Coherent interaction graphs . A nondeterministic geometry of - PowerPoint PPT Presentation

. . . . . . . . . . . . Coherent interaction graphs . A nondeterministic geometry of interaction for MLL Nguyễn Lê Thành Dũng 1,2 Thomas Seiller 2 1 École normale supérieure de Paris 2 Laboratoire d’informatique de Paris Nord, CNRS / Université Paris 13 Linearity/TLLA 2018 (FLoC workshop) Oxford, July 8 th , 2018 Nguyễn L. T. D. & T. Seiller Coherent interaction graphs Linearity/TLLA 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 / 22

. . . . . . . . . . . . . . . . . MLL proofs as matchings (i.e. fjxed-point-free involutions) Nguyễn L. T. D. & T. Seiller Coherent interaction graphs Linearity/TLLA 2018 . . . . . . . . . . . . . . . . . . 2 / 22 . . . . . 2 proofs of A ⊗ A ⊸ A ⊗ A : ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊥ , A ⊗ A ⊢ A ⊥ , A ⊥ , A ⊗ A ⊢ A ⊥ ` A ⊥ , A ⊗ A ⊢ A ⊥ ` A ⊥ , A ⊗ A

. . . . . . . . . . . . . . . . . MLL proofs as matchings (i.e. fjxed-point-free involutions) Nguyễn L. T. D. & T. Seiller Coherent interaction graphs Linearity/TLLA 2018 . . . . . . . . . . . . . . . . . . 2 / 22 . . . . . 2 proofs of A ⊗ A ⊸ A ⊗ A : ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊥ , A ⊗ A ⊢ A ⊥ , A ⊥ , A ⊗ A ⊢ A ⊥ ` A ⊥ , A ⊗ A ⊢ A ⊥ ` A ⊥ , A ⊗ A

. . . . . . . . . . . . . . . . Cut-elimination on matchings Geometry of Interaction: predict the normal form by following paths Nguyễn L. T. D. & T. Seiller Coherent interaction graphs Linearity/TLLA 2018 . . . . . . . . . . . . . . . . . . . . . 3 / 22 . . . ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊥ , A ⊗ A ⊢ A ⊥ , A ⊥ , A ⊗ A ⊢ A ⊥ ` A ⊥ , A ⊗ A ⊢ A ⊥ ` A ⊥ , A ⊗ A (Cut) ⊢ A ⊥ ` A ⊥ , A ⊗ A

. . . . . . . . . . . . . . . . Cut-elimination on matchings Geometry of Interaction: predict the normal form by following paths Nguyễn L. T. D. & T. Seiller Coherent interaction graphs Linearity/TLLA 2018 . . . . . . . . . . . . . . . . . . . . . 3 / 22 . . . ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊥ , A ⊗ A ⊢ A ⊥ , A ⊥ , A ⊗ A ⊢ A ⊥ ` A ⊥ , A ⊗ A ⊢ A ⊥ ` A ⊥ , A ⊗ A (Cut) ⊢ A ⊥ ` A ⊥ , A ⊗ A

. . . . . . . . . . . . . . . . Cut-elimination on matchings Geometry of Interaction: predict the normal form by following paths Nguyễn L. T. D. & T. Seiller Coherent interaction graphs Linearity/TLLA 2018 . . . . . . . . . . . . . . . . . . . . . 3 / 22 . . . ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊥ , A ⊗ A ⊢ A ⊥ , A ⊥ , A ⊗ A ⊢ A ⊥ ` A ⊥ , A ⊗ A ⊢ A ⊥ ` A ⊥ , A ⊗ A (Cut) ⊢ A ⊥ ` A ⊥ , A ⊗ A

. . . . . . . . . . . . . . . . Cut-elimination on matchings Geometry of Interaction: predict the normal form by following paths Nguyễn L. T. D. & T. Seiller Coherent interaction graphs Linearity/TLLA 2018 . . . . . . . . . . . . . . . . . . . . . 3 / 22 . . . ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊥ , A ⊗ A ⊢ A ⊥ , A ⊥ , A ⊗ A ⊢ A ⊥ ` A ⊥ , A ⊗ A ⊢ A ⊥ ` A ⊥ , A ⊗ A (Cut) ⊢ A ⊥ ` A ⊥ , A ⊗ A

. . . . . . . . . . . . . . . . Cut-elimination on matchings Geometry of Interaction: predict the normal form by following paths Nguyễn L. T. D. & T. Seiller Coherent interaction graphs Linearity/TLLA 2018 . . . . . . . . . . . . . . . . . . . . . 3 / 22 . . . ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ , A ⊥ , A ⊗ A ⊢ A ⊥ , A ⊥ , A ⊗ A ⊢ A ⊥ ` A ⊥ , A ⊗ A ⊢ A ⊥ ` A ⊥ , A ⊗ A (Cut) ⊢ A ⊥ ` A ⊥ , A ⊗ A

. . . . . . . . . . . . . . . . Cut-elimination on matchings: another example Alternating paths composition of strategies in game semantics Nguyễn L. T. D. & T. Seiller Coherent interaction graphs Linearity/TLLA 2018 . . . . . . . . . . . . . . . . . . . . . 4 / 22 . . . ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ ⊗ A ⊥ , A , A ⊢ A ⊥ , A ⊢ A ⊥ ` A ⊢ A ⊗ A ⊥ , A ⊥ ` A (Cut) ⊢ A ⊥ ` A

. . . . . . . . . . . . . . . . Cut-elimination on matchings: another example Alternating paths composition of strategies in game semantics Nguyễn L. T. D. & T. Seiller Coherent interaction graphs Linearity/TLLA 2018 . . . . . . . . . . . . . . . . . . . . . 4 / 22 . . . ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ ⊗ A ⊥ , A , A ⊢ A ⊥ , A ⊢ A ⊥ ` A ⊢ A ⊗ A ⊥ , A ⊥ ` A (Cut) ⊢ A ⊥ ` A

. . . . . . . . . . . . . . . . . Cut-elimination on matchings: another example Nguyễn L. T. D. & T. Seiller Coherent interaction graphs Linearity/TLLA 2018 . . . . . . . . . . . . . . . . . . 4 / 22 . . . . . ⊢ A ⊥ , A ⊢ A ⊥ , A ⊢ A ⊥ ⊗ A ⊥ , A , A ⊢ A ⊥ , A ⊢ A ⊥ ` A ⊢ A ⊗ A ⊥ , A ⊥ ` A (Cut) ⊢ A ⊥ ` A Alternating paths ≃ composition of strategies in game semantics

. . . . . . . . . . . . . . From matchings to Interaction Graphs Matchings are both a GoI and a sort of game semantics Execution between matchings can be extended to arbitrary graphs: Defjnition paths between G and H . Proposition Nguyễn L. T. D. & T. Seiller Coherent interaction graphs Linearity/TLLA 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . 5 / 22 Let G , H be two graphs. Their execution G :: H is the graph whose vertex set is V ( G ) △ V ( H ) , and whose edges correspond to alternating � − � : { MLL proofs } → { matchings } ⊂ { graphs } then enjoys: � cut ( π, ρ ) � = � π � :: � ρ �

. . . . . . . . . . . . Interaction graphs as a denotational semantics . Proposition (Associativity / Church–Rosser) Then it suffjces to defjne types as some sets of graphs with the same vertex set to get a model of MLL, that is: Theorem morphisms given by execution. In general, a whole family of models, depending on choices of Extension to MELL: generalize from graphs to graphings (cf. Luc Pellissier’s talk) to represent exponentials Nguyễn L. T. D. & T. Seiller Coherent interaction graphs Linearity/TLLA 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 / 22 If V ( F ) ∩ V ( G ) ∩ V ( H ) = ∅ , then ( F :: G ) :: H = F :: ( G :: H ) . Interaction graphs constitute a ∗ -autonomous category with composition of parameters (e.g. monoid of weights → quantitative semantics)

. Our goal: non-determinism / additives . . . . . . . . . . Let’s extend MLL with non-deterministic sums of (sub-)proofs: . How to interpret this rule in interaction graphs? Also relevant for additives : &-intro is a non-det. superposition A solution: coherent interaction graphs Originally introduced in Seiller’s PhD for a difgerent purpose Using a coherence relation is common for additives, e.g. confmict nets (Hughes–Heijltjes), Girard’s “Transcendental syntax 2”, etc. But we won’t treat additives here: technical issues common to all GoI approaches Nguyễn L. T. D. & T. Seiller Coherent interaction graphs Linearity/TLLA 2018 . . . . . . . . . . . . . . . 7 / 22 . . . . . . . . . . . . . ⊢ Γ . . . ⊢ Γ (Sum) ⊢ Γ Formal sums of graphs → size explosion

. . . . . . . . . . . . Our goal: non-determinism / additives . Let’s extend MLL with non-deterministic sums of (sub-)proofs: How to interpret this rule in interaction graphs? Also relevant for additives : &-intro is a non-det. superposition A solution: coherent interaction graphs Using a coherence relation is common for additives, e.g. confmict nets (Hughes–Heijltjes), Girard’s “Transcendental syntax 2”, etc. GoI approaches Nguyễn L. T. D. & T. Seiller Coherent interaction graphs Linearity/TLLA 2018 . . . . . . . . . . . . . . . 7 / 22 . . . . . . . . . . . . ⊢ Γ . . . ⊢ Γ (Sum) ⊢ Γ Formal sums of graphs → size explosion ▶ Originally introduced in Seiller’s PhD for a difgerent purpose ▶ But we won’t treat additives here: technical issues common to all

. . . . . . . . . . . . . . . Coherent graphs Defjnition Defjnition G Nguyễn L. T. D. & T. Seiller Coherent interaction graphs Linearity/TLLA 2018 . . . . . . . . . . . . . . 8 / 22 . . . . . . . . . . . A coherent graph is a graph G equipped with a coherence relation ¨ G on its edge set E ( G ) . i.e. ( E ( G ) , ¨ G ) is a coherent space (which we’ll identify with E ( G ) ) If V ( G ) = V ( H ) = V , then the incoherent sum of G and H is defjned as ⌣ + H = ( V , E ( G ) ⊕ E ( H )) . ( ⊕ : disjoint union of coherent spaces) ⌣ + interprets the Sum rule Think of a coherent graph ( V , E ) as the formal sum ∑ ( V , C ) ( C clique ) C ⊂ E

. . . . . . . . . . . . . . . . Execution of coherent graphs: example Incoherence: don’t take this path Nguyễn L. T. D. & T. Seiller Coherent interaction graphs Linearity/TLLA 2018 . . . . . . . . . . . . . . . . . . . . . . . . 9 / 22 Here red ¨ black, blue ¨ black, red ⌣ blue