Geometry of Interaction

Transcription

1 Linear logic, Ludics, Implicit Complexity, Operator Algebras Geometry of Interaction Laurent Regnier Institut de Mathématiques de Luminy

2 Disclaimer Syntax/Semantics Syntax = nite (recursive) sets Semantics = embedding of syntax into abstract (nonrecursive) framework in model theory: formula sets, or elements of a (innite) boolean algebra in denotational semantics: proofs morphism between domains etc.

3 1987: birth of the GoI The three levels of logic The level of formula (truth) model theory The level of proofs (provability) denotational semantics The level of interaction (cut elimination) geometry of interaction GoI programme = build a semantics for cut elimination.

4 GoI in litterature Instances of GoI: a game semantics interpretation (Abramsky-Jagadeesan-Malacaria game model) traced monoidal categories the semantics of sharing reduction (Abadi-Gonthier-Lévy context semantics) an abstract machine (Danos-Regnier): a regular paths computing device a reversible automaton Krivine's machine an abstract version of proof-nets experiments a precursor of ludics: GoI admits localisation (that's kind of the problem)

5 The basic schema of GoI Formula A space S(A) Proof Π : A operator π acting on S(A) (notation: π : A) Cut given π 1 : A B and π 2 : B C, dene Ex(π 1, π 2 ) : A C Note: π 1, π 2 are not functions, Ex(π 1, π 2 ) is not composition Simplied (but not so much) version: Given π 1 : A and π 2 : A, dene Ex(π 1, π 2 ) : π 1 π 2 if Ex(π 1, π 2 ) (see ludics...)

6 Basic schema: the a priori typed variant Categorical avor, implicit use of duality (eg game semantics) S(A) is built by induction on A: S(A B) = S(A B) = S(A) + S(B) π : A is an operator on S(A) (satisfying... ) Possibly get a denability theorem: π : A if Π actually is a proof of A full abstraction of AJM game model What about pure lambda-calculus (or system F )? Remark ( ) π : A B = π A,A π A,B π B,A π B,B

7 Basic schema: the a posteriori typed variant Girard's symmetric realisability construction (at work in: LL strong normalization, phase semantics, ludics, quantum coherent spaces... ) Fix a given (universal) space S (eg S = l 2 ) S(A) = S for all A: all operators act on S Fix a duality, eg π π i ππ is nilpotent T (A) is a set of operators dened by induction on A: T (A ) = T (A) T (A B) = {π 1 + π 2, π 1 T (A), π 2 T (B)} Thus T (A B) = (T (A) T (B )) Adequation lemma: if Π proof of A then π T (A) Remark What is π 1 + π 2?

8 The multiplicative case: MLL Follow the a priori typed scheme: operators = partial permutations on nite sets Formula A S(A) = {occurrences of atoms in A} S(A B) = S(A B) = S(A) + S(B) Proof axiom links permutation on S(A) Cut identify atoms in A to their dual in A ;

9 The multiplicative case (continued) π 1 : A B, π 2 : B C, π = π 1 + π 2 : (A B) (B C) σ : (A B) (B C) partial permutation on S(A ) + S(B) + S(B ) + S(C) exchanging dual (occurrences of) atoms in B and B Remark π and σ are partial symetries: π 2 and σ 2 are projectors

10 Execution formula Matrix representation: ( ) π1 0 π = π 1 + π 2 = = 0 π 2 π A,A 1 π A,B π B,A 1 π B,B π B,B 0 0 π C,B σ = π B,C 2 2 π C,C 2 The execution formula Ex(π 1, π 2 ) = (1 σ 2 )π k 0(σπ) k (1 σ 2 )

11 GoI and experiments Hint: let A be a MLL formula. A point in A may be viewed as a set (ie a vector over F 2 ) of localisations(moves) in S(A) For example: a = (a 1, a 2 ) A 1 A 2 = A 1 A 2 α S(A 1 A 2 ) = S(A 1 ) + S(A 2 ) α = (i, α i ), i = 1 or 2, α i S(A i ) (a 1, a 2 ) = a 1 a 2 = {(1, α 1 ), α 1 a 1 )} {(2, α 2 ), α 2 a 2 )} Theorem If π : A is the GoI of Π : A then a Π i π(a) = a

12 The multiplicative case in the a posteriori typed scheme Fixed space = N (or l 2 ) p, q : N N (iso N + N N), eg p(k) = 2k and q(k) = 2k + 1 π 1 + π 2 = pπ 1 p + qπ 2 q (π 1 + π 2 )(2k) = 2π 1 (k) and (π 1 + π 2 )(2k + 1) = 2π 2 (k) + 1 NB Operators act now on innite space

13 Relating the two schemes A priori typed ( ) π π : A B = A,A π A,B π B,A π B,B A posteriori typed pπ A,A p +qπ A,B p + qπ B,A p qπ B,B q

14 Adding exponentials: the a priori typed scheme S(!A) = N S(A), adding a copy index From π : A B construct!π :!A!B:!π!A,!A (k, a) = (k, π A,A (a))!π!a,!b (k, a) = (k, π A,B (a))!π!b,!a (k, b) = (k, π B,A (b))!π!b,!b (k, b) = (k, π B,B (b))

15 Exponentials continued d :!A A : d d!a,a : (0, a) a : d A,!A c :!A!A 1!A 2 : c c!a,!a!a : (p(k), a) (k, a) 1 : c!a!a,!a (q(k), a) (k, a) 2 : c!a!a,!a dig :!A!!A : dig dig!a,!!a : (τ(k, k ), a) (k, (k, a)) : dig!!a,!a where τ : N N N

16 Exponentials: the a posteriori typed scheme Fixed space = N (or l 2 ) Use.,. : N N N for exponentials: (k, a) k, a Dene π π if ππ nilpotent (ie computation is nite) T (!A) = {!π, π T (A)} Theorem if Π : A then π T (A) This is a strong normalisation theorem. In the setting of Hilbert spaces one can alternatively dene duality by means of weak nilpotency, allowing to account for non terminating terms eg xed points.

17 The GoI equationnal theory Monoid with 0 generated by p, q (multiplicatives), d (dereliction), r, s (contraction), t (digging) Involution: 0 = 0, 1 = 1, (uv) = v u Morphism:!(0) = 0,!(1) = 1,!(u)!(v) =!(uv),!(u) =!(u ) Annihilation equations: x y = δ xy Commutation equations:!(u)d = du!(u)x = x!(u) for x = r, s!(u)t = t!(!(u)) (x, y generators)

18 The theorem AB Orientate equations rewriting system Normal forms = 0 or AB Inverse semigroup structure

19 The path interpretation of GoI To a lambda-term M we associate a GoI weighted graph G n (M): Variable case: v n! n (d) n Abstraction case:! n (p) λ n! n (q)! n (p) λ n! n (q) G n (M) G n (M) G n (λxm):

20 The path interpretation of GoI Application case: G n (MN): c n! n n 1! n (p) G n (M) G n+1(n)! n (t)! n (t)! n (r)! n (s) n n n

21 The path interpretation of GoI Note: γ path in G n (M), w(γ) its weight is a GoI operator Denition Execution paths = invariant of beta-reduction = virtual redexes Regular path = non null weight path (w(γ) 0) Theorem γ is an execution path i γ is regular Theorem If M is a term (thus an MELL proof) then Ex(M) = γ R w(γ) where R = {regular paths G 0 (M)}

22 Interaction Abstract Machine Term (proof-net) weighted graph: token = element of S (the space in the a posteriori typed scheme) weighted edge = transition Term (proof-net) = automaton: the IAM Remark All transitions are reversible and have disjoint domains and codomains the automaton is bideterministic

23 IAM In order to make the abstract machine explicit, redene the space S of tokens: token (state) = (B, S) (really B.S): B = box stack of exponential signatures S = balanced stack of exponential signatures + multiplicative constants P and Q exponential signature = binary tree with leaves in {, R, S} Transitions = partial transformations on (B, S) Theorem KAM IAM

24 Conclusion A lot more to say GoI for additives Pointixion: relating GoI/AJM games with HO Coherence problems: Π Π 0 Ex(π) = π 0 GoI for other systems, eg interaction nets, π-calculus, dierential nets...