Preuves de logique linéaire sur machine, ENS-Lyon, Dec. 18, 2018

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1 Université de Lorraine, LORIA, CNRS, Nancy, France Preuves de logique linéaire sur machine, ENS-Lyon, Dec. 18, 2018

2 Introduction Linear logic introduced by Girard both classical and intuitionistic separate multiplicatives (,, ɛ) from additives (&,,, ) ILL via its sequent calculus multiplicatives split the context additives share the context formulas cannot be freely duplicated or discarded no weakening (C) or contraction rule (W) exponentials! A re-introduce controlled C&W generally undecidable Mechanized cut-elimination via phase semantics A relational phase semantics (no monoid) via Okada s lemma, both in Prop and Type

3 ILL sequent calculus (multiplicatives) A, B, Γ C A B, Γ C L Γ A B, C A B, Γ, C L Γ A ɛ, Γ A ɛ L Γ A B Γ, A B R A, Γ B Γ A B R ɛ ɛ R

4 ILL sequent calculus (additives) A, Γ C A & B, Γ C &1 L Γ A Γ B Γ A & B & R B, Γ C A & B, Γ C &2 L A, Γ C B, Γ C A B, Γ C Γ A Γ A B 1 R Γ B Γ A B 2 R Γ, A L Γ R L

5 ILL (exponentials and structural) A, Γ, B! A, Γ B! L! Γ A! Γ! A! R A A id Γ A A, B Γ, B Γ, B! A, Γ B W! A,! A, Γ B! A, Γ B cut C Γ A A Γ p

6 Relational (overview) It is an algebraic semantics Comparable to Lindenbaum construction Interpret formula by themselves (completeness) does not require cut (cut-admissibility) Usual phase semantics based on commutative monoidal structure (contexts) a stable closure Relational phase semantics a composition relation (no axiom) closure axioms absord the monoidal structure

7 Relational (details) Closure cl : (M Prop) (M Prop) with predicates X, Y : M Prop X cl X X Y cl X cl Y cl(cl X ) cl X Composition : M M M Prop, e : M extended to predicates M Prop by X Y := {x y x X, y Y} X Y := {z z X Y} x cl(e x) (neutral1) e x cl{x} (neutral2) x y cl(y x) (commutativity) x (y z) cl((x y) z) (associativity) Stability: (cl X ) Y cl(x Y)

8 Rel. Phase Sem. (exponential, soundness) Let J := {x x cl{e} x cl(x x)} Choose K J such that e cl K and K K K Semantics for variables: : Var M Prop which is closed: cl V V extended to formulas A B := cl( A B ) A B := A B A & B := A B A B := cl( A B ) := cl := M ɛ := cl{e}! A := cl(k A ) Γ 1,..., Γ n := Γ 1 Γ n Soundness: if Γ A has a proof then Γ A

9 Relational Phase Sem. (cut-admissibility) A syntactic model M := list Form for Γ,, Θ M Θ Γ Γ, p Θ K := {! Γ Γ M} ( K and K K K) contextual closure cl : (M Prop) (M Prop) cl X Γ A, X, Γ = A, Γ = A where = : list Form Form Prop = is a deduction relation such as provability or cut-free provability permutations: Γ p Γ = A = A

10 Rules as algebraic equations Define A := {Γ Γ = A}, then cl( A) A A B (A B) iff Γ = A = B for any Γ, Γ, = A B A B cl{ A, B } iff A, B, Γ = C A B, Γ = C for any Γ, C K J iff = closed under W and C.

11 Okada s lemma For = defined as cf closed under cut-free ILL A, A A A and Γ, Γ Γ By induction on A, then by induction on Γ By soundness, from a (cut using) proof of Γ A we deduce Γ Γ A A hence Γ cf A hence Γ A is cut-free provable Hence a semantic proof of cut-admissibility

12 Extensions, other logics, cut-elimination Extensions to other logics:, contextual closure very generic of course fragments of ILL, but also CLL ILL with modality, Linear time ILL Bunched Implications (BI) Relevance logic, prop. Intuitionistic Logic Display calculi (context = consecutions)? Computational content Prop Type gives cut-elimination algo. can be extracted (you do not want to read it...) Around 1300 loc for specs and 1000 loc for proofs 2/5 of which are libraries (lists, permutations...)