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¨

• hm 1968]

Let S, T be two closed βη-normal forms. Then, there exist U1, . . . , Un such that SU1 . . . Un →∗ λxy.x TU1 . . . Un →∗ λxy.y

• Syntactical meaning: normal λ-terms never contain useless structure.
• Semantical meaning: no consistent λ-theory can equate two βη-normal

forms.

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Separation in the λ-calculus

Theorem 1. [B¨

• hm 1968]

Let S, T be two closed βη-normal forms. Then, there exists F such that FS →∗ λxy.x FT →∗ λxy.y (F = λx.xU1 . . . Un, where the Ui 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.

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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 A1, . . . , An ⇒ X, where types are built

• n a single atom X. Then, there exist a type B and terms Ui of type

Ai[B/X], with 1 ≤ i ≤ n, such that SU1 . . . Un →∗ λxy.x TU1 . . . Un →∗ 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.

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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:

A⊥ A A⊥ A

axiom cut

• ?

? A A⊥ ?A⊥ ?A⊥ (?B⊥ A) ?B⊥ A ?B⊥

• Unlike sequent calculus, composition is associative (parallelism):

π1 π2 π3 A⊥ B B⊥ C C⊥ D 4

Proof-nets

• Cut-elimination becomes graph-rewriting, with rules of the form

ν is a net depending

• n the rule itself.

May be non-local.

. . . . . . → . . . . . . ν

• 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. . .

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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): However, π1 = π2 in relational semantics. . . For any interpretation of the atom X as a coherence space, we have π1 = π2, and the two proof-nets are interactively indistinguishable.

??X⊥ ? ? ? ? ? ???X⊥ ??X⊥ ?X⊥ ?X⊥ X⊥ X⊥ ? ??X ? ? ? ? ? ???X ??X ?X ?X X X ? ?X ?X⊥

π1 vs. π2 = dotted vs. dashed lines. Conjecture 1. Relational semantics is faithful for MELL proof-nets.

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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:

1 1 ⊗ A B A ⊗ B !A A A . . . ! ⊥ ⊥

• A

B A B ?A A A . . . ? 7

Differential Interaction Nets

• Two simple nets are σ-equivalent iff they can be rewritten one into the
• ther by applying any number of times the following equation (where e

stands for either ! or ?, and σ is any permutation):

e e σ . . . . . . . . . ≡

• A differential interaction net is a denumerable set of σ-equivalence classes
• f 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.

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Differential Interaction Nets

• Cut-elimination is defined as follows:

⊥ 1 1 ⊥ A A ! ? . . . . . . !A m z }| { n z }| { ( σ A A . . . . . . → → → 8 > > > > < > > > > : S σ∈Sn A⊥ A⊥ A⊥ A⊥

if m = n if m = n

?A⊥ ⊗

• A

B A ⊗ B A⊥ B⊥ B⊥ A⊥ A B A⊥ B⊥ )

• There are also rules for η-expansion. The rewriting relation →∗ formed

by β-reduction plus η-expansion can be shown to be confluent.

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Interaction between dual nets

• If α, β are two simple nets of resp. conclusions A1, . . . , An and

A⊥

1 , . . . , A⊥ n , we denote by α | β the net

α β A1 An A⊥ 1 A⊥ n . . . . . .

• If µ, ν are two nets with dual conclusions Γ, Γ⊥, we set

µ | ν = {α | β ; α ∈ µ, β ∈ ν}.

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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 →∗

• r vice versa.

Corollary 1. A non-trivial denotational semantics of DIN cannot identify two different normal nets.

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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
• ne given above) are easily separated once expanded in DIN. Indeed,
• ur 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 MELL proof-nets have different Taylor-Ehrhard expansions.

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