The Separation Theorem for Differential Interaction Nets Damiano - PowerPoint PPT Presentation

The Separation Theorem for Differential Interaction Nets Damiano Mazza ∗ Laboratoire d’Informatique de Paris Nord, Universit´ e Paris 13 joint work with Michele Pagani † Dipartimento di Filosofia, Universit` a Roma Tre LPAR, Yerevan, 19 October 2007 ∗ Post-doc ANR project “NOCoST”. † Post-doc research project “Ricerche sulla geometria della logica”.

Separation in the λ -calculus Theorem 1. [B¨ ohm 1968] Let S, T be two closed βη -normal forms. Then, there exist U 1 , . . . , U n such that → ∗ SU 1 . . . U n λxy.x → ∗ TU 1 . . . U n λxy.y • Syntactical meaning: normal λ -terms never contain useless structure. • Semantical meaning: no consistent λ -theory can equate two βη -normal forms. 1

Separation in the λ -calculus Theorem 1. [B¨ ohm 1968] Let S, T be two closed βη -normal forms. Then, there exists F such that → ∗ FS λxy.x → ∗ FT λxy.y ( F = λx.xU 1 . . . U n , where the U i are the terms from the previous slide). • Syntactical meaning: normal λ -terms never contain useless structure. • Semantical meaning: no consistent λ -theory can equate two βη -normal forms. 1

Separation and faithfulness with simple types • Modulo a few technical adjustments, separation holds also in the simply typed λ -calculus: Theorem 2. [Statman 1980-Joly 2000] Let S, T be two simply- typed closed terms of type A 1 , . . . , A n ⇒ X , where types are built on a single atom X . Then, there exist a type B and terms U i of type A i [ B/X ] , with 1 ≤ i ≤ n , such that → ∗ SU 1 . . . U n λxy.x → ∗ TU 1 . . . U n λxy.y • Syntactical separation and denotational faithfulness are strongly related: separation always implies faithfulness, and, in certain cases, from faithfulness one can infer separation. 2

Separation and faithfulness in logic • Thanks to the Curry-Howard isomorphism, Statman and Joly’s results can automatically be restated in terms of natural deduction for propositional minimal logic: type = formula normal term = “cut-free proof with atomic axioms” application = modus ponens β -reduction = cut-elimination • However, nothing can be said about classical logical systems (i.e., with involutive negation): no “good” syntax, no denotational semantics. . . unless we consider Girard’s linear logic. 3

Proof-nets • The most natural proof-system for linear logic is that of proof-nets , which are particular graph-like structures. Nodes are logical rules, and edges are occurrences of formulas involved in rules: axiom A ⊥ ? A A ⊥ ? B ⊥ A � ? ? B ⊥ � A ? A ⊥ cut � A ⊥ A ? A ⊥ � (? B ⊥ � A ) • Unlike sequent calculus, composition is associative ( parallelism ): π 1 π 2 π 3 A ⊥ B ⊥ C ⊥ B C D 4

Proof-nets • Cut-elimination becomes graph-rewriting, with rules of the form ν is a net depending . . . . . . . . . . . . on the rule itself. → ν May be non-local. • In the multiplicative-exponential fragment of linear logic ( MELL , which is enough to represent for example System F , i.e., the polymorphic λ -calculus), cut-elimination is confluent and strongly normalizing. • Several interesting denotational semantics of proof-nets exist: coherence spaces , relational semantics , games semantics . . . 5

Separation and faithfulness in linear logic • Unfortunately, faithfulness of coherence spaces fails in MELL proof-nets, and with it syntactical separation (Tortora de Falco, 2000): π 1 vs. π 2 = dotted vs. dashed lines. X ⊥ X ⊥ For any interpretation of the atom X X ? ? ? ? X as a coherence space, we have ? ? ? X ⊥ ? X ⊥ ? X ⊥ ? X � π 1 � = � π 2 � , and the two proof-nets ? X ? X ? ? ? ? are interactively indistinguishable. ?? X ⊥ ?? X ⊥ ?? X ?? X However, � π 1 � � = � π 2 � ? ? in relational semantics. . . ??? X ⊥ ??? X Conjecture 1. Relational semantics is faithful for MELL proof-nets. 6

Differential Interaction Nets (DIN) • DIN are a graphical formalism similar to MELL proof-nets, recently introduced by Ehrhard and Regnier. They sprang from Ehrhard’s finiteness spaces , a denotational semantics for linear logic based on certain topological vector spaces, with a notion of derivative operator. MELL proof-nets can be embedded in DIN thanks to a sort of Taylor expansion. • Simple nets are built out of the following typed cells : A B A . . . A 1 ⊗ ! 1 A ⊗ B ! A A B A . . . A � ⊥ ? A � B ⊥ ? A 7

Differential Interaction Nets • Two simple nets are σ -equivalent iff they can be rewritten one into the other by applying any number of times the following equation (where e stands for either ! or ? , and σ is any permutation): . . . . . . σ . . . ≡ e e • A differential interaction net is a denumerable set of σ -equivalence classes of simple nets, all having the same conclusions. The empty net is denoted by 0 , and the net containing only the empty simple net is denoted by 1 . 8

Differential Interaction Nets • Cut-elimination is defined as follows: 1 ⊥ → 1 ⊥ A ⊥ B ⊥ A ⊥ B ⊥ A B A B � ⊗ → A ⊥ � B ⊥ A ⊗ B m n z }| { z }| { 8 A ⊥ A ⊥ ( ) A ⊥ A ⊥ . . . . . . A . . . A . . . > A A > if m = n S > > ! ? < σ ∈ S n σ → > ? A ⊥ > ! A > if m � = n > : 0 • There are also rules for η -expansion. The rewriting relation → ∗ formed by β -reduction plus η -expansion can be shown to be confluent. 9

Interaction between dual nets • If α, β are two simple nets of resp. conclusions A 1 , . . . , A n and A ⊥ 1 , . . . , A ⊥ n , we denote by � α | β � the net α β . . . . . . A ⊥ A ⊥ A 1 An n 1 • If µ, ν are two nets with dual conclusions Γ , Γ ⊥ , we set � µ | ν � = {� α | β � ; α ∈ µ, β ∈ ν } . 10

The Separation Theorem • As in the simply typed λ -calculus, we consider formulas built on a single atomic pair X, X ⊥ , and we realize separation up to “Statman’s typical ambiguity”: Theorem 3. [Separation] For each pair of different normal nets µ , µ ′ with same conclusions Γ , there is a simple net ν with conclusions Γ[?1 /X ] ⊥ such that → ∗ � µ [?1 /X ] | ν � 1 � µ ′ [?1 /X ] | ν � → ∗ 0 or vice versa. Corollary 1. A non-trivial denotational semantics of DIN cannot identify two different normal nets. 11

Applying the separation theorem to MELL • As already said, MELL proof-nets can be embedded in DIN via what we call the Taylor-Ehrhard expansion . • All known examples of indistinguishable pairs of proof-nets (such as the one given above) are easily separated once expanded in DIN. Indeed, our work points out that separation fails in linear logic because of the fundamental asymmetry of its exponential connectives (the modalities ! and ? ), which are instead perfectly symmetrical in DIN. • Our work also gives the following reformulation of Tortora de Falco’s conjecture about the faithfulness of relational semantics over MELL proof-nets: Conjecture 2. [Faithfulness of Rel ] Different proof-nets MELL have different Taylor-Ehrhard expansions. 12