B¨ ohm’s Theorem for Resource Lambda Calculus through Taylor Expansion

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B¨ ohm’s Theorem for Resource Lambda Calculus through Taylor Expansion

^?

Giulio Manzonetto¹ and Michele Pagani²

1 Intelligent Systems, Radboud University [email protected]

2 Laboratoire LIPN, CNRS UMR7030 Universit´e Paris 13 [email protected]

Abstract. We study the resource calculus, an extension of theλ-calculus allowing to model resource consumption. We achieve an internal separation result, in analogy with B¨ohm’s theorem ofλ-calculus. We define an equivalence relation on the terms, which we prove to be the maximal non-trivial congruence on normalizable terms respectingβ-reduction. It is significant that this equivalence extends the usualη-equivalence and is related to Ehrhard’s Taylor expansion – a translation mapping terms into series of finite resources.

Keywords:differential linear logic, resource lambda-calculus, separation property, B¨ohm-out technique.

Introduction

Böhm’s theorem in the λ-calculus. Böhm’s theorem [1] is a fundamental result in the untyped λ-calculus [2] stating that, given two closed distinct βη- normalλ-termsM andN, there exists a sequence ofλ-terms~L, such thatM ~L β- reduces to the first projectionλxy.xandN ~L β-reduces to the second projection λxy.y. The original issue motivating this result was the quest for solutions of systems of equations betweenλ-terms: given closed termsM1, N1, . . . ,Mn, Nn, is there a λ-term S such that SM1 ≡β N1 ∧ · · · ∧ SMn ≡β Nn holds? The answer is trivial for n = 1 (just take S = λz.N1 for a fresh variable z) and Böhm’s theorem gives a positive answer for n = 2 and M1, M2 distinct βη- normal forms (apply the theorem to M1, M2 and set S = λf.f ~LN1N2). The result has been then generalized in [3] to treat every finite familyM1, . . . , Mn of pairwise distinctβη-normal forms. This generalization is non-trivial since each M_i may differ from the other ones at distinct addresses of its syntactic tree.

As an important consequence of B¨ohm’s theorem we have that the βη- equivalence is the maximal non-trivial congruence on normalizable terms extending the β-equivalence. The case of non-normalizable terms has been addressed by Hyland in [4]. Indeed, theβη-equivalence is not maximal on such terms, and one must consider the congruenceH^? equating twoλ-terms whenever they have

?This work is partly supported by NWO Project 612.000.936 CALMOC and the chaire CNRS “Logique lin´eaire et calcul”.

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the same B¨ohm tree up to possibly infinite η-expansions [2, §16.2]. Then one proves that for all closed λ-terms M 6≡_H^?N there is a sequence of λ-terms ~L such thatM ~L β-reduces to the identityλx.xwhileN ~Lis unsolvable (i.e., it does not interact with the environment [2,§8.3]) orvice versa. This property is called semi-separation because of the asymmetry between the two values: the identity, on the one hand, and any unsolvable term, on the other hand. In fact, non- normalizable terms represent partial functions, and one cannot hope to separate a term less defined than another one without sending the first to an unsolvable term (corresponding to the empty function). Despite the fact that Hyland’s semi- separability is weaker than the full separability achieved by B¨ohm’s theorem, it is sufficient to entail thatH^? is the maximal non-trivial congruence onλ-terms extending β-equivalence and equating all unsolvable terms [2, Thm. 16.2.6].

The resourceλ-calculus.We study B¨ohm’s theorem in the resourceλ-calculus (Λ^r for short), which is an extension of the λ-calculus along two directions.

First,Λ^r is resource sensitive. Following Girard’s linear logic [5], the λ-calculus application can be written as M(N^!) emphasizing the fact that the argument N is actually infinitely available for the function M, i.e. it can be erased or copied as many times as needed during the evaluation. Λ^r extends this setting by allowing also applications of the form M(Nⁿ) where Nⁿ denotes a finite resource that must be used exactlyn-times during the evaluation. If the number n does not match the needs ofM then the application evaluates to the empty sum 0, expressing the absence of a result. In fact, 0 is a β-normal form giving a notion of unsolvable different from the λ-calculus one represented by looping terms³. The second feature ofΛ^r is the non-determinism. Indeed, the argument of an application, instead of being a single term, is a bag of resources, each being either finite or infinitely available. In the evaluation several possible choices arise, corresponding to the different possibilities of distributing the resources among the occurrences of the formal parameter. The outcome is a finite formal sum of terms collecting all possible results.

Boudol has been the first to extend theλ-calculus with a resource sensitive application [8]. His resource calculus was designed to study Milner’s encoding of the lazyλ-calculus into theπ-calculus [9,10]. Some years later, Ehrhard and Reg- nier introduced the differential λ-calculus [11], drawing on insights gained from the quantitative semantics of linear logic, denoting proofs/terms as smooth (i.e.

infinitely differentiable) functions. As remarked by the authors, the differential λ-calculus is quite similar to Boudol’s calculus, the resource sensitive applica- tionM(Nⁿ) corresponds to applying the n-th derivative of M at 0 to N. This intuition was formalized by Tranquilli, who defined the present syntax ofΛ^rand showed a Curry-Howard correspondence between this calculus and Ehrhard and Regnier’s differential nets [12]. The main differences between Boudol’s calculus andΛ^rare that the former is equipped with explicit substitution and lazy operational semantics, while the latter is a true extension of the regular λ-calculus.

Since we cannot separate M from M +M we will conveniently suppose that

3 Denotational models ofΛ^rdistinguishing between 0 and the usual unsolvable terms are built in [6]. For more details on the notion of solvability inΛ^r see [7].

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the sum onΛ^ris idempotent as in [13]; this amounts to say that we only check whether a term appears in a result, not how many times it appears.

A resource conscious B¨ohm’s theorem.A notable outcome of Ehrhard and Regnier’s work has been to develop theλ-calculus application as an infinite series of finite applications, M(N^!) = P∞

n=0 1

n!M(Nⁿ), in analogy with the Taylor expansion of the entire functions. In [14], the authors relate the B¨ohm tree of a λ-term with its Taylor expansion, giving the intuition that the latter is a resource conscious improvement of the former. Following this intuition, we achieve the main result of this paper, namely a separation property in Λ^r that can be seen as a resource sensitive B¨ohm’s theorem (Theorem 2). Such a result states that for all closedβ-normalM, Nhavingη-different Taylor expansion, there is a sequence L, such that~ M ~L β-reduces toλx.xandN ~L β-reduces to 0, or vice versa.

This theorem reveals a first sharp difference betweenΛ^r and theλ-calculus, as our result is much similar to Hyland’s semi-separation than B¨ohm’s theorem, even if we consider the β-normal forms. This is due to the empty sum 0, the unsolvableβ-normal form, outcome of the resource consciousness ofΛ^r.

Taylor expansion is a semantical notion, in the sense that it is an infinite series of finite terms. It is then notable that we give a syntactic characterization of the Taylor equality introducing theτ-equivalence inDefinition 3(Proposition 1). As expected, our semi-separability is strong enough to entail that theητ-equivalence induces the maximal non-trivial congruence on β-normalizable terms extending theβ-equivalence (Corollary 1).

A crucial ingredient in the classic proof of B¨ohm’s theorem is the fact that it is possible to erase subterms in order to pull out of the terms their structural difference. This is not an easy task inΛ^r, since the finite resources must be con- sumed and cannot be erased. In this respect, our technique has some similarities with the one developed to achieve the separation for the λI-calculus (i.e., the λ-calculus without weakening, [2, §10.5]). Moreover, since the argument of an application is a bag of resources, comparing a difference between two terms may turn into comparing the differences between two multisets of terms, and this problem presents analogies with that of separating a finite set of terms [3].

Basic definitions and notations.We letNdenote the set of natural numbers.

Given a setX,M_f(X) is the set of all finite multisets overX. Given a reduction

→r we let←,^r −^r∗→and≡_r denote its transpose, its transitive-reflexive closure and its symmetric-transitive-reflexive closure, respectively.

An operatorF(−) (resp.F(−,−)) isextended by linearity (resp.bilinearity) by settingF ΣiAi

=ΣiF(Ai) (resp. F ΣiAi, ΣjBj

=Σi,jF(Ai, Bj)).

1 Resource Calculus

Syntax.Theresource calculus has three syntactic categories:terms that are in functional position,bagsthat are in argument position and represent unordered lists of resources, and finite formal sums that represent the possible results of a computation. Figure 1(a) provides the grammar for generating the set Λ^r of terms and the setΛ^b of bags, together with their typical metavariables.

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Λ^r: M, N, L ::=x|λx.M |M P terms Λ^b: P, Q, R ::= [M1, . . . , Mn,M^!] bags

Λ^e: A, B ::=M |P expressions

M,N∈2hΛ^ri P,Q∈2hΛ^bi A,B∈2hΛ^ei:=2hΛ^ri ∪2hΛ^bi sums (a) Grammar of terms, resources, bags, expressions, sums.

λx.(P

iMi) :=P

iλx.Mi

(P

iMi)P:=P

iMiP

M(P

iPi) :=P

iMPi

[(P

iMi)]·P:=P

i[Mi]·P (b) Notation on2hΛ^ei.

yhN/xi:=

(N ify=x, 0 otherwise, [M^!]hN/xi:= [MhN/xi,M^!],

(λy.M)hN/xi:=λy.(MhN/xi),

(M P)hN/xi:=MhN/xiP+M(PhN/xi), ([M]·P)hN/xi:= [MhN/xi]·P+ [M]·PhN/xi, (c) Linear substitution, in the abstraction case we supposey /∈FV(N)∪ {x}.

Fig. 1:Syntax, notations and linear substitution of resource calculus.

A bag [M ,~ M^!] is a compound object, consisting of a multiset of linear resources [M~] and a set of terms M presented in additive notation (see the dis- cussion on sets and sums below) representing the reusable resources. Roughly speaking, the linear resources in M~ must be used exactly once during a reduction, while the reusable ones in M can be used ad libitum (hence, following the linear logic notation, M is decorated with a ! superscript). We shall deal with bags as if they were multisets presented in multiplicative notation, defining union by [M ,~ M^!]·[N ,~ N^!] := [M , ~~ N ,(M+N)^!]. This operation is commuta- tive, associative and has the empty bag 1 := [0^!] as neutral element. To avoid confusion with application we will never omit the dot “·”. To lighten the notations we write [L₁, . . . , L_k] for the bag [L₁, . . . , L_k,0^!], and [M^k] for the bag [M, . . . , M] containing k copies of M. Such a notation allows to decompose a bag in several ways, and this will be used throughout the paper. As for example:

[x, y,(x+y)^!] = [x]·[y,(x+y)^!] = [x^!]·[x, y, y^!] = [x, y]·[(x+y)^!] = [x, x^!]·[y, y^!].

Expressions (whose set is denoted by Λ^e) are either terms or bags and will be used to state results holding for both categories.

Let 2be the semiring {0,1} with 1 + 1 = 1 and multiplication defined in the obvious way. For any setX, we write2hX ifor the free 2-module generated by X, so that 2hX i is isomorphic to the finite powerset of X, with addition corresponding to union, and scalar multiplication defined in the obvious way.

However we prefer to keep the algebraic notations for elements of 2hX i, hence set union will be denoted by + and the empty set by 0. This amounts to say that 2hΛ^ri (resp. 2hΛ^bi) denotes the set of finite formal sums of terms (resp.

bags), with an idempotent sum. We also set2hΛ^ei=2hΛ^ri ∪2hΛ^bi. This is an abuse of notation, as 2hΛ^eihere does not denote the 2-module generated over

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Λ^e=Λ^r∪Λ^b but rather the union of the two2-modules; this amounts to say that sums may be taken only in the same sort.

Thesize of A∈ 2hΛ^ei is defined inductively by: size(Σ_iA_i) = Σ_isize(A_i), size(x) = 1, size(λx.M) = size(M) + 1, size(M P) = size(M) + size(P) + 1, size([M1, . . . , Mk,M^!]) =Σ_i=1^k size(Mi) + size(M) + 1.

Notice that the grammar for terms and bags does not include any sums, but under the scope of a (·)^!. However, as syntactic sugar – andnot as actual syntax – we extend all the constructors to sums as shown in Figure 1(b). In fact all constructors except the (·)^! are (multi)linear, as expected. The intuition is that a reusable sum (M +N)^! represents a resource that can be used several times and each time one can choose non-deterministicallyM or N.

Observe that in the particular case of empty sums, we getλx.0 := 0,M0 := 0, 0P:= 0, [0] := 0 and 0·P := 0, but [0^!] = 1. Thus 0 annihilates any term or bag, except when it lies under a (·)^!. As an example of thisextended (meta-)syntax, we may write (x₁+x₂)[y₁+y₂,(z₁+z₂)^!] instead ofx₁[y₁,(z₁+z₂)^!]+x₁[y₂,(z₁+ z₂)^!] +x₂[y₁,(z₁+z₂)^!] +x₂[y₂,(z₁+z₂)^!]. This kind of meta-syntactic notation is discussed thoroughly in [14].

Theα-equivalence and the set FV(A) of free variables are defined as in or- dinaryλ-calculus. From now on expressions are considered up toα-equivalence.

Concerning specific terms we set:

I:=λx.x, Xn:=λx1. . . λxnλx.x[x^!₁]. . .[x^!_n] forn∈N, whereXn is called then-th B¨ohm permutator.

Due to the presence of two kinds of resources, we need two different notions of substitutions: the usual λ-calculus substitution and a linear one, which is particular to differential and resource calculi (see [14,15]).

Definition 1 (Substitutions).We define the following substitution operations.

1. A{N/x} is the usual capture-free substitution of N for x in A. It is extended to sums as in A{N/x} by linearity in A, and using the notations ofFigure 1(b)forN.

2. AhN/xi is the linear substitution defined inductively in Figure 1(c). It is extended toAhN/xiby bilinearity in both AandN.

Intuitively, linear substitution replaces the resource toexactly onelinear free occurrence of the variable. In presence of multiple occurrences, all possible choices are made and the result is the sum of them. E.g., (x[x])hI/xi=I[x] +x[I].

Notice the difference between [x, x^!]{M+N/x}= [(M +N),(M +N)^!] = [M,(M+N)^!]+[N,(M+N)^!] and [x, x^!]hM+N/xi= [x, x^!]hM/xi+[x, x^!]hN/xi= [M, x^!] + [x, M, x^!] + [N, x^!] + [x, N, x^!].

Linear substitution bears resemblance to differentiation, as shown clearly in Ehrhard and Regnier’s differential λ-calculus [11]. For instance, it enjoys the following Schwarz lemma, whose proof is rather classic and is omitted.

Lemma 1 (Schwarz Lemma [14,11]). Given A∈2hΛ^ei, M,N∈2hΛ^riand y /∈FV(M)∪FV(N) we have AhM/yihN/xi =AhN/xihM/yi+AhMhN/xi/yi.

In particular, if x /∈FV(M)the two substitutions commute.

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M RM

λx.M Rλx.M lam M RM

M P RMP appl P RP M P RMP

appr

M RM

[M]·P R[M]·P lin M RM [M^!]·P R[M^!]·P

bng ARA

A+BRA+B sum

(a) Rules defining the context closure of a relationR⊆Λ^e×2hΛ^ei.

MRN

λx.MRλx.N lam MRN PRQ MPRNQ

app

MRN PRQ

[M]·PR[N]·Q lin MRN

[M^!]R[N^!] bng ARB A⁰RB⁰ A+A⁰RB+B⁰

sum

(b) Rules defining a compatible relationR⊆2hΛ^ei ×2hΛ^ei.

Fig. 2:Definition of context closure and compatible relation.

Operational semantics. Given a relation R⊆Λê×2hΛêi itscontext closure is the smallest relation in2hΛêi ×2hΛêicontainingR and respecting the rules ofFigure 2(a). The main notion of reduction of resource calculus isβ-reduction, which is defined as the context closure of the following rule:

(β) (λx.M)[L1, . . . , Lk,N^!]→^β MhL1/xi · · · hLk/xi{N/x}.

Notice that theβ-rule is independent of the ordering of the linear substitutions, as shown by the Schwarz lemma above. We say thatA∈Λ^eis inβ-normal form (β-nf, for short) if there is no Asuch that A →^β A. A sum Ais in β-nf if all its summands are. Notice that 0 is a β-nf. It is easy to check that a termM is in β-nf iffM =λx₁. . . x_n.yP₁. . . P_k for n, k ≥ 0 and, for every 1≤i ≤k, all resources inP_i are in β-nf. The variabley in M is calledhead-variable.

The regularλ-calculus [2] can be embedded into the resource one by translat- ing every applicationM N intoM[N^!]. In this fragment theβ-reduction defined above coincides with the usual one. Hence the resource calculus has usual looping terms like Ω:= (λx.x[x^!])[(λx.x[x^!])^!], but also terms like I1 or I[y, y] reducing to 0 because there is a mismatch between the number of linear resources needed by the functional part of the application and the number it actually receives.

Theorem 1 (Confluence [15]). Theβ-reduction is Church-Rosser onΛ^r. The resource calculus isintensional, indeed just like in theλ-calculus there are different programs having the same extensional behaviour. In order to achieve an internal separation, we need to consider theη-reductionthat is defined as the contextual closure of the following rule:

(η) λx.M[x^!]→^η M, ifx /∈FV(M).

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(P

iAi)^◦:=S

iA^◦_i x^◦:={x} (λx.M)^◦:=λx.M^◦ (M P)^◦:=M^◦P^◦ ([M^!])^◦:=Mf(M^◦) ([M]·P)^◦:= [M^◦]·P^◦

Fig. 3:Taylor expansionA^◦ofA.

The Taylor expansion.Thefinite resource calculusis the fragment of resource calculus having only linear resources (every bag has the set of reusable resources empty). The terms (resp. bags, expressions) of this sub-calculus are calledfinite and their set is denoted by Λ^r_f (resp. Λ^b_f, Λ^e_f). Notice that the bags of Λ^b_f are actually finite multisets. It is easy to check that the above sets are closed under β-reduction, while η-reduction cannot play any role here.

InDefinition 2, we describe the Taylor expansion as a map (·)^◦ from 2hΛ^ri (resp.2hΛ^bi) to possibly infinite sets of finite terms (resp. finite bags). The Taylor expansion defined in [11,14], in the context ofλ-calculus, is a translation devel- oping every application as an infinite series of finite applications with rational coefficients. In our context, since the coefficients are in2, the Taylor expansion of an expression is a (possibly infinite) set of finite expressions. Indeed, for all sets X, the set of the infinite formal sums2hX i∞with coefficients in2is isomorphic to the powerset ofX. Our Taylor expansion corresponds to the support⁴ of the Taylor expansion taking rational coefficients given in [11,14].

To lighten the notations, we adopt for sets of expressions the same abbrevia- tions introduced for finite sums inFigure 1(b). E.g., givenM ⊆Λ^r_f andP,Q ⊆ Λ^b_f we haveλx.M={λx.M |M ∈ M}andP · Q={P·Q| P∈ P, Q∈ Q}.

Definition 2. Let A∈ 2hΛ^ei. The Taylor expansion of A is the set A^◦ ⊆Λ^e_f which is defined (by structural induction onA) in Figure 3.

As previously announced, the Taylor expansion of an expressionAcan be infinite, e.g., (λx.x[x^!])^◦={λx.x[xⁿ]|n∈N}. Different terms may share the same Taylor expansion: (x[(z[y^!])^!])^◦= (x[(z1 +z[y, y^!])^!])^◦. The presence of linear resources permits situations whereM^◦(N^◦, e.g. M :=x[x, x^!], N :=x[x^!]. The presence of non-determinism allows to build terms likeM1:=x[(y+z)^!], M2:=x[(y+h)^!] such that M₁^◦∩M₂^◦={x[yⁿ]|n∈N}is infinite. However the intersection can also be finite as inN1:=x[y, z^!], N2:=x[z, y^!] whereN₁^◦∩N₂^◦={x[y, z]}.

2 A Syntactic Characterization of Taylor Equality

In Λ^r, there are distinct βη-nf’s which are inseparable, in contrast with what we have in the regularλ-calculus. In fact, all normal forms having equal Taylor expansions are inseparable, since the first author proved in [16] that there is a non-trivial denotational model of resource calculus equating all terms having the same Taylor expansion. For example,x[(z[y^!])^!] andx[(z1 +z[y, y^!])^!] are distinct inseparableβη-nf’s since they have the same Taylor expansion.

4 I.e., the set of those finite terms appearing in the series with a non-zero coefficient.

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Because of its infinitary nature, the property of having the same Taylor expansion is more semantical than syntactical. In this section, we provide an al- ternative syntactic characterization (Definition 3,Proposition 1).

A relation R ⊆ 2hΛêi ×2hΛêi is compatible if it satisfies the rules in Fig- ure 2(b). The congruence generated by a relation R, denoted ≡R, is the smallest compatible equivalence relation containing R. Given two relations R,S ⊆ 2hΛêi ×2hΛêi, we write≡RS for the congruence generated by their unionR∪S.

Definition 3. The Taylor equivalence≡τ is the congruence generated by:

(τ) [M^!]≡τ1 + [M, M^!]

Moreover, we set Avτ BiffA+B≡τ B.

It is not difficult to check thatvτ is a compatible preorder. We now prove that it captures exactly the inclusion between Taylor expansions (Proposition 1).

E.g.,z[x, x^!] +z1≡τ z[x^!]vτ z[x, x^!] +z[y^!], whilex[x^!]6≡τ x[y^!]6vτx[y, y^!].

Note that all elements ofA^◦share the same minimum structure, called here skeleton, obtained by taking 0 occurrences of every reusable resource.

Definition 4. Given A∈Λ^e, its skeleton s(A)∈Λ^e_f is obtained by erasing all the reusable resources occurring in A. That is, inductively:

s(x) :=x, s(λx.M) :=λx.s(M), s(M P) :=s(M)s(P), s([M₁, . . . , M_n,M^!]) := [s(M₁), . . . ,s(M_n)].

Obviouslys(A)∈A^◦. In general it is false thats(A)∈B^◦entailsAvτ B. Take for instanceA:=x[x^!] andB:=x[y^!]; indeeds(A) =x1∈ {x[yⁿ]|n∈N}=B^◦. The above implication becomes however true whenA is “expanded enough”.

Definition 5. Given k∈N, we say that A∈Λ^e isk-expandedif, whenever it contains a bag that can be decomposed into [M^!]·P, we have P = [M^k]·P⁰ for someP⁰ k-expanded. A sumA∈2hΛ^ei isk-expanded if all its summands are.

E.g.,x,x1,x[y⁴, x³,(x+y)^!] are 3-expanded, but the latter is not 4-expanded.

Lemma 2. Let A∈Λ^e be k-expanded for somek∈N. Then for every B ∈Λ^e such that size(B)≤k, we have thats(A)∈B^◦ entailsAvτ B.

Proof. By induction onA. The only significant case is when A is a bag, which splits in three subcases, depending on how such a bag can be decomposed.

Case I (A= [M^k, M^!]·P,P k-expanded).By definitions(A) = [s(M)^k]·s(P).

Froms(A)∈B^◦, we deduce thatB=Q1·Q2·Q3whereQ1= [L1, . . . , L`],Q2= [(N1+· · ·+Nn)^!] for some `, n≥0 and Q1, Q2, Q3 are such that [s(M)^`]∈Q^◦₁, [s(M)^k−`]∈Q^◦₂ ands(P)∈(Q2·Q3)^◦. From size(B)≤kwe get` < k and this entailsn >0. We then have thats(M)∈L^◦_i, for everyi≤`, and there is ajsuch that s(M) ∈N_j^◦. By induction hypothesis, we haveM vτ Li for every i ≤ `, M vτ NjandP vτ Q2·Q3. Hence, by compatibility ofvτ, we derive [M^`]vτQ1

and [M^!] vτ [N_j^!], that entails [M^k, M^!] vτ [M^`, M^!] vτ Q1·[N_j^!]. From this, using [N_j^!]·Q2≡τQ2andP vτQ2·Q3, we concludeAvτ Q1·[N_j^!]·Q2·Q3≡τ B.

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Case II (A = [M]·P, P without reusable resources). By definition s(A) = [s(M)]·s(P). Supposes(A)∈B^◦, then two subcases are possible.

Case B = [N]·Q such that s(M) ∈ N^◦ and s(P) ∈ Q^◦. By induction hypothesisM vτ NandP vτ Q. By compatibility,A= [M]·P vτ [N]·Q=B.

CaseB = [N^!]·Q such that s(M) ∈ N^◦ and s(P)∈ ([N^!]·Q)^◦. Then by induction hypothesis,M vτN andP vτ [N^!]·Q, and, always by compatibility ofvτ, we conclude thatA= [M]·P vτ[N, N^!]·Qvτ B.

Case III (A= 1). Froms(A) = 1∈B^◦, we deduceB is a bag containing only reusable resources, hence triviallyA= 1vτB. ut Lemma 3. For allA∈Λ^eandk∈Nthere is ak-expandedAsuch thatA≡τ A. Proof. By immediate structural induction onA. The crucial case is whenA= [M^!]·P. Then by induction hypothesisM ≡_τ MandP ≡_τ Pfor somek-expanded M,P. Let us explicit the first sum intoM=M1+· · ·+Mm. Then, we have:

[M^!]·P ≡τ [M^!]·P≡τ[M₁^k, . . . , M_m^k,M^!]·P+Pk−1

n₁=0· · ·Pk−1

n_m=0[M₁ⁿ¹, . . . , M_mⁿ^m]·P. Note that all summands in the last sum arek-expanded, since so areM,P. ut Proposition 1. For allA,B∈2hΛ^eiwe have thatAvτBiff A^◦⊆B^◦. Proof. (⇒) By a trivial induction on a derivation ofA+B≡τ B, remarking that all rules defining≡_τ preserve the property of having equal Taylor expansion.

(⇐) By induction on the number of terms in the sumA. IfA= 0, then clearly AvτBby the reflexivity of≡τ. IfA=A+A⁰, then by induction hypothesis we haveA⁰vτ B. As forA, letk≥max(size(A),size(B)), byLemma 3, we have ak- expanded sumA⁰⁰=A1+· · ·+Aa≡τA. FromA⁰⁰≡τ Aand the already proved left-to-right direction of the proposition we get (A⁰⁰)^◦ =A^◦ ⊆B^◦. This means that A^◦_i ⊆B^◦ for alli≤a. In particulars(Ai)∈B^◦_j_i, for a particular summand Bj_i ofB. Since we are supposing that Ai is k-expanded and size(Bj_i)≤k, we can applyLemma 2, so arguingAivτBj_ivτB. Since this holds for everyi≤a, we getA⁰⁰vτ B. Then we can concludeA≡τ A⁰+A⁰⁰vτ B+B≡τ B. ut We conclude that ≡τ deserves the name of Taylor equivalence.

3 Separating a Finite Term from Infinitely Many Terms

We now know that two βη-nf’sM,Nsuch thatM≡τ Nare inseparable; hence, in order to achieve an internal separation, we need to consider theητ-difference.

One may hope that M 6≡_ητ N is equivalent to first compute the η-nf of two β-normalM,Nand then check whether they areτ-different. Unfortunately, this is false as shown in the following counterexample (which is a counterexample to the confluence ofη-reduction modulo≡τ [17,§14.3]).

For all variables a, bwe setMa,b :=v[λz.a1, λz.b[z^!]] +v[λz.a[z, z^!], λz.b[z^!]]

andNa,b:=v[λz.a1, b] +v[λz.a[z, z^!], b]. Note thatMa,b

η∗→Na,b, hence:

Nx,y

←η∗Mx,y ≡τv[λz.x[z^!], λz.y[z^!]]≡τ My,x

η∗→Ny,x.

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Ifx6=y,Nx,yandNy,xare two inseparable but distinctβη-nf’s s.t.Nx,y6≡_τNy,x. This means that, to study theητ-difference, we cannot simply analyze the structure of the η-nf’s. Our solution will be to introduce a relation _s such that M6≡ητ Nentails either M6sNor vice versa (Definition 7). Basically, this approach corresponds to first compute the Taylor expansion of M,N and then compute theirη-nf pointwise, using the following partialη-reduction onΛ^r_f. Definition 6. The partial η-reduction →^ϕ is the contextual closure of the rule λx.M[xⁿ]−→^ϕ M ifx /∈FV(M).

We define the relation s corresponding to the Smith extension of ^ϕ∗→ and τ

corresponding to the relation “^η@_∼^η” of [2, Def. 10.2.32] keeping in mind the analogy between Taylor expansions and B¨ohm trees discussed in [14].

Definition 7. Given M,N∈2hΛ^riwe define:

−MsNiff∀M ∈M^◦,∃N ∈N^◦ such thatM −→^ϕ∗ N.

−MτNiff∃M⁰−→^η∗ M,∃N⁰−→^η∗ Nsuch that M⁰vτN⁰.

It is easy to check thats is a preorder, and we conjecture that alsoτ is (we will not prove it because unnecessary for the present work).

Remark 1. It is clear that M

→η N impliesM s N. More generally, by transitivity ofswe have thatM→^η∗ NentailsMsN.

Lemma 4. Let M,N∈2hΛ^ri. ThenMτ NandNτM entailsM≡ητ N Proof. By hypothesis we have M⁰,M⁰⁰ −→^η∗ M, N⁰,N⁰⁰ −→^η∗ N, such that M⁰ vτ

N⁰ and N⁰⁰ vτ M⁰⁰. Then, N⁰ ≡τ M⁰+N⁰ −→^η∗ M+N, hence N ≡ητ M+N. Symmetrically, M⁰⁰ ≡_τ M⁰⁰ +N⁰⁰ −→^η∗ M+N, hence M ≡_ητ M+N, and we

conclude M≡ητ N. ut

proof in

tech. app. ← Lemma 5. Let M,N∈2hΛ^ri. ThenM_sNentails M_τN.

To sum up,M 6≡_ητ Nimplies that, say, M6_sN, which means ∃M ∈ M^◦ such that ∀N ∈N^◦ we haveM 6^ϕ∗→N. Hence, whatLemma 7below does, is basically to separate such finite term M from all N’s satisfying the condition ∀N ∈ N^◦ M 6^ϕ∗→N(that are infinitely many). For technical reasons, we will need to suppose that M has a number ofλ-abstractions greater than (or equal to) the number ofλ-abstractions ofN, at any depth where its syntactic tree is defined.

Definition 8. LetM, N∈Λ^rbeβ-nf ’s of the shapeM =λx₁. . . λx_a.yP₁. . . P_p, N =λx₁. . . λx_b.zQ₁. . . Q_q. We say that M isλ-wider thanN ifa≥band each (linear or reusable) resource L in P_i is λ-wider than every (linear or reusable) resource L⁰ ∈ Q_i, for alli ≤q. Given M,N in β-nf we say that M isλ-wider thanNiff each summand of Misλ-wider than all summands ofN.

Notice that empty bags inM make it λ-wider thanN, independently from the corresponding bags inN. For example,y1 isλ-wider thanzQfor any bagQ.

The term x[λy.I,I]1 isλ-wider thanx[I][I] but not than himself.

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Remark 2. IfMisλ-wider thanNthen every M ∈M^◦ isλ-wider thanN. Lemma 6. For allM,Ninβ-nf there is aβ-normal M⁰ such thatM⁰

η∗→Mand M⁰ isλ-wider than bothM andN.

Proof. (Outline) Form >size(M), defineE_m^h(M) by induction as follows:

E_m⁰(A) =A E_m^h+1(A+A) =E_m^h+1(A) +E_m^h+1(A) E_m^h+1(λx1. . . λxn.y ~P) =λx1. . . λxm.yE_m^h(P~)E_m^h([x^!_n+1]). . .E_m^h([x^!_m]) E_m^h+1([M1, . . . , Mk,M^!]) = [E_m^h+1(M1), . . . ,E_m^h+1(Mk),(E_m^h+1(M))^!]

Consider thenM⁰=E_k^k(M) for somek >max(size(M),size(N)). ut The next lemma will be the key ingredient for proving the resource B¨ohm’s theorem (Theorem 2, below).

Lemma 7. LetM ∈Λ^r_f be a finite β-nf andΓ ={x₁, . . . , x_d} ⊇FV(M). Then, there exist a substitution σ and a sequence R~ of closed bags such that, for all β-normal N∈2hΛ^ri such thatM isλ-wider thanNandFV(N)⊆Γ, we have:

(1) Nσ ~R−→^β∗

(I if∃N⁰ ∈N^◦, M −→^ϕ∗ N⁰, 0 otherwise.

Proof. The proof requires an induction loading, namely the fact that σ = {Xk1/x1, . . . ,Xk_d/xd}such that for all distincti, j≤dwe haveki,|ki−kj|>2k for some fixedk >size(M). Recall thatX_n has been defined insection 1.

The proof is carried by induction on size(M). Let M =λxd+1. . . λxd+a.xhP1. . . Pp

where a, p ≥ 0, h ≤ d+a and, for every i ≤ p, Pi = [Mi,1, . . . , Mi,mi] with m_i ≥0. Notice that, for every j ≤ m_i, FV(M_i,j) ⊆Γ ∪ {xd+1, . . . , x_d+a} and k > size(M_i,j). So, define σ⁰ = σ·

X_k_d+1/x_d+1, . . . ,X_k_d+a/x_d+a such that k_i,|ki−k_j|>2k, for every differenti, j≤d+a.

By induction hypothesis on Mi,j, we have a sequence R~i,j of closed bags satisfying condition (1) for every N⁰ such that Mi,j is λ-wider than N⁰ and FV(N⁰)⊆Γ∪ {xd+1, . . . , xd+a}.

We now build the sequence of bags R~ starting from such R~i,j’s. First, we define a closed termH as follows (settingm= max{k1, . . . , kd+a}):

H :=λz1. . . λzpλw1. . . λwm+k_h−p.I[z1R~1,1]. . .[z1R~1,m₁]. . .[zpR~p,1]. . .[zpR~p,m_p].

Then we set: R~ := [X^!_k

d+1]. . .[X^!_k

d+a] 1. . .1

| {z }

k_h−ptimes

[H^!] 1. . .1

| {z }

mtimes

.

Notice the base of induction is when p= 0 or for every i ≤ p, mi = 0. In these casesH will be of the formλ~zλ ~w.I.

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We prove condition (1) for any N satisfying the hypothesis of the lemma.

Note that we can restrict to the case N is a single term N, the general case follows by distributing the applicationNσ ~Ron every summand of N. So, let

N=λxd+1. . . λxd+b.xh⁰Q1. . . Qq

and let us prove that N σ ~R −→^β∗ I if there is N⁰ ∈ N^◦, M ^ϕ∗→ N⁰, otherwise N σ ~R −→^β∗ 0. SinceM is λ-wider than N, we have a≥ b. We then get, setting σ⁰⁰=σ· {Xkd+1/x_d+1, . . . ,X_k_d+b/x_d+b}:

(2) N σ ~R−→^β∗ Xk_h0Q1. . . Qqσ⁰⁰[X^!_k_d+b+1]. . .[X^!_k_d+a] 1. . .1

| {z }

k_h−p

[H^!] 1. . .1

| {z }

m

.

Indeed, sincea≥b, the variablesxd+b+1, . . . , xd+a can be considered not occurring inQ1, . . . ,Qq, so thatσ⁰acts onQ1, . . . ,Qq exactly asσ⁰⁰. Moreover, since k > aandk > q, we havekh⁰ ≥2k > q+ (a−b), so, settingg=q+ (a−b), we have (2)β-reduces to:

(3) λy_g+1...λy_k

h0λy.yQ₁...Q_q[X^!_k

d+b+1]...[X^!_k

d+a][y^!_g+1]...[y_k^!

h0] σ⁰1...1

|{z}

k_h−p

[H^!] 1...1

|{z}m

We consider now three cases.

Case I (h⁰ 6=h). This meansM andN differ on their head-variable, in particular, for every N⁰ ∈N^◦,M 6^ϕ∗→ N⁰. We prove then (3)−→^β∗ 0. By the hypothesis onkh, kh⁰, we have either kh> kh⁰+ 2k orkh⁰ > kh+ 2k. In the first case, we get (by the hypothesis onkandN)kh> kh⁰+p+q+a+b > kh⁰+p−g, so that (3)−→^β∗ 0 since the head-variabley will get 0 from an empty bag of the bunch of thekh−pempty bags. In the second case, we getm≥kh⁰−g > kh−p, so that (3)−→^β∗ 0 since the head-variabley will get 0 from an empty bag of the bunch of themempty bags.

Case II (h⁰=handp−a6=q−b).This means that, for everyN⁰ ∈N^◦,M^ϕ∗6→ N⁰ (in fact, note that (·)^◦ preserve the length of the head prefix of abstractions and that of the head sequence of applications, while →^ϕ preserves the difference between the two). We prove then(3)−→^β∗ 0. As before, we have two subcases. If p−a < q−b, thenkh⁰ −g =kh−g < kh−pso (3)−→^β∗ 0, the head variable y getting 0 from an empty bag of the bunch of thekh−pempty bags. Otherwise, p−a > q−bimpliesk_h⁰−g=k_h−g > k_h−pand so(3)−→^β∗ 0, the head variable y getting 0 from an empty bag of the bunch of themempty bags.

Case III (h⁰ =handp−a=q−b).In this case we havek_h⁰−g=k_h−g=k_h−p, so that (3) β-reduces to HQ1. . . Qq[X^!_k

d+b+1]. . .[X^!_k

d+a] 1. . .1

| {z }

kh−p

1. . .1

| {z }

m

σ⁰. Notice that, by the definition of the substitutionσ⁰, we can rewrite this term as (4) HQ₁. . . Q_q[x^!_k

d+b+1]. . .[x^!_k

d+a] 1. . .1

| {z }

k_h−p

1. . .1

| {z }

m

σ⁰.

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SinceN can be inΛ^r−Λ^r_f, the bagsQ_i’s may contain reusable resources. Hence, for everyi≤q, let us explicitQ_i= [N_i,1, . . . , N_i,`_i,(N_i,`_i₊₁+· · ·+N_i,n_i)^!], where n_i≥`_i≥0. We split into three subcases. Notice thatp≥q, indeeda≥b(asM is λ-wider thanN) and we are considering the casep−a=q−b. Also, recall thatmi is the number of resources inPi.

Subcase III.a (∃i ≤ q, m_i < `_i). In this case, for every N⁰ ∈ N^◦, we have M 6^ϕ∗→N⁰. In fact, anyN⁰∈N^◦is of the formλxd+1. . . λxd+b.xh⁰Q⁰₁. . . Q⁰_q, with Q⁰_j∈Q^◦_j for every j≤q. In particular,Q⁰_i has at least`i> mi linear resources, hence Pi6^ϕ∗→Q⁰_i, and henceM 6^ϕ∗→N⁰.

We have (4)^β∗→0. Indeed, applying the β-reduction to(4)will eventually match the abstraction λzi in H with the bag Qi. The variable zi has mi linear occurrences in H and no reusable ones. This means there will be not enough occurrences of zi to accommodate all the `i linear resources of Qi, so giving (4)^β∗→0.

Subcase III.b (∃i ≤ q, mi 6= `i andni = `i). This case means that Qi has no reusable resources and a number of linear resources different fromPi. Hence M ^ϕ∗6→N⁰, for everyN⁰∈N^◦. Also,(4)^β∗→0 since the number of linear resources in Qi does not match the number of linear occurrences of the variablezi in H.

Subcase III.c (∀i≤q, m_i ≥`_i, andm_i> `_i entailsn_i > `_i). The hypothesis of the case says that`_i < m_i entails that Q_i has some reusable resources.

LetF_i be the set of mapss:{1, . . . , m_i} → {1, . . . , `_i, `_i+ 1, . . . , n_i} such that

`i-injectivity: for everyj, h≤mi, ifs(j) =s(h)≤`i, thenj =h,

`_i-surjectivity: for everyh≤`_i, there isj≤m_i,s(j) =h.

Intuitively,Fidescribes the possible ways of replacing themioccurrences of the variable zi in H by theni resources inQi: the two conditions say that each of the `i linear resources of Qi must replace exactly one occurrence of zi. Notice that, being under the hypothesis thatp−a=q−b, we havep−q=a−b, hence (4)β-reduces to the following sum

(5) P

s₁∈F1

...

sq∈Fq

I[N_1,s₁₍₁₎R~1,1]...[N_1,s₁_(m₁₎R~1,m₁]...[N_q,s_q₍₁₎R~q,1]...[N_q,s_q_(m_q₎R~q,m_q] [xd+b+1R~q+1,1]...[xd+b+1R~q+1,m₁]...[xd+aR~p,1]...[xd+aR~p,m_p]σ⁰ Notice that for every i≤q, si ∈ Fi, j ≤mi the term Mi,j is λ-wider than N_i,s_i_(j)and FV(N_i,s_i_(j))⊆Γ∪ {xd+1, . . . , xd+a}, so by the induction hypothesis (6) N_i,s_i_(j)σ⁰R~_i,j−→^β∗

(I if∃N⁰ ∈N_i,s^◦

i(j), Mi,j ϕ∗→N⁰, 0 otherwise.

Also, for everyi,1≤i≤p−q=a−b,j≤mq+i, we have thatMq+i,jisλ-wider thanxd+b+i∈Γ∪ {xd+1, . . . , xd+a}, so by induction hypothesis

(7) xd+b+iσ⁰R~q+i,j

−→β∗

(I ifMi,j

ϕ∗→xd+b+i, 0 otherwise.

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From(6)and(7) we deduce that(5)−→^β∗ I if, and only if,

(i) for all i≤q, there exists si ∈ Fi such that, for allj ≤mi,∃N⁰ ∈N_i,s^◦

i(j)

satisfyingM_i,j^ϕ∗→N⁰, and

(ii) for alli < p−q=a−band for allj≤mq+i, we haveMq+i,j

ϕ∗→xd+b+i. Thanks to the conditions on the function si, item (i) is equivalent to say that for alli≤q,∃Q⁰_i∈Q^◦_i, P_i^ϕ∗→Q⁰_i, while item (ii) is equivalent to say that for all i < p−q=a−b,Pq+i

ϕ∗→[x^m_d+b+i^q+i ]. This means that (5)−→^β∗ I if, and only if, M =λxd+1. . . λxd+a.xhP1. . . Pp

ϕ∗→λxd+1. . . λxd+bλxd+b+1. . . λxd+a.xhQ⁰₁. . . Q⁰_q[x^m_d+b+1^q+1 ]. . .[x^m_d+a^p ]

ϕ∗→λxd+1. . . λxd+b.xhQ⁰₁. . . Q⁰_q

where the last term is inN^◦. To sum up, Nσ ~R β-reduces toIif∃N⁰ ∈N^◦ such thatM ^ϕ∗→N⁰ and to 0, otherwise. We conclude that condition(1)holds. ut

4 A Resource Conscious B¨ ohm’s Theorem

In this section we will prove the main result of our paper, namely B¨ohm’s theorem for the resource calculus. We first need the following technical lemma.

Lemma 8. Let M,N∈2hΛ^ri. IfM6τ NthenM⁰ 6τ Nfor allM⁰ −→^η∗ M. Proof. Suppose, by the way of contradiction, that there is an M⁰ −→^η∗ M such thatM⁰τN. Then, there areM⁰⁰−→^η∗ M⁰andN⁰−→^η∗ N, such thatM⁰⁰vτN⁰. By transitivity of−→,^η∗ M⁰⁰−→^η∗ Mholds, so we get MτNwhich is impossible. ut

We are now able to prove the main result of this paper.

Theorem 2 (Resource B¨ohm’s Theorem).LetM,N∈2hΛ^ribe closed sums in β-nf. If M 6≡ητ Nthen there is a sequence P~ of closed bags such that either MP~ −→^β∗ IandNP~ −→^β∗ 0, or vice versa.

Proof. Let M 6≡ητ N, then M 6τ N or vice versa (Lemma 4): say M 6τ N. ApplyingLemma 8we haveM⁰6τNfor allM⁰

←η∗M; in particular, byLemma 6, M⁰ 6τ N holds for an M⁰ λ-wider than both M and N. ByLemma 5 there is M⁰ ∈(M⁰)^◦ such that for allN ∈N^◦ we haveM⁰ 6^ϕ∗→N; such a term M⁰ is in β-nf since M⁰ is inβ-nf and isλ-wider than bothMandNbyRemark 2.

FromLemma 7, recalling thatM⁰,M,Nare closed, there is a sequenceP~ of closed bags such that: (i)NP~ ^β∗→0, since for allN ∈N^◦ we haveM⁰6^ϕ∗→N, and (ii)MP~ ^β∗→I, sinceM⁰ ^η∗→Mand hence byRemark 1 there is anM ∈M^◦ such thatM⁰^ϕ∗→M. This concludes the proof of our main result. ut

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Corollary 1. Let ∼be a congruence on2hΛ^eiextending β-equivalence. If there are two closed M,N∈2hΛ^riin β-nf such thatM6≡_ητ Nbut M∼N, then ∼is trivial, i.e. for all sums L∈2hΛ^ri,L∼0.

Proof. Suppose M 6≡_ητ N but M ∼ N. From Theorem 2 there is P~ such that MP~ −→^β∗ IandNP~ −→^β∗ 0, or vice versa. By the congruence of∼, we haveMP~ ∼ NP~. By the hypothesis that ∼extends β-equivalence, we getI∼0. Now, take any termL∈2hΛ^ri, we haveL≡βIL∼0L= 0. ut Acknowlegements.We thank Stefano Guerrini and Henk Barendregt for stim- ulating discussions and remarks.

References

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lambda calculus. J. London Math. Soc.2(12) (1976) 361–370 5. Girard, J.Y.: Linear logic. Th. Comp. Sc.50(1987) 1–102

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Computer Science Logic. Volume 6247 of LNCS., Springer (2010) 170–184 7. Pagani, M., Ronchi Della Rocca, S.: Linearity, non-determinism and solvability.

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calculus. Inf. Comput.126(1) (1996) 83–102

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http://www.dsi.unive.it/∼gmanzone/papers/bcem11.pdf

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A Technical Appendix

This technical appendix is devoted to provide the full proof ofLemma 5. To prove this result we will need some preliminary definitions and technical lemmas.

Definition 9. The sliced size of an expression A∈Λ^e is the number size^sl(A) defined by structural induction onA as follows:

– size^sl(x) := 1,

– size^sl(M P) := size^sl(M) + size^sl(P), – size^sl(λx.M) := size^sl(M) + 1,

– size^sl([M1, . . . , Mm,(Σ_i=m+1^m+k Mi)^!]) := maxi∈{1,...,m+k}(size^sl(Mi)) + 1.

Lemma 9. ForA,B∈2hΛ^eiandA, B∈Λ^e_f, we have:

(i) ∀A⁰ ∈A^◦,size^sl(A⁰)≤size(A);

(ii) ifA→^ϕ B, thensize^sl(A)≥size^sl(B);

(iii) ifA

→η B, then∀A∈A^◦,∃B∈B^◦ such thatsize^sl(A)≥size^sl(B)and vice versa,∀B ∈B^◦,∃A∈A^◦ such thatsize^sl(A)≥size^sl(B).

Proof. (i) By a straightforward induction onA.

(ii) By a straightforward inspection of the rules definingA→^ϕ B.

(iii) By an easy inspection of the rules defining A →^η B, one gets ∀A ∈ A^◦,∃B ∈ B^◦ such that A →^ϕ B (hence by (ii), size^sl(A) ≥size^sl(B)) andvice versa ∀B ∈B^◦,∃A∈A^◦ such thatA→^ϕ B, hence size^sl(A)≥size^sl(B). ut Lemma 10. Let A, B∈Λ^e_f such thatA−→^ϕ∗ B, then:

(i) ifB =xthen either A=xorA−→^ϕ∗ λy.x[y^k], for somek∈N;

(ii) ifB =M P then there areM⁰, P⁰ such thatM⁰−→^ϕ∗ M,P⁰−→^ϕ∗ P and either A=M⁰P⁰ or A−→^ϕ∗ λy.M⁰P⁰[y^k]for somey /∈FV(M⁰P⁰)andk∈N;

(iii) if B =λx.M, then there exists M⁰ such that M⁰ −→^ϕ∗ M and either A = λx.M⁰ orA−→^ϕ∗ λy.(λx.M⁰)[y^k]for somey /∈FV(λx.M⁰)andk∈N;

(iv) ifB= [N₁, . . . , N_n]thenA= [M₁, . . . , M_n]withM_i−→^ϕ∗ N_ifor everyi≤n.

Proof. By induction on the length of the reduction chainA−→^ϕ∗ B. ut Lemma 5is a direct consequence of the following result. To ease the formulation of its statement we defines on setsA,B ⊆Λ^e_f by setting A sB iff∀A∈ A,

∀B ∈ Bwe haveA−→^ϕ∗ B.

Lemma 11. Let A ⊆ Λ^e_f be such that sup_A∈A size^sl(A)

is finite and let B ∈ 2hΛ^ei. Then A sB^◦ implies there is B⁰

η∗→Bsuch that A ⊆(B⁰)^◦. Proof. The proof is performed by induction on the triplet

sup

A∈A

size^sl(A) ,sup

A∈A

size^sl(A)

− inf

B∈B^◦

size^sl(B)

,size(B)

, lexicographically ordered. We split in several cases, depending onB.