A linear-non-linear model for a computational call-by-value lambda calculus (extended abstract)

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1 A linear-non-linear model or a computational call-by-value lambda calculus (extended abstract) Peter Seliner 1 and Benoît Valiron 2 1 Dalhousie University, seliner@mathstat.dal.ca 2 University o Ottawa, bvali087@uottawa.ca Abstract. We ive a cateorical semantics or a call-by-value linear lambda calculus. Such a lambda calculus was used by Seliner and Valiron as the backbone o a unctional prorammin lanuae or quantum computation. One eature o this lambda calculus is its linear type system, which includes a duplicability operator! as in linear loic. Another main eature is its call-by-value reduction stratey, toether with a side-eect to model probabilistic measurements. The! operator ives rise to a comonad, as in the linear loic models o Seely, Bierman, and Benton. The side-eects ive rise to a monad, as in Moi s computational lambda calculus. It is this combination o a monad and a comonad that makes the present paper interestin. We show that our cateorical semantics is sound and complete. 1 Introduction In the last ew years, there has been some interest in the semantics o quantum prorammin lanuaes. [18] ave a denotational semantics or a low-chart lanuae, but this lanuae did not include hiher-order types. Several authors deined quantum lambda calculi [21, 19] as well as quantum process alebras [11, 12], which had hiher-order eatures and a well-deined operational semantics, but lacked denotational semantics. [20] ave a cateorical model or a hiherorder quantum lambda calculus, but omitted all the non-linear eatures (i.e., classical data). Meanwhile, Abramsky and Coecke [2, 9] developed cateorical axiomatics or Hilbert spaces, but there is no particular lanuae associated with these models. In this paper, we ive the irst cateorical semantics o an unabrided quantum lambda calculus, which is a version o the lanuae studied in [19]. For the purposes o the present paper, an understandin o the precise mechanics o quantum computation is not required. We will ocus primarily on the type system and lanuae, and not on the structure o the actual builtin quantum operations (such as unitary operators and measurements). In this sense, this paper is about the semantics o a eneric call-by-value linear lambda calculus, which is parametric on some primitive operations that are not urther explained. It should be understood, however, that the need to support primitive quantum operations motivates particular eatures o the type system, which we briely explain now. The irst important lanuae eature is linearity. This arises rom the wellknown no-clonin property o quantum computation, which asserts that quantum data cannot be duplicated [23]. So i x : qbit is a variable representin a quantum bit, and y : bit is a variable representin a classical bit, then it is leal to write (y, y), but not (x, x). In order to keep track o duplicability at hiherorder types we use a type system based on linear loic. We use the duplicability operator! to mark classical types. In the cateorical semantics, this operator ives rise to a comonad as in the work o [16] and [6]. Another account o mixin copyable and non-copyable data is [10], where the copyability is internal to objects. A second eature o quantum computation is its probabilistic nature. Quantum physics has an operation called measurement, which converts quantum data to classical data, and whose outcome is inherently probabilistic. Given a quantum state α 0 + β 1, a measurement will yield output 0 with probability α 2 and 1 with probability β 2. To model this probabilistic eect in our call-by-value settin, our semantics requires a computational monad in the sense o [14]. The coexistence o the computational monad T and the duplicability comonad! in the same cateory is what makes our semantics interestin and novel. It diers rom the work o [7], who considered a monad and a comonad one two dierent cateories, arisin rom a sinle adjunction. The computational aspects o linear loic have been extensively explored by many authors, includin [8, 6, 5, 1, 22]. However, these works contain explicit lambda terms to witness the structural rules o linear loic, or example, x :!A derelict(x) : A. By contrast, in our lanuae, structural rules are implicit at the term level, so that!a is rearded as a subtype o A and one writes x :!A x : A. As we have shown in [19], linearity inormation can automatically be inerred by the type checker. This allows the prorammer to proram as in a non-linear lanuae. This use o subtypin is the main technical complication in our proo o welldeinedness o the semantics. This is because one has to show that the denotation is independent o the choice o a potentially lare number o possible derivations o a iven typin judment. We are orced to introduce a Church-style typin system, and to prove that the semantics inally does not depend on the additional type annotations. Another technical choice we made in our lanuae concerns the relation between the exponential! and the pairin operation. Linear loic only requires!a!b!(a B) and not the opposite implication. However, in our prorammin lanuae settin, we ind it natural to identiy a classical pair o values with a pair o classical values, and thereore we will have an isomorphism!a!b =!(A B). The plan o the paper is the ollowin. First, we describe the lambda calculus and equational axioms we wish to consider. Then, we develop a cateorical model, called linear cateory or duplication, which is inspired by [8] and [14].

2 We then show that the lanuae is an internal lanuae or the cateory, thus obtainin soundness and completeness. 2 The lanuae We will describe a linear typed lambda calculus with hiher-order unctions and pairs. The lanuae is desined to manipulate both classical data, which is duplicable, and quantum data, which is non-duplicable. For simplicity, we assume the lanuae is strictly linear, and not aine linear as in [19]. This means duplicable values are both copyable and discardable, whereas non-duplicable values must be used once, and only once. 2.1 The type system The set o types is iven as ollows: Type A, B ::= α (A B) (A B)!A. Here α ranes over type constants. While the remainder o this paper does not depend on the choice o type constants, in our main application [19] this is intended to include a type qbit o quantum bits, and a type bit o classical bits. A B stands or unctions rom A to B, A B or pairs, or the unit type, and!a or duplicable objects o types A. We denote!!!a with n! s by! n A. The intuitive deinition o!a is the key to the spirit in which we want the lanuae to be understood: The! on!a is understood as speciyin a property, rather than additional structure, on the elements o A. Thereore, we will have!a =!!A. Whether or not a iven value o type A is also o type!a should be somethin that is inerred, rather than speciied in the code. Since a term o type!a can always be seen as a term o type A, we equip the type system with a subtypin relation as ollows: Provided that (m = 0) (n 1),! n α <:! m α (ax ),! n <:! m ( ), A <: A B <: B! n (A B) <:! m (A B ) ( ), A <: A B <: B! n (A B) <:! m (A B ). ( ). This relation encapsulates the main properties terms should satisy with respect to duplicability. 2.2 Terms The lanuae consists o the ollowin typed terms, divided into values on the one hand, and eneral terms, or computations, on the other. Both share a subset o the values, the core values. CoreValue U ::= x A c A n λ n x A.M, Value V, W ::= U V, W let x A = V in W let x A, y B n = V in W let = V in W, Term M, N ::= U M, N (MN) let x A, y B n = M in N let = M in N, where n is an inteer, c ranes over a set o constant terms, x over a set o term variables and α over a set o constant types. We abbreviate (λ 0 x A.M)N by let x A = N in M, λ n x!m. let = x in M by λ n m.m and we omit numerical indexes when they are null. Note that the above terms carry Church-style type annotations, as well as inteer superscripts; we call these terms indexed terms. We also deine a notion o untyped terms as terms with no index: PureTerm M, N ::= x c λx.m (MN) M, N let x, y = M in N let = M in N. The erasure operation Erase : Term PureTerm is deined as the operation o removin the types and inteers attached to a iven indexed term. I M = Erase( M), we say that M is an indexation o M Finally, we deine an α-equivalence on terms, denoted by = α, in the usual way (see or example [3]). 2.3 Duplicable pairs and pairs o duplicable elements Beore we ormally present the type system, let us inormally motivate our choice o typin rules. One non-obvious choice we had to make is or the interaction o pairs and duplicability. Unlike previous works with comonads [8, 5], we want to think o the type!(a B) as a type o pairs o elements o type A and B: we want to use the same operation to access the components as one would use or a pair o type A B, without havin to use a dereliction operation. This immediately raises a concern: consider a pair o elements x, y o type!(a B). Are x and y duplicable? In the usual linear loic interpretation, they are not. Havin a ininite supply o pair o shoes does not mean one has an ininite supply o riht shoes: we cannot discard the let shoes. On the other hand, in our interpretation o classical data as residin in classical memory and thereore bein duplicable, i the strin x, y is duplicable, then so should be the elements x and y. In other words, we want the duplication to permeate the pairin. The choice o such a permeable pairin is more or less orced on us by our desire to have no explicit term syntax or structural rules. Consider the ollowin untyped terms, which can be typed i t is o type!(a!(b C)): let x, u = t in let y, z = u in z, y, x, (1) let x, u = t in let y, z = u in z, y, x. (2) First, we expect these two terms to be axiomatically equal. Term (2) should be o type!(!(c B) A), reardless o the permeability o the pairin: i y, z is duplicable, so should be z, y. Now, consider the term (1) with a non-permeable pairin. In the naive type system, u ends up bein o type B C, and the variables y and z in the inal recombination end up bein respectively o type B and C. It is not possible to make z, y o the duplicable type!(c B). We thereore choose a permeable pairin, which will be relected, albeit subtly, in the typin rule (.I) and (.E) in the ollowin section.

3 A <: B!A (ax!, x : A x B 1) : B, x : A M : B λ 0 x A.M : A B (λ1)!, Γ 1 M 1 :! n A 1!, Γ 2 M 2 :! n A 2!, Γ 1, Γ 2 M 1, M 2 n :! n (A 1 A 2) c <: B! c B : B (ax 2) Γ 1,! M : A B Γ 2,! N : A (app) Γ 1, Γ 2,! MN : B!, x : A M : B! λ n+1 x A.M :! n+1 (A B) (λ2)! n :! n (.I) (.I)!, Γ 1 M :!, Γ 2 N : A!, Γ 1, Γ 2 let = M in N : A (.E)!, Γ 1 M :! n (A 1 A 2)!, Γ 2, x 1 :! n A 1, x 2:! n A 2 N : A!, Γ 1, Γ 2 let x A1 1, (.E) xa2 2 n = M in N : A 2.4 Typin judments Table 1. Typin rules A typin judment is a tuple M : A, where M is an indexed term, A is a type, and is a typin context. To each constant term c we assin a type!a c. A valid typin judment is a typin judment that can be derived rom the typin rules in Table 1. We use the notation! or a context where all variables have a type o the orm!a. Finally, when we write a context Γ,, we assume the contexts Γ and to be disjoint. The ollowin lemmas are proved by structural induction on terms or type derivations as appropriate. Lemma 1. I V is a value such that V :!A is a valid typin judment, then =! or some context. Lemma 2. Consider the ollowin valid typin judment:, x : A M : B. Then or every ree instance x A in M, A <: A. Deinition 1. In a typin judment M : A, a term variable x is called dummy i x FV (M). Lemma 3. Any dummy variable x in M : B satisies (x) =!A, or some A. Conversely, i M : B is valid and i x FV (M), then or all types A the typin judment, x :!A M : B is valid. Typin derivations are not unique per se. However or a iven valid typin judement M : A two typin derivations will only dier with respect to the placement o dummy variables, namely the unused variables in context. 2.5 Type castin and substitution Lemma Lemma 4. Suppose M : A is a valid typin judment, and suppose <: and A <: A. Then there exists a canonical valid typin judment M : A such that Erase(M) = Erase(M ). Moreover, i M is a value, so is M. Proo. By induction on M. We will denote this M with { <: M : A <: A }. I = or A = A, we omit them or clarity. (β λ) let x = V in M ax M[V/x] : A (β ) let x, y n = V, W n in M ax M[V/x, W/y] : A (β ) let = in M ax M : A (η λ) λ n x A.{V :! n (A B) <: A B}x A ax V :! n (A B). (βλ) 2 let x A = N in x A ax N : A. (η ) let x A, y B n = N in x!na, y!nb n ax N :! n (A B). (η ) let = N in n ax N :! n. (let 1) let 1 = (let 2 = M in N) in P ax let 2 = M in let 1 = N in P : A (let 2) let 1 = V in let 2 = W in M ax let 2 = W in let 1 = V in M : A (let app ) let x A B = M in let y A = N in xy ax MN : B (let λ ) let x D = V in λ n y A.M ax λ n y A. let x D = V in M :! n (A B) (let ) let x A = M in let y B = N in x A, y B n ax M, N n :! n (A B) (app <: ) {M :! n (A D) <: B D }{N : C <: B} ax {{M :! n (A D) <: A D}{N : C <: A} : D <: D } (let <:) let x A, y B n = {M :! n (A B) <:! n (A B )} in N ax let x A, y B n = M in {, x :! n A, y :! n B <:, x :! n A, y :! n B N} (let x <:) let x A = {M : A <: A } in N ax let x A = M in {, x : A <:, x : A N} (let <:) let = {M :! m <:! n } in N ax let = M in N Table 2. Axiomatic equivalence axioms (α let), x : A let y A = x A in M : B ax, y : A M : B (let!λ )! let x!c = V in λy.m ax λ n+1 y. let x!c = V in M :! n+1 (A B) (let 1 ) V, let = M in N ax let = M in V, N :! n (A B) (let 2 ) let = M in N, V ax let = M in N, V :! n (A B) (let app 1 ) V (let = M in N) ax let = M in V N : B (let app 2 ) (let = M in N)V ax let = M in NV : B Table 3. Axiomatic equivalence: derived rules Deinition 2. Given two valid typin judments!, Γ 1 V : A and!, Γ 2, x : A M : B where V is a value, we deine the substitution (with capture avoidin)!, Γ 1, Γ 2 M[V/x] : B as ollows: we replace each ree instance x A (where A <: A rom Lemma 2) in M by { V : A <: A }. Lemma 5 (Substitution Lemma). In Deinition 2,!, Γ 1, Γ 2 M[V/x] : B is well-typed. Also, i M is a value, so is M[V/x]. Proo. Proo by structural induction on M, usin Lemmas 2 and Axiomatic equivalence We deine an equivalence relation on (indexed) typin judments. We write M ax M : A, or simply M ax M, to indicate that M : A and M : A are equivalent. Axiomatic equivalent is deined as the relexive, symmetric, transitive, and conruence closure o the rules rom Tables 2, so lon as both sides o the equivalences are well-typed. The symbol is a place holder or x,, or x, y. Lemma 6. The equivalences o Table 3 are derivable. The ollowin result stipulates that all the indexations o a iven erasure live in the same axiomatic class. In other words, the axiomatic equivalence class o a term is independent o its indexation.

4 Theorem 1. I Erase(M) = Erase(M ) and i M, M : A are valid typin judments, then M ax M. Proo (Sketch). The actual proo is lon and technical, and is omitted here or space reasons. We proceed by irst deinin a special subset o terms, called neutral terms, or which the Theorem is obvious. We then prove that every term is axiomatically equivalent to a neutral term via a series o rewrite systems. 3 Linear cateory or duplication As it was advertised, the structure o the cateorical semantics will closely ollow the one proposed by Bierman [8], but with the added twist o a computational monad à la Moi [14]. Indeed, since one has tensor products and a tensor unit, one can expect the cateorical model to be symmetric monoidal. Since one can construct candidate maps or buildin a comonad, a comonoid structure or each!a and coherence maps or the comonad, we have a linear cateory. Finally, the computational aspect will be taken care by Moi s computational monad. 3.1 Linear exponential comonads In his Ph.D. thesis, Bierman [8] ives the deinition o a linear cateory. We preer here the terminoloy iven in [15], and use the concept o linear exponential comonad. Deinition 3. Let (C,, ) be a symmetric monoidal cateory [13], where α A,B,C : A (B C) (A B) C, λ A : A A, ρ A : A A and σ A,B : A B B A are the usual associativity, let unit, riht unit and symmetry morphisms. Let (L, δ, ɛ, d L, d L ) be a monoidal comonad [8], where ɛ A : LA A, δ A : LA LLA, d L A,B : LA LB L(A B) and dl : L. We say that L is a linear exponential comonad [15] provided that 1. each object in C o the orm LA is equipped with a commutative comonoid (LA, A, A ), where A : LA LA LA and A : LA ; 2. A and A are monoidal natural transormations; 3. A : (LA, δ A ) (LA LA, (δ A δ A ); d A ) and A : (LA, δ A ) (, d L ) are L-coalebra morphisms; 4. Every map δ A is a comonoid morphism (LA, A, A ) (L 2 A, LA, LA ). The equations or 2 4 are to be ound in Table Stron monad and T -exponentials. To capture the computational eect o the probabilistic measurement, we use the notion o stron monad, as in [14]. Recall that a monad over a cateory C is a triple (T, η, µ) where T : C C is a unctor, η : id T and µ : T 2 T are natural transormations and such that T µ A ; µ A = µ T A ; µ A and η T A ; µ A = id T A = T η A ; µ A. Given a map : A T B, we deine the map : T A T B by T ; µ B. LA LB d L A,B L(A B) δ A LA L 2 A A L A A B (LA LA) (LB LB) sw (LA LB) (LA LB) λ 1 d L L d L dl L L LA LB A B d L A,B L(A B) A B d L A,B dl A,B ; L(A B) L(A B) d L (A B) L A and A are monoidal natural transormations. LA LA δ A δ A L 2 A L 2 A d L LA,LA L(LA LA), LA A δ A d L L 2 A L ; L A A and A are L-coalebra maps. LA A δ A id L 2 A, LA LA LA δa δ A L 2 A L 2 A δ A is a comonoid morphism. Table 4. Equations or a linear exponential comonad δ A λ LA L 2 A. A LA Deinition 4. A stron monad over a monoidal cateory C is a monad (T, η, µ) toether with a natural transormation t A,B : A T B T (A B), called the tensorial strenth, subject to a number o coherence conditions. Remark 1. I the cateory C is symmetric, the tensorial strenth t induces two natural transormations T A T B T (A B), namely Ψ 1 : T A T B Ψ 2 : T A T B σt A,T B T B T A tt B,A T (T B A) (σt B,A;tA,B) T (A B), tt A,B (σt A,B;tB,A) T (T A B) T (B A) T σb,a T (A B). Note that Ψ 1 and Ψ 2 miht not be equal: the map Ψ 1 evaluates the irst variable and then the second one. The map Ψ 2 does the opposite. The strenth is called commutative i Ψ 1 = Ψ 2. Lemma 7. I (T, η, µ, t) is a commutative stron monad on a symmetric monoidal cateory C, then (T, η, µ, Ψ 1 ) and (T, η, µ, Ψ 2 ) are monoidal monads. Deinition 5. A symmetric monoidal cateory (C,, ) toether with a stron monad (T, η, µ) is said to have T -exponentials [14], or Kleisli exponentials, i it is equipped with a biunctor : C op C C, and a natural isomorphism Φ : C(A, B C) = C(A B, T C). Lemma 8. The map Φ induces a natural transormation ε A,B : (A B) A T B deined by Φ(id A B ). 3.3 Idempotent, stron monoidal comonad A comonad (L, ɛ, δ) on some cateory is said to be idempotent i δ : L LL is an isomorphism. A monoidal comonad (L, δ, ɛ, d L, d L ) is stron monoidal i d L and d L A,B are isomorphisms.

5 Deinition 6. Given a monoidal cateory (C,, ) with an idempotent, stron monoidal comonad (L, ɛ, δ), a biunctor : C op C C, we deine a canonical arrow or C with respect to duplication by induction: For all objects A, the arrows id A, ɛ A, δ A, d L and dl A,B are canonical. All expansions o canonical arrows with respect to duplication are also canonical. An expansion o an arrow : A B is deined to be either or any o L,X, X, X, X, where is an expansion o and X ranes over the objects o the cateory. Finally, compositions o canonical arrows are also canonical. Theorem 2 (Coherence or idempotent comonads). Given a cateory C with the structure in Deinition 6, i, : A B are two canonical arrows with respect to duplication, then they are equal. 3.4 Linear cateory or duplication We now have enouh backround to deine a candidate or the cateorical model o the lanuae we describe in Section 2. Deinition 7. A linear cateory or duplication is a cateory C with the ollowin structure: a symmetric monoidal structure (,, α, λ, ρ, σ); an idempotent, stronly monoidal, linear exponential comonad (L, δ, ɛ, d L, d L,, ); a stron monad (T, µ, η, t); a Kleisli exponential. The computational linear cateory is deined to be the Kleisli cateory C T, as deined in [14]. Remark 2. A linear cateory or duplication ives rise to a double adjunction C L U L F L C U T C T,. F T Here the let adjunction arises rom the co- Kleisli cateory C L o the comonad L. It is as in the linear-non-linear models o [4], and C L is a cateory o classical (non-quantum) values. The riht adjunction arises rom the Kleisli cateory C T o the computational monad T, as in [14]. Here C T is a cateory o (eectul) quantum computations. 3.5 The cateory C λ Deinition 8. We can deine a cateory C λ as ollows: Objects are types, and arrows A B are axiomatic classes o valid typin judments o the orm x : A V : B, where V is a value. We deine the composition o arrows x : A V : B and y : B W : C to be x : A let y = V in W : C. The identity on A is set to be the arrow x : A x : A. Lemma 9. The cateory C λ is well-deined. α A,B,C = x : A (B C) let y, z = x in let t, u = z in y, t, u : (A B) C λ A = x : A let y, z = x in let = y in z : A ρ A = x : A let y, z = x in let = z in y : A σ A,B = x : A B let y, z = x in z, y : B A η A = x : A λ.x : A µ A = x : ( A) λ.(x ) : A t A,B = z : A ( B) let x, y = z in λ. x, y : (A B) ɛ A = x :!A x A : A δ A = x :!A x!2 A :! 2 A d! A,B = z :!A!B let x, y = z in x, y :!(A B) d! = z : let = z in :! A = x :!A x, x :!A!A A = x :!A : (x : A V : B) (y : C W : D) = z : A B let x, y = z in V, W : C D (x : A V : B) (y : C W : D) = z : B C λx.(let y = zv in W ) : A D (x : A V : B) = y : A λ. let x = (y ) in (V ) : B Φ A,B,C (x : A V : B C) = t : A B λ. let x, y = t in V y : C Table 5. Deinitions o maps and operations on maps in C λ Proo. The composition o two arrows yields an arrow axiomatically equivalent to a value due to Axiom (β λ ) and Lemma 5. Composition is associative due to Axiom (let 1 ). The arrow x : A x : A is indeed the identity on A due to axioms (α let ) and (βλ 2 ). Lemma 10. Given a valid typin judment V : A where V is a value, there exists a canonical value V such that Erase(V ) = Erase(V ) and such that! V :!A. We denote this V by {! <: V : A :> A }. Proo. By induction on V. Lemma 11. I V ax W : A, and i V = {! <: V : A :> A } and W = {! <: W : A :> A }, then V ax W. Proo. By induction on V ax W. Theorem 3. I we deine T (A) := A and L(A) =!A, toether with the maps and the operations on maps deined in Table 5, C λ is a linear cateory or duplication. Proo. The proo is mainly a lon list o veriications. It uses Theorem 1, Lemmas 9, 10 and Denotational semantics 4.1 Interpretation o the lanuae The lambda-calculus deined in Section 2 is thouht as a computational lambdacalculus. Usin Moi s technique, we split the interpretation o the lanuae

6 into the interpretation o the values in a linear cateory or duplication C and the interpretation o the computations, i.e. eneral terms, in its Kleisli cateory C T. Without loss o enerality, or notation purposes, we assume the cateory to be strictly monoidal. We deine an interpretation o the type system to be a map that assins to each constant type α an object (α). Each type A is interpreted as an object o C: [α] = (α), [ ] =, [!A] = L[A], [A B ] = [A] [B ] and [A B ] = [A] [B ]. Given a valid subtypin A <: B, there exists a canonical arrow [A] [B ] in C with respect to duplication, as deined in Deinition 6. Moreover, this arrow is unique by Theorem 2. We extend the map to interpret A<:B as this unique arrow and we denote it by I A,B. We use the ollowin straihtorward shortcut deinitions, where A, A, B, B are types and, Γ and Γ are typin contexts: Split!,Γ,Γ : [! ] [Γ ] [Γ ] [! ] [Γ ] [! ] [Γ ]. Given : [! ] [Γ ] [A] and : [! ] [Γ ] [B ], we deine the map! : [! ] [Γ ] [Γ ] A B. Given a natural transormation n A : F A GA, i = {x 1 : A 1... x n : A n } we deine n = n [[A1]]... n [[An]]. Deinition 9. The map is said to be an interpretation o the lanuae i moreover it assins to each constant term c : A c an arrow (c) : [A c ] in C. Given a linear cateory or duplication C, it is possible to interpret the typin derivation o a well-typed value as a map in C and the typin derivation o a valid computation as a map in the Kleisli cateory C T. We deine them inductively. I x 1 : A 1,... x n : A n V : B is a value with typin derivation π, its value interpretation [π ] v is an arrow [A 1 ]... [A n ] C [B ]; i x 1 : A 1,... x n : A n M : A is a term with typin derivation π, its computational interpretation [π ] c is an arrow [A 1 ]... [A n ] C T ([B ]). Table 6 ormulates the deinition in the simple case where the contexts, Γ 1 and Γ 2 contain only one variable. One can easily extend this to the eneral settin. As we already noted in Section 2.4, a valid typin judment does not have a unique typin tree per se. However the ollowin result holds: Theorem 4. Given a valid typin judment with two typin derivations π and π, or any interpretation we have [π ] c = [π ] c (and [π ]v = [π ] v i the typin judment is a value). Proo. The proo is done by showin that iven any typin judment M : A with denotation one can actor as!γ, where is the denotation o M : A, where,!γ = and Γ is the set o dummy variables. Deinition 10. Given a interpretation o the lanuae in a cateory C, we deine the denotation o a valid typin judment M : A with typin derivation π to be [ M : A] c = [π ]c and [ M : A]v = [π ]v i M is a value. Interpretation o core values: [!, x : A x : B ] v I A,B = [! ] [A] [B ] [! c : B ] v = [! ] (c) I Ac,B [A c ] [B ] [! :! n ] v = [! ] d L L I!,! n L n [, x : A M : B ] c = [ ] [A] T ([B ]) [ λx.m : A B ] v = [ ] Φ 1 () [A] [B ] [!, x : A M : B ] c = [! ] [A] T ([B ]) [! λx.m :! n+1 (A B)] v = [! ] L(Φ 1 );I!(A B),! n+1 (A B) L n+1 ([A] [B ]) Interpretation o extended values: [!, Γ 1 V : A] v = [! ] [Γ 1 ] [A] [!, Γ 2, x : A W : B ] v = [! ] [Γ 2 ] [A] [B ] [!, Γ 2, Γ 1 let x = V in W : B ] v id = [! ] [Γ 2 ] [Γ 1 ]! [! ] [Γ 2 ] [A] [B ] [!, Γ 1, V :! n (A 1 A 2)] v = [! ] [Γ 1 ] L n ([A 1 ] [A 2 ]) [!, Γ 2, x :! n A 1, y :! n A 2 W : C ] v = [! ] [Γ 2 ] Ln [A 1 ] L n [A 2 ] [C ] [!, Γ 2, Γ 1 let x, y n = V in W : C ] v id = [! ] [Γ 2 ] [Γ 1 ]! [! ] [Γ 2 ] L n ([A 1 ] [A 2 ]) ) id (d Ln 1 [[A 1 ]],[[A 2 ]] [! ] [Γ 2 ] L n [A 1 ] L n [A 2 ] [C ] [!, Γ 2 V : ] v = [! ] [Γ 2 ] [!, Γ 1 W : C ] v = [! ] [Γ 1 ] [C ] [!, Γ 1, Γ 2 let = V in W : C ] v = [! ] [Γ 1 ] [Γ 2 ] id! [! ] [Γ 1 ] [C ] [!, Γ 1 V :! n A] v = [! ] [Γ 1 ] L n [A] [!, Γ 2 W :! n B ] v = [! ] [Γ 2 ] L n [B ] [!, Γ 1, Γ 2 V, W n :! n (A B)] v = [! ] [Γ 1 ] [Γ 2 ]! L n [A] L n d Ln A,B [B ] L n ([A] [B ]) Interpretation o computations: First, i U is a core value, [ U : A] c = [ U : A]v ; ηa. [!, Γ 1 M : A B ] c = [! ] [Γ 1 ] T ([A] [B ]) [!, Γ 2 N : A] c = [! ] [Γ 2 ] T ([A]) [!, Γ 1, Γ 2 MN : B ] c = [! ] [Γ 1 ] [Γ 2 ]! T ([A] [B ]) T ([A]) Ψ1 T (([A] [B ]) [A]) ε A,B T ([B ]) [!, Γ 1 M :! n (A 1 A 2)] c = [! ] [Γ 1 ] T L n ([A 1 ] [A 2 ]) [!, Γ 2, x :! n A 1, y :! n A 2 N : C ] v = [! ] [Γ 2 ] Ln [A 1 ] L n [A 2 ] T ([C ]) [!, Γ 2, Γ 1 let x, y n = M in N :! n C ] c = [! ] [Γ 2 ] [Γ 1 ] id! [! ] [Γ 1 ] T L n ([A 1 ] [A 2 ]) t;t ( id (d Ln ) 1 ) T ([! ] [Γ 1 ] L n [A 1 ] L n [A 2 ]) T [C ] [!, Γ 2 M : ] c = [! ] [Γ 2 ] T ( ) [!, Γ 1 N : C ] c = [! ] [Γ 1 ] T ([C ]) [!, Γ 1, Γ 2 let = M in N : C ] c id = [! ] [Γ 1 ] [Γ 2 ]! [! ] [Γ 1 ] T ( ) t; T ([C ]) [!, Γ 1 M :! n A] c = [! ] [Γ 1 ] T L n [A] [!, Γ 2 N :! n B ] c = [! ] [Γ 2 ] T L n [B ] [!, Γ 1, Γ 2 M, N n :! n (A B)] c = [! ] [Γ 1 ] [Γ 2 ]! T L n [A] T L n Ψ1;T d Ln A,B [B ] T L n ([A] [B ]) Table 6. Interpretation o values and computations. Lemma 12. Suppose that V : A is a valid typin judment where V is a value. Then [ V : A] c [[ V :A]]v = [ ] [A] η [[A]] T ([A]). Proo. Proo by induction on V, usin the biunctoriality o LA and the equations or stron monadicity in Deinition Soundness o the denotation The axiomatic equivalence and the cateorical semantics are two aces o the same coin. Indeed, as we will prove in this section, two terms in the same axiomatic equivalence class have the same denotation. A corollary is that the indexation o terms does not inluence the denotation. This proves semantically the act that it is sae to work with untyped terms. An alternate justiication o this act is o course the operational semantics, which was iven in [19].

7 Lemma 13. Suppose M = { <: M : A <: A }. Then [ M : A ] c = I, ; [ M : A] c ; T (I A,A ). I M = V is a value, rom Lemma 4, M = V is a value. Then [ V : A ] v = I, ; [ V : A] v ; I A,A. Proo. Proo by structural induction on M. Lemma 14 (Substitution). Given two valid typin judments!, Γ 1, x : A M : B and!, Γ 2 V : A, the typin judment!, Γ 1, Γ 2 M[V/x] : B is valid. Let h be [!, Γ 1, Γ 2 M[V/x] : B ] c and h be [!, Γ 1, Γ 2 W [V/x] : B ] v, in the case where M = W is a value. Then they are deined by [! ] [Γ 1 ] [Γ 2 ] Split!,Γ1,Γ 2 id [[!,Γ2 V :A]]v [! ] [Γ 1 ] [! ] [Γ 2 ] h T ([B ]) [[!,Γ1,x:A M:B]] c [! ] [Γ 1 ] [A], [! ] [Γ 1 ] [Γ 2 ] Split!,Γ1,Γ 2 id [[!,Γ2 V :A]]v [! ] [Γ 1 ] [! ] [Γ 2 ] h [B ] [[!,Γ1,x:A W :B]] v [! ] [Γ 1 ] [A]. Proo. Proo by induction on M, usin Lemma 1, Lemma 12 and the naturality o Φ. Theorem 5. I M ax M : A then [ M : A] c = [ M : A] c (and [ M : A] v = [ M : A] v i M is a value) or every interpretation. Proo. Proo by induction on M ax M, usin Lemmas 13 and 14. Corollary 1. I Erase(M) = Erase(M ) and i M, M : A are valid typin judments, then [M ] c = [M ] c (and [M ] v = [M ] v i M is a value). Proo. Corollary o Theorems 1 and Completeness The cateory C λ bein a linear cateory or duplication, one can interpret the lanuae in it. This section states that the deined lambda-calculus is an internal lanuae o linear cateories or duplication. Since the cateory C λ is a monoidal cateory, one can w.l.o.. eneralize the notion o pairin to inite tensor products o terms. Then the ollowin results are true: Lemma 15. In C λ, a valid typin judment x 1 : A 1,... x n : A n M : B has or computational denotation (t : A 1 A n let x 1,... x n = t in λ.m : B). I M = V is a value, the value interpretation is (t : A 1 A n let x 1,... x n = t in V : B). Proo. Proo by structural induction on M and V. Theorem 6. In C λ, bein the identity, one has [x : A M : B ] c ax (x : A λ.m : B) and [x : A V : B ] v ax (x : A V : B). Proo. Corollary o Lemma 15 5 Towards a denotational model o quantum lambda calculus As noted in the introduction, this paper is mostly concerned with the cateorical requirements or modelin a eneric call-by-value linear lambda calculus, i.e., its type system (which includes subtypin) and equational laws. We have not yet specialized the lanuae to a particular set o built-in operators, or example, those that are required or quantum computation. However, since the quantum lambda calculus [19] is the main motivation behind our work, we will comment very briely on what additional properties would be required to interpret its primitives. The quantum lambda calculus is obtained by instantiatin and extendin the call-by-value lanuae o this paper with the ollowin primitive types, constants, and operations: Types: bit, qbit Constants: 0 :!bit, 1 :!bit new :!(bit qbit), U :!(qbit n qbit n ), meas :!(qbit!bit) Operations: Γ 1,! P : bit Γ 2,! M : A Γ 2,! N : A (i ) Γ 1, Γ 2,! i P then M else N : A Here, U ranes over a set o built-in unitary operations. In the intended semantics,!bit = bit, while!qbit is empty. new creates a new qubit, and meas measures a qubit. The denotational semantics o these operations is already well-understood in the absence o hiher-order types. They can all be interpreted in the cateory Q o superoperators rom [18]. The part that is not yet well-understood is how these eatures interact with hiher-order types. In liht o our present work, we can conclude that a model o the quantum lambda calculus consists o a linear cateory or duplication (C, L, T, ), such that the associated cateory o computations C T contains the cateory Q o [18] as a ull monoidal subcateory. To construct an actual instance o such a model is still an open problem. 6 Conclusion We have developed a call-by-value, computational lambda-calculus or manipulatin duplicable and non-duplicable data, toether with an axiomatic equivalence relation on typed terms. We use a subtypin relation in order to have implicit promotion, dereliction, copyin and discardin. Then we developed cateorical model or the lanuae, inspired by the work o [8] and [14]. We inally showed that the model is sound and complete with respect to the axiomatic equivalence. Reerences 1. Abramsky, S.: Computational interpretations o linear loic. Theoretical Computer Science 111 (1993) 3 57

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