M, N ::= x λp : A.M MN (M, N) () c A. x : b p x : b

Transcription

1 A PATTERN-MATCHING CALCULUS FOR -AUTONOMOUS CATEGORIES ABSTRACT. This article sums up the details of a linear λ-calculus that can be used as an internal language of -autonomous categories. The coherent isomorphisms associated with the autonomous (= symmetric monoidal) structure are represented by pattern matching, and the inverse to the evident natural transformation η A : A ((A ) ) is represented by a type-indexed family of constant C A : ((A ) ) A. The calculus presented in this article is no great novelty. It is essentially DILL [1] minus additives, plus extended pattern-matching and a type-indexed family of constants C A : ((A ) ) A. It can also be seen as Hasegawa s DCLL [2] minus the additive functions space, plus pattern matching and the multiplicative conjunction. The main purpose of our calculus is the rapid verification of equalities in autonomous and -autonomous categories. To this end, we emphasize pattern matching, which in is very useful in practice, more strongly than in [1] or [2]. (A note to the impatient: the translation of categorical expressions into our calculus is presented in Table 5.) Autonomous category is another word for symmetric monoidal category. -autonomous category is an autonomous category together with an object, such that the evident natural transformation η A : A (A ) has an inverse. We shall present a calculus for autonomous categories, the autonomous calculus, and discuss the equational laws for the family of constants C A : ((A ) ) A which represents the inverse of the natural transformation η : A ((A ) ). The syntax of the autonomous calculus is as follows: Types A, B ::= A B A B b Patterns P, Q ::= x (P, Q) () Terms M, N ::= x λp : A.M MN (M, N) () c A where b ranges over base types and c A over constants of type A. The typing rules for patterns are x : b p x : b x : A B p x : A B Γ p P : A p Q : B if dom(γ) dom( ) = Γ, p (P, Q) : A B p () : (The capital greeks letters Γ, range over non-repetitive sequences x 1 : A 1, x 2 : A 1,..., x n : A n of typed variables, and dom(γ) is the set of variables that occur in Γ.) Note that in a pattern we allow only variables whose type is not of the form or A B. Also, note that for every type A, there a unique pattern P A for which some judgement Γ p P A : A is derivable up to renaming the variables in dom(γ). The typing rules for terms are presented in Table 1. The notation Γ stands for any merging of 1

2 2 A PATTERN-MATCHING CALCULUS FOR -AUTONOMOUS CATEGORIES x : A x : A Γ, M : B p P : A Γ λp : A.M : A B Γ M : A B N : A if dom(γ) dom( ) = Γ MN : B Γ M : A N : B if dom(γ) dom( ) = Γ (M, N) : A B () : TABLE 1. Typing judgments of the autonomous calculus Γ and. For example, we allow Thus, the permutation rule x 1 : A 1, x 2 : A 2 M : A y 1 : B 2, y 2 : B 2 N : B y 1 : B 1, x 1 : A 1, y 2 : B 2, x 2 : A 2 (M, N) : A B Γ M : A if Γ is a permutation of Γ Γ M : A is derivable by induction over typing judgements. The autonomous calculus is linear in the sense that, in a derivable judgement Γ M : A, every variable in Γ has exactly one free occurrence in M. So every judgement Γ M : A has a unique derivation, because in the binary typing rules (application and pairing) it is determined for each free variable whether it comes from the left premise, or from the right. We use let P : A be M in N as syntactic sugar for (λp : A.N)M. The equations of the autonomous calculus are presented in Table 2. A signature Σ consists of a collection of base types b and a collection of constants c A. Definition 1. An autonomous theory over a signature Σ is a set of judgments Γ M N : A, where Γ M : A and Γ N : A are derivable typing judgements over Σ, such that is a congruence that contains all equations described in Table 2. Proposition 0.1. The equations in Table 2 are interderivable with the equations in Table 3. Proposition 0.2. In every autonomous theory, the β-rule let x be M in N N[M/x] holds. Proof. By induction over the derivation of N.

3 A PATTERN-MATCHING CALCULUS FOR -AUTONOMOUS CATEGORIES 3 ASSOC let Q be (let P be L in M) in N let P be L in let Q be M in N if FV(P ) FV(N) = ID let P be M in P M PAT let (P 1, P 2 ) be (M 1, M 2 ) in N let P 1 be M 1 in let P 2 be M 2 in N if FV(P 1 ) FV(M 2 ) = PAT let (P 1, P 2 ) be (M 1, M 2 ) in N let P 2 be M 2 in let P 1 be M 1 in N if FV(P 2 ) FV(M 1 ) = PAT let () be () in N N η λp.mp M if FV(P ) FV(M) = LET λ let Q be M in λp.n λp.let Q be M in N if FV(P ) FV(M) = TABLE 2. Equations of the autonomous calculus ID let P be M in P M PAT let (P 1, P 2 ) be (M 1, M 2 ) in N let P 1 be M 1 in let P 2 be M 2 in N if FV(P 1 ) FV(M 2 ) = PAT let () be () in N N η λp.mp M if FV(P ) FV(M) = LET λ let Q be M in λp.n λp.let Q be M in N if FV(P ) FV(M) = LET app let P be L in MN M(let P be L in N) if FV(P ) FV(M) = LET app let P be L in MN (let P be L in M)N if FV(P ) FV(N) = LET pair let P be L in (M, N) (M, let P be L in N) if FV(P ) FV(M) = LET pair let P be L in (M, N) (let P be L in M, N) if FV(P ) FV(N) = TABLE 3. Alternative equations for the autonomous calculus Now for the semantics of the autonomous calculus. Definition 2. An interpretation of typing judgments over a signature Σ in an autonomous category C sends each type A of T homomorphically to an object A of C, each environment Γ = x 1 : A 1,..., x n : A n to the object A 1 A n (where associates to the left, say, and the empty product is ), and each judgement Γ M : A to a morphism Γ M : A : Γ A according to the rules in Table 4. (Thus, an interpretation is uniquely determined by its effect on base types and constants.) A model of an autonomous theory T is an interpretation such that, for every equation Γ M N : A in T, it holds that Γ M : A = Γ N : A.

4 4 A PATTERN-MATCHING CALCULUS FOR -AUTONOMOUS CATEGORIES x : A x : A = id A Γ, M : B = f : Γ, ( B Γ λp : A.M : A B = λ Γ A = Γ, ) f B Γ M : A B = f : Γ A B N : A = g : A Γ MN : B = Γ, = Γ f g ( A B ) A apply B Γ M : A = f : Γ A N : B = g : B Γ (M, N) : A B = Γ, = Γ f g A B () : = id TABLE 4. Semantics of the autonomous calculus Proposition 0.3 (Soundness). For every interpretation of typing judgements over a signature Σ, the well-typed equations Γ M N : A such that Γ M : A = Γ N : A form an autonomous theory. Theorem 0.4 (Completeness). Suppose that Γ M : A and Γ N : A are typing judgments of a theory T. If the equation Γ M N : A holds in every model of T, then it is in T. Proof. We present a term model of T. A categorical language is given by the grammar Types A, B ::= A B A B b Terms f, g ::= id A g f g f α A,B,C λ A ρ A σ A,B λf apply A,B h A B (with the evident typing rules) where b ranges over base types and h ranges over generators of type A B. A reverse interpretation sends every type A of the categorical language to a type A of the autonomous calculus, homomorphically (i.e., it does nothing but exchange base types) every term f : A B of the categorical language to a term f : A B of the autonomous calculus, according to Table 5. It is easy to check that, for every reverse interpretation, the categorical language, modulo the induced equality f g f g, forms an autonomous category. To prove completeness, we start with the categorical language whose base types are those of T, and whose generators are of the form h c : A for every constant c : A of T. Let be the reverse interpretation that sends each base type to itself, and each generator h c : A to

5 A PATTERN-MATCHING CALCULUS FOR -AUTONOMOUS CATEGORIES 5 id A : A A = λp : A.P g f : A C = λp : A. g ( f P ) f 1 f 2 : A 1 A 2 B 1 B 2 = λ(p 1, P 2 ) : A 1 A 2.( f 1 P 1, f 2 P 2 ) α : A (B C) (A B) C = λ(p, (Q, R)) : A ( B C ).((P, Q), R) λ : A A = λ((), P ) : A.P ρ : A A = λ(p, ()) : A.P σ : A B B A = λ(p, Q) : A B.(Q, P ) λf : A (B C) = λp : A.λQ : B. f (P, Q) applya,b : (A B) A B = λ(f, P ) : ( A B ) A.fP TABLE 5. Reverse interpretation (used e.g. in the term-model construction) λ() :.c. Then it is not hard to show that the induced autonomous category forms an initial model of T. Now we make the step from autonomous categories to -autonomous categories. A -autonomous category is an autonomous category together with an object such that, for every object A, the evident map η A : A ((A ) ) has an inverse. In the autonomous calculus, η A is λp : A.λk : A.kP. We represent its inverse by a constant C A : ((A ) ) A. Thus, the categorical equations C A η A = id A η A C A = id (A ) (after simplification) correspond to λp : A.C A (λk : A.kP ) λp : A.P λh : (A ).λk : A.k(C A h) λh : (A ).h in the autonomous calculus. The first equation is interderivable with C A (λk : A.kM) M C-APP where M ranges over terms of type A (with no free occurrence of k), and the second equation is interderivable with K(C A M) MK C-ETA where M ranges over terms of type (A ) and K ranges over terms of type A. (Of course, we must have FV(M) FV(K) =.) Thus we obtain: Theorem 0.5. The autonomous calculus together with a type and a type-indexed family of constants C A : ((A ) ) A satisfying the rules C-APP and C-ETA is sound and complete for - autonomous categories.

6 6 A PATTERN-MATCHING CALCULUS FOR -AUTONOMOUS CATEGORIES The use of the law C-ETA is unpleasantly restricted because K must have type A rather than A B for arbitrary B. This restriction is lifted by the following law: N(C A M) C B (λk : B.M(k N)) To see that this law holds, consider N(C A M) C B (λk.b.k(n(cm))) = C B (λk.b.(k N)(CM)) C B (λk.b.m(k N)) C-NAT C-APP C-ETA (The law C-NAT is called so because it essentially encodes the fact that the type-indexed family C A represents a natural transformation.) REFERENCES [1] A. Barber. Dual intuitionistic linear logic. Technical Report ECS-LFCS , University of Edinburgh, [2] Masahito Hasegawa. Classical linear logic of implications. In Proc. 11th Annual Conference of the European Association for Computer Science Logic (CSL 02), Edinburgh, volume 2471 of LNCS. Springer-Verlag, September 2002.