Constructive approach to relevant and affine term calculi

Transcription

1 Constructive approach to relevant and affine term calculi Jelena Ivetić, University of Novi Sad, Serbia Silvia Ghilezan,University of Novi Sad, Serbia Pierre Lescanne, University of Lyon, France Silvia Likavec, University of Torino, Italy CM:FP, Niš, June 2013.

2 Computational interpretations of logics The Curry-Howard correspondence captures the computational content of logic, by observing a multilevel connection between formal systems (logic) and some term calculi (computation). the λ-calculus intuitionistic natural deduction the λµ-calculus (Parigot, 1992) classical natural deduction the λµ µ-calculus (Herbelin, Curien, 2000) classical sequent calculus the λ Gtz -calculus (Espírito Santo, 2006) intuitionistic sequent calculus In all these formal systems, structural rules are implicit.

3 ND with explicit structural rules A A (Ax) Γ, A B Γ A B ( Γ A A B intro) ( elim ) Γ, B Γ B Γ, A B (Weak) Γ, A, A B Γ, A B (Cont) Implicit SR contexts are sets Axiom: Γ, A A additive (context-sharing) rules Explicit SR contexts are multisets Axiom: A A multiplicative (context-splitting) rules

4 goal? To propose term calculi (CH) corresponding to the formal systems with exp. str. rules. motivation? theoretical - to obtain an insight into a part of the computation process that is usually hidden; practical - controlling enables optimization. solution: λ R (resource control lambda calculus) ND with exp.str.rules λ Gtz R (resource control lambda Gentzen calculus) LJ with exp.str.rules (will not be presented here) why "resource control"? structural rules in logic duplication and erasure of variables in terms.

5 Syntax of the λ R -calculus pre-terms of the λ R -calculus: Pre-terms f ::= x λx.f ff x f x < x 1 f λx.f is an abstraction, ff is an application, x f is a weakening and x < x 1 f is a contraction. λ R -terms are only those pre-terms that satisfy the following two conditions: every free variable occurs exactly once in a term; every binder binds exactly one occurrence of a free variable. Example: pre-terms are not λ R -terms. λx.y, λx.xx, x < y z (xy)

6 λ R -terms - formally The set Λ R is defined by the following inference rules: x Λ R f Λ R x Fv(f ) λx.f Λ R f Λ R g Λ R Fv(f ) Fv(g) = fg Λ R f Λ R x / Fv(f ) f Λ R x 1, x 1, Fv(f ) x / Fv(f ) \ {x 1, } x f Λ R x < x 1 f Λ R

7 Although some λ-terms are not λ R -terms (like λx.y, xx,...) every λ-term has a corresponding λ R -term. Example λ term λ R term λx.y λx.x y λx.xx λx.x < x 1 (x 1 ) observe that contraction corresponds to variable duplication, while weakening corresponds to variable erasure.

8 Computing in the λ R -calculus Operational semantics: β-reduction - the key step: (β) (λx.m)n M[N/x] substitution - implicit i.e. meta-operator: x[n/x] N (λy.m)[n/x] λy.m[n/x], x y (MP)[N/x] M[N/x]P, x Fv(P) (MP)[N/x] MP[N/x], x Fv(M) (y M)[N/x] y M[N/x], x y (x M)[N/x] Fv(N) M (y < y 1 y 2 M)[N/x] y < y 1 y 2 M[N/x], x y (x < x 1 M)[N/x] Fv(N) < Fv(N 1) Fv(N 2 ) M[N 1/x 1, N 2 / ] Substitution satisfies interface preservation.

9 Computing in the λ R -calculus γ-reductions - contraction propagation: (γ 1 ) x < x 1 (λy.m) λy.x < x 1 (γ 2 ) x < x 1 (MN) (x < x 1 (γ 3 ) x < x 1 (MN) M(x < x 1 M M)N, if x 1, Fv(N) N), if x 1, Fv(M) ω-reductions - weakening extraction: (ω 1 ) λx.(y M) y (λx.m), x y (ω 2 ) (x M)N x (MN) (ω 3 ) M(x N) x (MN) γω-reductions - interaction of the structural operators: (γω 1 ) x < x 1 (y M) y (x < x 1 M), y x 1, (γω 2 ) x < x 1 (x 1 M) M[x/ ]

10 Computing in the λ R -calculus equivalencies: (ɛ 1 ) x (y M) y (x M) (ɛ 2 ) x < x 1 M x < x 1 M (ɛ 3 ) x < y z (y < u v M) x <y u (y < z v M) (ɛ 4 ) x < x 1 (y < y 1 y 2 M) y < y 1 y 2 (x < x 1 M), x y 1, y 2, y x 1, α-equivalence - for both binders: λx.m α λy.m[y/x] x < y z M α x < y 1 z 1 M[y 1 /y, z 1 /z]

11 The type system λ R x :α x :α (Ax) Γ, x :α M :β Γ λx.m :α β ( I) Γ M :α β N :α Γ, MN :β ( E ) Γ, x :α, y :α M :β Γ, z :α z < x y M :β (Cont) Γ M :α Γ, x :β x M :α (Weak) Theorem (Ghilezan et al. (2009)) If a λ R -term is typeable in the system λ R, then it is strongly normalizing (terminating).

12 The type system λ R x :α x :α (Ax) Γ, x :α M :β Γ λx.m :α β ( I) Γ M :α β N :α Γ, MN :β ( E ) Γ, x :α, y :α M :β Γ, z :α z < x y M :β (Cont) Γ M :α Γ, x :β x M :α (Weak) Theorem (Ghilezan et al. (2009)) If a λ R -term is typeable in the system λ R, then it is strongly normalizing (terminating).

13 The type system λ R x : σ x : σ (Ax) Γ, x : α M : σ Γ λx.m : α σ ( I) Γ M : n i=1 τ i σ 0 N : τ 0... n N : τ n Γ, n MN : σ ( E ) Γ, x : α, y : β M : σ Γ, z : α β z < x y M : σ (Cont) Γ M : σ Γ, x : x M : σ (Weak) Theorem (Ghilezan et al. (2011)) A λ R -term is strongly normalizing (terminating) if and only if it is typeable in the system λ R.

14 Towards substructural term calculi In order to obtain term calculi corresponding in the CH way to the intuitionistic implicative logic without (either explicit or implicit) weakening / contraction, we: start from the λ R -calculus; remove the weakening / contraction operator; remove all corresponding reduction and equivalence rules; but keep the related constraints in the definition of terms and in the type assignment rules. The obtained calculi are: the λ w -calculus, corresponding to a variant of the relevant logic, the λ c -calculus, corresponding to a variant of the affine logic.

15 The calculus without weakening - λ w Pre-terms: f ::= x λx.f ff x < x 1 f Terms: x Λ w f Λ w x Fv(f ) λx.f Λ w f Λ w g Λ w Fv(f ) Fv(g) = fg Λ w f Λ w x 1, x 1, Fv(f ) x / Fv(f ) \ {x 1, } x < x 1 f Λ w

16 Operational semantics: reductions: β, γ1, γ 2, γ 3 ; equivalencies: α, ɛ 2, ɛ 3, ɛ 4 ; reduced substitution definition. This is a strict sub-calculus of λ R, hence there are λ-terms and λ R -terms that cannot be represented in the λ w calculus, i.e. λx.y and z < x y x.

17 The type system λ w x :α x :α (Ax) Γ, x :α M :β Γ λx.m :α β ( I) Γ M :α β N :α Γ, MN :β ( E ) Γ, x :α, y :α M :β Γ, z :α z < x y M :β (Cont) Simply typed λ w -calculus corresponds in the Curry-Howard way to the intuitionistic natural deduction with explicit contraction and without weakening.

18 The type system λ w x :α x :α (Ax) Γ, x :α M :β Γ λx.m :α β ( I) Γ M :α β N :α Γ, MN :β ( E ) Γ, x :α, y :α M :β Γ, z :α z < x y M :β (Cont) Simply typed λ w -calculus corresponds in the Curry-Howard way to the intuitionistic natural deduction with explicit contraction and without weakening.

19 The type system λ w x : σ x : σ (Ax) Γ, x : α M : σ Γ λx.m : α σ ( I) Γ M : n i=1 τ i σ 1 N : τ 1... n N : τ n Γ, 1... n MN : σ ( E ) Γ, x : α, y : β M : σ Γ, z : α β z < x y M : σ (Cont) To be proved: A λ w -term is strongly normalizing if and only if it is typeable in the system λ w.

20 The calculus without contraction - λ c Pre-terms: f ::= x λx.f ff x f Terms: x Λ c f Λ c x Fv(f ) λx.f Λ c f Λ c g Λ c Fv(f ) Fv(g) = fg Λ c f Λ c x / Fv(f ) x f Λ c

21 Operational semantics: reductions: β, ω1, ω 2, ω 3 ; equivalencies: α (just for lambda abstraction), ɛ1 ; reduced substitution definition. This is a strict sub-calculus of λ R, hence there are λ-terms and λ R -terms that cannot be represented in the λ c calculus, i.e. λx.xx and x < x 1 (x 1 y).

22 The type system λ c x :α x :α (Ax) Γ, x :α M :β Γ λx.m :α β ( I) Γ M :α β N :α Γ, MN :β ( E ) Γ M :β Γ, x :α x M :β (Weak) Simply typed λ c -calculus corresponds in the Curry-Howard way to the intuitionistic natural deduction with explicit weakening and without contraction.

23 The type system λ c x :α x :α (Ax) Γ, x :α M :β Γ λx.m :α β ( I) Γ M :α β N :α Γ, MN :β ( E ) Γ M :β Γ, x :α x M :β (Weak) Simply typed λ c -calculus corresponds in the Curry-Howard way to the intuitionistic natural deduction with explicit weakening and without contraction.

24 The type system λ c x : σ x : σ (Ax) Γ, x : α M : σ Γ λx.m : α σ ( I) Γ M : n i=1 τ i σ 0 N : τ 0... n N : τ n Γ, n MN : σ ( E ) Γ M : σ Γ, x : x M : σ (Weak) To be proved: A λ c -term is strongly normalizing if and only if it is typeable in the system λ c.

25 References S. Ghilezan, J. Ivetić, P. Lescanne, D. Žunić: Intuitionistic sequent-style calculus with explicit structural rules. TbiLL th International Symposium on Language, Logic and Computation, Lecture Notes in Artificial Intelligence 6618: (2011). S. Ghilezan, J. Ivetić, P. Lescanne, S. Likavec: Intersection Types for the Resource Control Lambda Calculi. ICTAC th International Colloquium on Theoretical Aspects of Computing, Lecture Notes in Computer Science 6916: (2011). S. Ghilezan, J. Ivetić, P. Lescanne, S. Likavec: Intersection types for explicit substitution with resource control. ITRS Sixth Workshop on Intersection Types and Related Systems. S. Ghilezan, J. Ivetić, P. Lescanne, S. Likavec: Computational interpretations of some substructural logics. UNILOG The Fourth World Congress on Universal Logic.