ ⊢ let g be ∗ in f : P ⊢ ∗ : I Γ ⊢ f : P ∆ ⊢ g : Q Γ ⊢ f : P ⊗ Q ∆ , x : P , y : Q ⊢ g : R ( ⊗ I ) ( ⊗ E ) Γ , ∆ ⊢ f ⊗ g : P ⊗ Q Γ , ∆ ⊢ let f be x ⊗ y in g : R Table: Term Assignment for MILL Natural Deduction
Introduction MILL MLL MALL Exponentials Conclusion Natural Deduction and Term Calculus Good Features of This Formulation Substitution property Subject reduction theorem (with commuting conversions added to β ) Normalization and uniquenesss of normal form
Introduction MILL MLL MALL Exponentials Conclusion Natural Deduction and Term Calculus Bad Features of This Formulation No subformula property (because of ⊗ -E) Unnecessarily extends term calculus (with let construction) [Min98] proves a uniqueness of normal form theorem for the {⊗ , & , ⊸ } fragment using an extended notion of substitution.
Introduction MILL MLL MALL Exponentials Conclusion Categorical Semantics Closed Symmetric Monoidal Categories In the same way that intuitionistic propositional logic is the logic of Cartesian Closed Categories [Min00, Gol06, TS00], MILL is the logic of closed symmetric monoidal categories .
Introduction MILL MLL MALL Exponentials Conclusion Categorical Semantics Closed Symmetric Monoidal Categories In the same way that intuitionistic propositional logic is the logic of Cartesian Closed Categories [Min00, Gol06, TS00], MILL is the logic of closed symmetric monoidal categories . I will fly through the relevant definitions; feel free to pursue them when more time is available.
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