Böhm trees, Krivine machine and the Taylor expansion of ordinary lambda-terms

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1 Böhm trees, Krivine machine and the Taylor expansion of ordinary lambda-terms Thomas Ehrhard and Laurent Regnier Institut de Mathématiques de Luminy CNRS and Université de la Méditerranée UMR 6206 and June 6, 2005 Abstract We show that, given an ordinary lambda-term and a normal resource lambda-term which appears in the normal form of its Taylor expansion, the unique resource term of the Taylor expansion of the ordinary lambda-term reducing to this normal resource term can be obtained by running a version of the Krivine abstract machine. Introduction After having introduced the dierential lambda-calculus in [ER03], we studied in [ER04b] a subsystem of the dierential lambda-calculus which turns out to be very similar to resource oriented versions of the lambda-calculus previously introduced and studied by various authors [Bou93, BCL99, Kfo00]. Resource lambda-calculus and Taylor expansion. Our viewpoint on this resource lambda-calculus is that it is the sublanguage of the dierential lambda-calculus where the complete 1 Taylor expansions of ordinary lambda-terms can be written. Indeed, the only notion of application available in this calculus consists in taking a term s (of type A B if the calculus is typed) and a nite number of terms s 1,..., s n (of type A) and applying s to the multiset consisting of the terms s i, written multiplicatively s 1... s n. This application is written s (s 1... s n ). In ordinary dierential calculus, this operation corresponds to taking the nth derivative of s at 0, which is a symmetric n-linear map, and applying this derivative to (s 1,..., s n ). Dening the β-reduction in this calculus (as in the original dierential lambda-calculus) requires the possibility of adding terms, because the analogue of substitution is a notion of formal partial derivative whose inductive denition is based on Leibniz rule 2, and the expression s (s 1... s n ) is linear in s, s 1,..., s n ; the connection between ordinary linearity and this syntactical notion of linearity is discussed in the introduction of [ER03]. The logical signicance of this derivative, and the analogue in linear logic of this resource lambda-calculus are discussed in [ER04a], where dierential interaction nets are introduced. The striking fact is that this new structure appears in this linear logic setting as new operations associated to the exponentials, completely dual to the traditional structural operations (weakening, contraction), and to dereliction. In constrast, ordinary lambda-calculus has a notion of application which is linear in the function but not in the argument, for which we used the notation (M) N (parenthesis around the function, not around the argument). The connection between these two applications is given by the Taylor formula. This work has been supported by the ACI project GEOCAL. 1 By complete, we mean that all applications in the lambda-terms are Taylor expanded. 2 In [ER04a], it is shown that Leibniz rule is more pecisely connected to the interaction between derivation and contraction. 1

2 Taylor expansion and normalization. In [ER04b], we explained how to Taylor expand arbitrary ordinary lambda-terms as (generally innite) linear combinations of resource lambda-terms with rational coecients. We showed moreover that, when normalizing the resource terms which occur in such a Taylor expansion, one gets generally innitely many nite linear combinations of normal resource terms (with positive integers as coecients) which do not overlap; so it makes sense to sum up all these linear combinations. Moreover, the numerical coecients behave well during the reduction, in a sense which is made precise in the corresponding statement, Theorem 4 in the present paper. Overview We show that this sum s of normal resource terms obtained by normalizing the Taylor expansion of an ordinary lambda-term M is simply the Taylor expansion of the Böhm tree of M (the extension of Taylor expansion to Böhm trees is straightforward). Thanks to the results obtained in [ER04b], this reduces to showing that a normal resource term appears in s with a nonzero coecient i it appears with a nonzero coecient in the Taylor expansion of the Böhm tree of M. The only if part of this equivalence is fairly straightforward, whereas the if part requires the introduction of a version of the Krivine machine which also gives a appealing computational interpretation of the result. The Krivine machine. Usually, the Krivine machine [Kri05] is described as an abstract environment machine which performs the weak linear head reduction on lambda-terms: given a term M which is β- equivalent to a variable x, starting from the state (M,, ) (empty environment and empty stack 3 ), after a certain number of steps, the machine will produce the result (x, E, ) where the resulting variable x is not bound by the environment E. This computation can be understood as a special kind of reduction of lambdaterms which cannot be described exactly as a beta-reduction because, at each reduction step, only the leftmost occurrence of variable in the term is substituted. Rather than giving a formal denition 4, it is simpler to contemplate a simple example, so consider the term M 0 = (λx (x) (x) y) λz z. After one step of reduction, we get M 1 = (λx (λz z) (x) y) λz z. Observe that the argument and the lambda of the main redex are still there, and that the function still contains an occurrence of the variable x. Now the leftmost variable occurrence is z and the term M 1 reduces to M 2 = (λx (λz (x) y) (x) y) λz z. The leftmost occurrence of variable is x again and we get M 3 = (λx (λz (λz z) y) (x) y) λz z which reduces to M 4 = (λx (λz (λz y) y) (x) y) λz z. We arrive to a term M 4 whose redexes are all K-redexes 5 and reduces to the variable y. This is exactly this kind of computation that the Krivine machine performs, with the restriction that one does not reduce under the lambda's, in some sense (whence the word weak). We extend the Krivine machine in two directions 6. First, we accept to reduce under lambda's. Second, when the Krivine machine arrives to a state (x, E, Π) where the environment E does not bind x and Π is a non-empty stack, it classically stops with an error. Here instead we continue the computation by running the machine on each element of Π. This corresponds, in the linear head reduction process, to reducing within the arguments of the head variable when a head normal form has been reached. We call K this extended machine. When fed with a triple (M, E, ) where E does not bind the free variables of M, this machine produces the Böhm tree of M (all nite approximations being obtained in a nite number of steps). 3 The stack is there as usual for pushing the arguments of applications. 4 By the way, the best formal denition available is certainly the Krivine machine itself. 5 A K-redex is a redex (λx M) N such that x does not occur free in M. In M4, the outermost redex is not a K-redex, but becomes a K-redex after reduction of the internal K-redexes. 6 These extensions are fairly standard and part of the folklore. 2

3 A more informative version of the machine. Then we dene a version K of that machine where a tracing mechanism is added. The idea is to count precisely how many times the various parts of the term M have been used, starting from the state (M,, ), for reaching the state (x, E, ) (when one knows that M is equivalent to the variable x). This information is summarized as a resource term which has the same shape as M (or, equivalently, appears in the Taylor expansion of M with a nonzero coecient). For example, in the example of M 0, the corresponding resource term is λx x x y (λz z) 2, which appears with coecient 1 2 in the Taylor expansion of M 0. But there is no reason for limiting our attention to lambda-terms equivalent to a variable: when M reduces to a Böhm tree B, we just add a parameter to our Krivine machine, which is a resource term u occurring in the Taylor expansion of B. Then K(M,,, u) produces a resource term s which appears in the Taylor expansion of M and, in some sense, counts how much of M the machine uses for producing u. This machine also gives us a proof for the if part of our main result (see the beginning of this Overview section), because u appears with a nonzero coecient in the normal form of the resource term s produced by the machine. 1 Ordinary notions Böhm trees. We recall a few denitions, more or less standard in lambda-calculus. An elementary Böhm tree (EBT) is a normal term in the lambda-calculus extended with the constant Ω subject to the following equations: (Ω) M = Ω and λx Ω = Ω. In other words: Ω is an elementary Böhm tree; if x, x 1,..., x n are variables and B 1,..., B k are elementary Böhm trees, then λx 1... x n (x) B 1... B k is an elementary Böhm tree. The following clauses dene an order relation on EBTs: Ω B for all EBT B; λx 1... x n (x) B 1... B k C if c = λx 1... x n (x) C 1... C k with B j C j for all j. A (general) Böhm tree is now dened as an ideal of elementary Böhm trees, in other word, it is a set B of EBTs such that: if B C B then B B; B (equivalently, Ω B); if B, B B, there exists C B such that B, B C. To any ordinary lambda-term M is associated its Böhm tree. We dene rst a family of functions from lambda-terms to EBTs. BT 0 (M) = Ω; BT n+1 (λx 1... x p (x) M 1... M k ) = λx 1... x p (x) BT n (M 1 )... BT n (M k ); BT n+1 (λx 1... x p ((λy P ) Q) M 1... M k ) = BT n (λx 1... x p (P [Q/y]) M 1... M k ) It is straightforward to check that BT n (M) is a non decreasing sequence of EBTs. Then the Böhm tree of M is the downwards closure of the set {BT n (M) n N}, which is an ideal of EBTs. 3

4 The Krivine Abstract Machine. If f : S S is a partial function, a S and b S, we denote by f a b the partial function g : S S which is dened like f but for a, where it is dened and takes the value b. By simultaneous induction, we dene the two following concepts: A closure is a pair Γ = (M, E) where M is a lambda-term and E is an environment such that FV(M) Dom E. An environment is a nite partial function on variables, taking closures or the distinguished symbol free as value. We use Dom c E for the subset of Dom E whose elements are not mapped to free. We need also an auxiliary concept: a stack is a nite list Π of closures. We rst dene a sequence of functions from states to EBTs. K 0 (Γ, Π) = Ω; K n+1 (x, E, Π) = K n (E(x), Π) if x Dom c (E); K n+1 (x, E, Π) = (x) K n (Γ 1, )... K n (Γ k, ) where Π = (Γ 1,..., Γ n ), if E(x) = free; K n+1 (λx M, E, ) = λx K n (M, E x free, ) (assuming that x / Dom(E) and that x does not appear free in any of the terms mentioned in E); K n+1 (λx M, E, Γ :: Π) = K n (M, E x Γ, Π) (with similar assumptions for x); K n+1 ((M) N, E, Π) = K n (M, E, (N, E) :: Π). Observe that the denition is correct in the sense that, in all recursive calls of the function K, the closures are well formed (the domain of their environment contains the free variables of their term). Lemma 1 Let S = (Γ, Π) be a state. Then (K n (S)) n N is a non decreasing sequence of EBTs. This is easy to check. We dene K(S) as the downwards closure of the set {K n (S) n N}; this set is a Böhm tree. We dene another toral function T, from closures to lambda-terms. Given a closure Γ = (M, E), we set T(Γ) = M [T(E(x))/x] x Domc E This is a denition by induction on the height of closures, seen as nitely branching trees. We extend this mapping to states: T(Γ, (Γ 1,..., Γ n )) = (T(Γ)) T(Γ 1 )... T(Γ n ). The main, (almost) standard, property of the Krivine machine is as follows. Theorem 2 Let S be a state. Then K(S) = BT(T(S)). This soundness result shows in particular that the Krivine machine computes the Böhm tree of lambdaterms: BT(M) = K(M, E, ), where E is any environment mapping all the free variables of M to the value free. However the way it computes it is not the standard head reduction, but the linear head reduction of [DR99], also known as De Bruijn mini-reduction [DB87]. This reduction is much more elementary than the standard notion, since each substitution performed by the machine is linear (only head occurrences of variables are substituted). 4

5 2 Resource notions 2.1 The resource lambda-calculus. We give a short account of the resource lambda-calculus, as developped in [ER04b]. We recall the syntax and terminology of [ER04b]. As usual we are given a countable set of variables. Simple terms and poly-terms. If x is a variable, then x is a simple term. If x is a variable and t is a simple term, then λx t is a simple term. If t is a simple term and T is a simple poly-term, then t T is a simple term. A simple poly-term is a multiset of simple terms. We use multiplicative notations for these multisets: 1 denotes the empty poly-term, if t is a simple term, we use also t for denoting the simple poly-term whose only element is t, and if S and T are simple poly-terms, we use ST for the multiset union (sum) of S and T. We use the greek letters σ, τ... for simple terms or poly-terms when we do not want to be specic. The size of a simple resource (poly-)term σ is a nonnegative integer S(σ), dened by induction: S(x) = 1, S(λx s) = 1 + S(s), S( s S) = 1 + S(s) + S(S) and last, for poly-terms, S(s 1... s n ) = S(s 1 ) + + S(s n ). Observe that S(σ) = 0 holds i σ is the poly-term 1. We use for the set of all simple terms and! for the set of all simple poly-terms. Linear combinations and reduction. Let R be a set of scalars, a commutative semi-ring with multiplicative unit 1 (so that there is a canonical semi-ring homomorphism from N to R, under which we consider nonnegative integers as elements of R, although this homomorphism has no reason to be injective). If A is a set, we use R A for the free module over R generated by A. If α R A, we use Supp(α) for the set of all a A such that α a 0. A redex is a simple term of the shape r = λx s S. There is a small-step notion of reduction for these redexes, but we do not consider it here, we focus on the big-step reduction which stipulates that r reduces to 0 R if the cardinality of the multiset S is distinct from the number of free occurrences of x in s, and otherwise reduces to x (s, S) = [ ] s1,..., s d /x f(1),..., x f(d) R f S d s where S = s 1... s d and x 1,..., x d are the d free occurrences of x in s. In this expression, S d stands for the group of all permutations on the set {1,..., d}. This notion of reduction extends to all simple (poly-)terms, using the fact that all constructions of the syntax are linear. For instance, if s 1,..., s n and for each i, s i reduces to s i R, then the simple poly-term s 1... s n reduces to n i=1 s i R!. This notion of reduction is a relation from (!) to R (!) ; it is extended to a relation from R (!) to itself by linearity. This relation is conuent, and strongly normalizing when R = N. We use 0 for the set of all normal simple terms, and NF for the normalization map N (!) N (!) 0, which is linear. The following lemma is straightforward and will be useful in the sequel. Lemma 3 Let s 1,..., s p be simple terms and T 1,..., T p be simple poly-terms. Then p Supp( x (s i, T i )) Supp( x (s 1... s p, T 1... T p )). i=1 5

6 Taylor expansion of ordinary lambda-terms. Let us give an intuition of the resource lambda-calculus, explaining why it is related to the idea of Taylor expansion. Usually, when f is a suciently regular function from a vector space E to a vector space F (nite dimensional spaces, or Banach spaces, typically), at all point x E, f has nth derivatives for all n N, and these derivatives are maps f (n) : E E n F with the same regularity as f and such that f (n) (x, u 1,..., u n ) = f (n) (x) (u 1,..., u n ) is n-linear and symmetric in u 1,..., u n. When one is lucky, and at least locally around 0, the Taylor formula holds: f(x) = n=0 1 n! f (n) (0) (u,..., u). If we want to Taylor-expand lambda-terms, which after all are functions, we need to extend the language with explicit dierentials, or more precisely a construction of dierential application of a term M to n terms N 1,..., N n, as we did in [ER03] (a simplied version of that caclulus will be available soon in [Vau05]). The idea is that if M represents a function f from E to F and if N 1,..., N n represent n vectors u 1,..., u n E, then this new construction D n M (N 1,..., N n ) will represent the function from E to F which maps x to f (n) (x) (u 1,..., u n ), and therefore this linear and symmetric in the N i 's. So the Taylor expansion of a single lambda-calculus application (M) N would read n=0 1 n! (Dn M (N,..., N)) 0. If we want to Taylor expand all the applications occurring in a lambda-term, we see that the usual lambdacalculus application in its generality will become useless: only application to 0 is needed. This is exactly the purpose of the construction s s 1... s n of the resource lambda-calculus; with the notations of the dierential lambda-calculus, the expression s s 1... s n stands for (D n s (s 1,..., s n )) 0. So the resource lambda-calculus is a language for writing complete Taylor expansions of ordinary lambdaterms. The expansion of a term M will be an innite linear combination of resource terms, with rational coecients (actually of inverses of positive integers), let us use M for this expansion. By what we said, it should obey (M) N 1 = n! M (N ) n n=0 as well as x = x and (λx M) = λx M. From these equations, we obtain, applying the multinomial formula, that M = 1 m(s) s s T (M) where T (M) is dened inductively by T (x) = {x}, T (λx M) = {λx s s T (M)} and T ((M) N) = { s S s T (M) and S M fin (T (N))}. The positive number m(σ) associated to each (poly-)term σ is called is multiplicity coecient; see its denition and properties in [ER04b]. We can recall now the main result proven in that paper. Theorem 4 Let M be an ordinary lambda-term. 1. If s, s T (M), then Supp(NF(s)) Supp(NF(s )) = or s = s (syntactic equality, up to α-conversion only). 2. If s T (M) and u Supp(NF(s)), then NF(s) u = m(s)/m(u). For proving the rst part of this, we have been led to introduce a coherence relation on resource (poly- )terms, in such a way that T (M) is a maximal clique for this coherence relation, for any ordinary term M. 6

7 So, given an ordinary lambda-term M, it makes sense to apply NF to each of the simple terms occurring in its Taylor expansion, dening NF(M ) = 1 m(s) NF(s) s T (M) 1 and then what we get is a sum of normal simple resource terms u, each of them with coecient m(u). We want to prove that this sum is equal to BT(M), the Taylor expansion of the Böhm tree of M. To give a meaning to this notion, we need rst to dene T (B) when B is an EBT: the denition is the same as for ordinary lambda-terms, with the additional clause that T (Ω) =. For instance T ((x) Ω) = { x 1}. Observe by the way that B C T (B) T (C). We generalize this notion to arbitrary Böhm trees: T (B) = B B T (B) (this a directed union since B is an ideal). Of course, all these resource terms are normal. The obtained set is still a clique of (, ), but this clique has no reason to be maximal anymore (think of the clique associated to Ω!). Given a Böhm tree B, it makes sense nally to dene its Taylor expansion, as we did for ordinary lambda-terms: B = 1 m(b) b. b T (B) 2.2 Resource closures and resource stacks. We adapt now the concepts of closure and stack to the framework of the resource lambda-calculus, introducing multi-set based versions thereof. We stick to our multiplicative conventions for denoting multi-sets. A resource environment is a total function e on variables, taking resource closures or the symbol free as values. We extend pointwise the multi-set notations to resource environments, e.g. (ee )(x) = e(x)e (x) (equal to free when one of these two values is equal to free). For an environment e, we require moreover e(x) = 1 for almost all variables x, where 1 is the unit resource closure (see below). If x is a variable and c is a resource closure, we denote by [x c] the resource environment which takes the value 1 for all variables but for x, for which it takes the value c. If e is a resource environment, e \ x denotes the resource environment which takes the same values as e but for x where it takes the value free. We use Dom c e for the (co-nite) set of all variables where e does not take the value free. A resource closure is a pair γ = (T, e) where T is a simple resource poly-term and e is a resource environment, or is the special unit closure 1. Intuitively, this special closure is equal to any closure of the shape (1, e) where e maps all variables to free, to the unit closure 1 or to any closure of the shape we are now describing. Poly-term multiplication is extended to closures in the obvious way: the unit closure 1 is neutral, and (T, e)(t, e ) = (T T, ee ). A resource closure (T, e) will be said to be elementary if T has exactly one element. closures are product (in many dierent ways, usually) of elementary resource closures. All resource Finally, a resource stack π is a nite sequence of resource closures. A resource state is a triple (t, e, π) where t is a simple resource term, e is a resource environment and π is a resource stack. In such a resource state, the pair (t, e) will be considered as an elementary resource closure. By mutual induction, we dene T (E) and T (Γ), the set of all resource environments and resource closures of shape E and Γ respectively: T (E) is the set of all resource environments e such that if E(x) = free, then e(x) = free; otherwise and if E(x) is dened, then e(x) T (E(x)); if E(x) is undened, then e(x) = 1. 7

8 If Γ = (M, E), then T (Γ) = M fin (T (M)) T (E) {1}. This extends to standard stacks and resource stacks in the obvious way, dening π T (Π). Last we set T (Γ, Π) = T (Γ) T (Π). As we did for the ordinary lambda-calculus, we associate to each resource closure γ a (generally not simple) resource poly-term T D (γ) R! by the following inductive denition { 1 if γ = 1 T D (γ) = x1,...,x n (T, T D (e(x 1 )),..., T D (e(x n ))) if γ = (T, e) where x 1,..., x n is any repetition-free sequence of variables which contains all the variables Dom c e which are free in T or satisfy e(x) 1 (in particular, this expression is equal to 0 if there exists a variable x not free in T and such that e(x) 1). Due to the basic properties of partial derivatives explained in [ER04b], the expression above does not depend on the choice of this sequence of variables. Last, we extend this denition to resource states (γ, π) where π = (γ 1,..., γ k ) is a resource stack (the π i 's are therefore multisets of resource closures), setting T D (γ, π) = T D (γ) T D (γ 1 ) T D (γ k ) N. 3 A resource driven Krivine machine We dene a new version K of the Krivine machine which, fed with an ordinary closure Γ, an ordinary stack Π and a normal resource term u, will return a pair (γ, π) T (Γ, Π) where γ is an elementary resource closure, or will be undened. As we shall see, this partial function K will be dened exactly on the triples (Γ, Π, u)) such that u T (BT(T(Γ, Π))). We use the symbol for the result of the function when it is undened. As before, we dene by induction on n an increasing sequence of partial functions K n and we set K = K n=0 n. The base case is trivial: K 0 (Γ, Π, t) =, always. The inductive step is by case on the shape of the rst element of the closure Γ = (M, E) (remember that we assume that FV(M) Dom E). If M = x is a variable, we have two subcases. Assume rst that x Dom c (E). If K n (E(x), Π, u) =, then K n+1 (Γ, Π, u) = and otherwise, let (γ, π) = K n (E(x), Π, u), then γ if y = x K n+1 (M, E, Π, u) = (x, e, π) where e(y) = free if E(y) = free 1 otherwise. Otherwise, we have x Dom(E) and E(x) = free. The stack Π is a sequence (Γ 1,..., Γ k ) of closures. If u = x V 1 V k and for each j = 1,..., k and v supp(v j ), there exists an elementary resource closure γ j (v) such that K n (Γ j,, v) = (γ j (v), ), then { free if E(y) = free K n+1 (M, E, Π, u) = (x, e, π) where e(y) = 1 otherwise. and where π = (c 1,..., c k ) with c j = v supp(v j) γ j(v) Vj(v) (this product has to be understood as a product of resource closures, in the sense dened above). 8

9 Otherwise, K n+1 (M, E, Π, u) =. Assume now that M = λx N. Without loss of generality, we can assume that E(x) =. Again, we have two subcases. Assume rst that Π = is the empty stack. If u = λx v and K n (N, E x free,, v) = (t, e, ) with e(x) = free, then K n+1 (M, E,, u) = (λx t, e x 1, ) and otherwise, K n+1 (M, E,, u) =. Assume next that Π = Γ :: Π. If K n (N, E x Γ, Π, u) = (t, e, π ) with e(x) free, then and otherwise, K n+1 (M, E,, u) =. K n+1 (M, E, Π, u) = (λx t, e x 1, e(x) :: π ) Last assume that M = (P ) Q. If K n (P, E, (Q, E) :: Π, u) = (t, e, (T, e ) :: π), then and otherwise, K n+1 (M, E,, u) =. K n+1 (M, E, Π, u) = ( t T, ee, π) Lemma 5 Let Γ be an ordinary closure, Π be an ordinary stack and u be a simple resource term. If K(Γ, Π, u) is dened, then u is normal and K(Γ, Π, u) is a resource state (γ, π) which belongs to T (Γ, Π). The proof is by simple inspection of the denition above. Lemma 6 Let Γ be an ordinary closure, Π be an ordinary stack and u be a normal simple resource term. For each n N, we have the following equivalence: u T (K n (Γ, Π)) i K n (Γ, Π, u) is dened. Proof. By induction on n. For n = 0, this is trivial. Assume that the result is true for n, and consider the various following cases, according to the shape of Γ and Π. When needed, we give names to the elements of Π: (Γ 1,..., Γ k ) = Π. Assume that Γ = (x, E) with E(x) = free. Then u T (K n+1 (Γ, Π)) holds i u = x V 1 V k with V j M fin (K n (Γ j, )) for each j = 1,..., k. By inductive hypothesis, this latter property holds i for each j and each v supp(v j ), we have that K n (Γ j,, v) is dened. But this in turn is equivalent to saying that K n+1 (Γ, Π, u) is dened. Assume that Γ = (x, E) with E(x) free. Then u T (K n+1 (Γ, Π)) holds i u T (K n (E(x), Π)) which, by inductive hypothesis, holds i K n (E(x), Π, u) is dened, and this latter condition is equivalent to saying that K n+1 (Γ, Π, u) is dened. The cases Γ = ((M) N, E) and Γ = (λx t, E) with Π are handeled similarly. We are left with the case Γ = (λx M, E) with Π = which is straightforward. Lemma 7 Let Γ be an odinary closure, Π be an ordinary stack and u be a normal simple resource term. Let n N. If K n (Γ, Π, u) = (γ, π), then u Supp(NF(T D (γ, π))). 9

10 Proof. When needed, we use (Γ 1,..., Γ k ) = Π. The proof is by simple induction on n. For n = 0, this is trivial. So assume that K n+1 (Γ, Π, u) = (γ, π). We have several cases to consider, depending on Γ and Π. Assume rst that Γ = (x, E) with E(x) = free. Then by our hypothesis and by the denition of K, we know that u = x V 1 V k and that for each j = 1,..., k and v supp(v j ), there is a resource closure γ j (v) such that K n (Γ j,, v) = (γ j (v), ). Moreover, we know that γ = (x, e) with { free if E(y) = free e(y) = 1 otherwise, so that T D (γ) = x. By inductive hypothesis, for each j = 1,..., k and v supp(v j ), we have v Supp(NF(T D (γ j (v), ))). But we also know that π = (c 1,..., c k ) with c j = v supp(v γ j) j(v) Vj(v), so that V j Supp(NF(T D (c j ))) and nally u Supp(NF(T D (γ, π))) as required. Assume next that Γ = (x, E) with x Dom c E. Then we know that K n (E(x), Π, u) is dened, with value (γ, π) and that γ = (x, e) with e given by γ if y = x e(y) = free otherwise and if E(y) = free 1 otherwise. By inductive assumption, we have u Supp(NF(T D (γ, π))) and we conclude since T D (γ) = T D (γ ). The case Γ = (λx M, E) and Π = is straightforward. Assume that Γ = (λx M, E) and Π = Γ :: Π. Assume also that E(x) is undened. Then we know that K n (M, E x Γ, Π, u) is dened, with value (t, e, π ), and K n+1 (Γ, Π, u) = (λx t, e x 1, e(x) :: π ) (that is γ = (λx t, e x 1 ) and π = e(x) :: π ). Let us set (γ 1,..., γ k ) = π and e = e x free. By inductive assumption we have u Supp(NF(T D (t, e, π ))). We have NF(T D (λx t, e x 1, e(x) :: π )) = NF(... λx T D (t, e ) T D (e(x)) T D (γ 1 )... T D (γ k )). Let x 1,..., x p be an enumeration without repetitions of all the free varibles of t which do belong to Dom c e, so that the variable x does not occur in that list. We have T D (t, e ) = x1,...,x p (t, T D (e(x 1 )),..., T D (e(x p ))) by denition of T D and of e and hence NF(T D (λx t, e x 1, e(x) :: π )) = NF( x1,...,x p,x(t, T D (e(x 1 )),..., T D (e(x p )), T D (e(x))) T D (γ 1 ) T D (γ k )) = NF(T D (t, e, π )) so that u Supp(NF(T D (γ, π))). Assume last that Γ = ((M) N, E). We know that K n (M, E, (N, E) :: Π, u) is dened, with value (t, e, (S, e ) :: π), and K n+1 (Γ, Π) = ( t S, ee, π), that is γ = ( t S, ee ). By inductive hypothesis, we know that u Supp(NF(T D (t, e, (S, e ) :: π))). Applying Lemma 3 and the denition of T D we get the inclusion Supp(T D (t, e, (S, e ) :: π)) Supp(T D ( t S, ee, π)), which is strict in general. We conclude, applying the NF operator to both sides. 10

11 4 Normal form of the Taylor expansion Lemma 8 Let M and N be ordinary lambda-terms and let x be a variable. M fin (T (N)). Then Supp( x (s, S)) T (M [N/x]). Let s T (M) and S The proof is a straightforward induction on M, using Lemma 3 when M is an application. Lemma 9 Let M be an ordinary lambda-term and let s T (M). Then Supp(NF(s)) T (BT(M)). Proof. One denes a family of function NF n : N, mimicking the denition of BT. We set NF 0 (t) = 0 and NF n+1 (λx 1... x p x S 1 S k ) = λx 1... x p x NF n (S 1 ) NF n (S k ) and NF n+1 (λx 1... x p λx s S S 1 S k ) = NF n (λx 1... x p x (s, S) S 1 S k ), where NF n is extended multiplicatively to simple poly-terms, and linearly to linear combinations of simple (poly-)terms. One checks easily that for any s, there exists n such that NF(s) = NF n (s) (take any n S(s)). One concludes by proving by induction on n that Supp(NF n (s)) T (BT n (M)), using Lemma 8. Now we can prove the main theorem of the paper. Theorem 10 Let M be an ordinary lambda-term and let u be a normal simple resource term. Then u T (BT(M)) if and only if there exists s T (M) such that u Supp(NF(s)). Moreover, when this simple term s exists, it is unique. Proof. Assume rst that u T (BT(M)). Let E be an environment such that E(x) = free for all x FV(M). Since BT(M) = K(M, E, ) by Theorem 2, we obtain by Lemma 6 that K(M, E,, u) is dened ant takes a value (s, e, ) with (s, e) T (M, E) by Lemma 5. Moreover, by Lemma 7 we have u Supp(NF(T D (s, e))) = Supp(NF(s)) since e takes the value free for all free variables of s (indeed, FV(s) FV(M)). Assume conversely that u Supp(NF(s)) for some s T (M). By Lemma 9, we have u T (BT(M)), so again K(M, E,, u) is dened ant takes a value (s, e, ) with (s, e) T (M, E) and hence u Supp(NF(s )). But both s and s belong to T (M) and hence they must be equal by Theorem 4 and we conclude. The unicity statement also results from Theorem 4. As a consequence of this theorem and of the quantitative part of Theorem 4, we get immediately the announced commutation statement. Theorem 11 Let M be an ordinary lambda-term. Then BT(M) = NF(M ) = s T (M) 5 Concluding remarks and perspectives 1 m(s) NF(s). By Theorem 10, there exists a partial function E : Λ 0 such that E(M, u) is dened if and only if u T (BT(M)) and then takes as value the unique simple term s T (M) such that u Supp(NF(s)). In the proof of that theorem, we have seen how this function E can be dened, using a modied version of Krivine machine. When BT(M) is a variable, the situation is particularly simple: we have T (BT(M)) = { } and E(M, ) is the unique s T (M) which has a non-zero normal form, and the normal form of s must be m(s). In that particular case, it is interseting to observe that S(s) is the number of steps in the reduction of M to 11

12 by the Krivine machine, which in terms seems to be a sensible measure of the complexity of the reduction of M. The map S E : Λ 0 N seems therefore to provide more generally a way of measuring the complexity of the reduction of lambda-terms. The interesting point is that this measure is associated to the algebraic property stated by Theorems 10 and 11. The next step would be to adapt these constructions to a denotational setting, observing that the size of a normal simple term can be read in its relational denotational semantics. This might lead to a new denotational way of dealing with the computational complexity of lambda-terms. References [BCL99] Gérard Boudol, Pierre-Louis Curien, and Carolina Lavatelli. A semantics for lambda calculi with resource. Mathematical Structures in Computer Science, 9(4):437482, [Bou93] [DB87] [DR99] Gérard Boudol. The lambda calculus with multiplicities. Technical Report 2025, INRIA Sophia- Antipolis, N.G. De Bruijn. Generalizing Automath by means of a lambda-typed lambda calculus. In D.W. Kueker, E.G.K. Lopez-Escobar, and C.H. Smith, editors, Mathematical Logic and Theoretical Computer Science, Lecture Notes in Pure and Applied Mathematics, pages Marcel Dekker, Reprinted in: Selected papers on Automath, Studies in Logic, volume 133, pages , North- Holland, Vincent Danos and Laurent Regnier. Reversible, irreversible and optimal lambda-machines. Theoretical Computer Science, 227(1-2):273291, [ER03] Thomas Ehrhard and Laurent Regnier. The dierential lambda-calculus. Theoretical Computer Science, 309(1-3):141, [ER04a] Thomas Ehrhard and Laurent Regnier. Dierential interaction nets. In Proceedings of WoLLIC'04, Electronic Notes in Theoretical Computer Science. Elsevier Science, To appear. [ER04b] Thomas Ehrhard and Laurent Regnier. Uniformity and the Taylor expansion of ordinary lambdaterms. Technical report, Institut de mathématiques de Luminy, [Kfo00] Assaf J. Kfoury. A linearization of the lambda-calculus. Journal of Logic and Computation, 10(3):411436, [Kri05] [Vau05] Jean-Louis Krivine. A call-by-name lambda-calculus machine. Higher-Order and Symbolic Computation, To appear. Lionel Vaux. The dierential lambda-mu calculus. Technical report, Institut de Mathématiques de Luminy, To appear. 12