Standardization of a call-by-value lambda-calculus

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1 Standardization of a call-by-value lambda-calculus Giulio Guerrieri 1, Luca Paolini 2, and Simona Ronci Della Rocca 2 1 Laboratoire PPS, UMR 7126, Université Paris Diderot, Sorbonne Paris Cité F Paris, France giulio.guerrieri@pps.univ-paris-diderot.fr 2 Dipartimento di Informatica, Università degli Studi di Torino C.so Svizzera 185, Torino, Italia {paolini,ronci}@di.unito.it Abstract We study an extension of Plotkin s call-by-value lambda-calculus by means of two commutation rules (sigma-reductions). Recently, it as been proved tat tis extended calculus provides elegant caracterizations of many semantic properties, as for example solvability. We prove a standardization teorem for tis calculus by generalizing Takaasi s approac of parallel reductions. Te standardization property allows us to prove tat our calculus is conservative wit respect to te Plotkin s one. In particular, we sow tat te notion of solvability for tis calculus coincides wit tat for Plotkin s call-by-value lambda-calculus ACM Subject Classification D.3.1 Formal Definitions and Teory, F.3.2 Semantics of Programming Language, F.4.1 Matematical Logic. Keywords and prases standardization, sequentialization, lambda-calculus, sigma-reduction, parallel reduction, call-by-value, ead reduction, ernal reduction, solvability, potential valuability, observational equivalence. 1 Introduction Te λ v -calculus (λ v for sort) as been roduced by Plotkin in [15], in order to give a formal account of te call-by-value evaluation, wic is te most commonly used parameter passing policy for programming languages. λ v sares te syntax wit te classical, call-by-name, λ-calculus (λ for sort), but its reduction rule, β v, is a restriction of β, firing only in case te argument is a value (i.e., a variable or an abstraction). Wile β v is enoug for evaluation, it turned out to be too weak to study operational properties of terms. For example, in λ, te β-reduction is sufficient to caracterize solvability and (using extensionality) separability, but, in order to caracterize similar properties for λ v, it as been necessary to roduce different notions of reduction unsuitable for a correct call-by-value evaluation (see [13, 14]): tis is disappoing and requires complex reasoning. In tis paper we study λ σ v, te extension of λ v proposed in [3]. It keeps te λ v (and λ) syntax and it adds to te β v -reduction two commutation rules, called σ 1 and σ 3, wic unblock β v -redexes tat are idden by te yper-sequential structure of terms. It is well-known (see [14, 1]) tat in λ v tere are normal forms tat are unsolvable, e.g. (λyx.xx)(zz)(λx.xx). Te more evident benefit of λ σ v is tat te commutation rules make all normal forms solvable (indeed (λyx.xx)(zz)(λx.xx) is not a λ σ v normal form). More generally, te so obtained language, allows us to caracterize Tis work was partially supported by LINTEL TO_Call1_2012_0085, i.e. a Researc Project funded by te Compagnia di San Paolo. Giulio Guerrieri, Luca Paolini and Simona Ronci Della Rocca; licensed under Creative Commons License CC-BY Leibniz International Proceedings in Informatics Scloss Dagstul Leibniz-Zentrum für Informatik, Dagstul Publising, Germany

2 2 Standardization of a call-by-value lambda-calculus operational properties, like solvability and potential valuability, in an ernal and elegant way (see [3]). In tis paper we prove a standardization property in λ σ v, and some of its consequences, namely its soundness wit respect to te semantics of λ v. Let us recall te notion of standardization, wic as been first studied in te ordinary λ-calculus (see, for example [5, 8, 2]). A reduction sequence is standard if its redexes are ordered in a given way, and te corresponding standardization teorem establises tat every reduction sequence can constructively be transformed o a standard one. Standardization is a key tool to grasp te way in wic reductions work, tat seds some ligt on redexes relationsips and teir dependencies. It is useful for caracterization of semantic properties troug reduction strategies (te proof of operational semantics adequacy is a typical use). In te λ v setting standardization teorems ave been proved by Plotkin [15], Paolini and Ronci Della Rocca [14, 12] and Crary [4]. Te definition of standard sequence of reductions considered by Plotkin and Crary coincides, and it imposes a partial order on redexes, wile Paolini and Ronci Della Rocca define a total order on tem. All tese proofs are developed by using te notion of parallel reduction roduced by Tait and Martin-Löf (see Takaasi [17] for details and eresting tecnical improvements). We empasize tat tis metod does not involve te notion of residual of a redex, on wic many classical proofs for te λ-calculus are based (see for example [8, 2]). As in [15, 17, 14, 4], we use a suitable notion of parallel reduction for developing our standardization teorem for λ σ v. In particular we consider two groups of redexes, ead β v -redexes and ead σ-redexes (putting togeter σ 1 and σ 3 ), and we induce a total order on ead redexes of te two groups, witout imposing any order on ead σ-redexes temselves. More precisely, wen σ-redexes are missing, tis notion of standardization coincides wit tat presented in [14]. Moreover, we sow tat it is not possible to strengten our standardization by (locally) ordering σ 1 -reduction to σ 3 -reduction (or viceversa). As usual, our standardization proof is based on a sequentialization result: inner reductions can always be postponed after te ead ones, for a non-standard definition of ead reduction. Sequentialization as eresting consequences: it allows us to prove tat fundamental operational properties in λ σ v, like observational equivalence, potential valuability and solvability, are conservative wit respect to te corresponding notions of λ v. Tis fully justifies te project in [3] were λ σ v as been roduced as a tool for studying te operational beaviour of λ v. Oter variants of λ v ave been roduced in te literature for modeling te call-by-value computation. We would like to cite ere at least te contributions of Moggi [10], Felleisen and Sabry [16], Maraist et al. [9], Herbelin and Zimmerman [7], Accattoli and Paolini [1]. All tese proposals are based on te roduction of new constructs to te syntax of λ v, so te comparison between tem is not easy wit respect to syntactical properties (some detailed comparison is given in [1]). We po out tat te calculi roduced in [10, 16, 9, 7] present some variants of our σ 1 and/or σ 3 rules, often in a setting wit explicit substitutions. Outline. In Section 2 we roduce te language λ σ v and its operational beaviour; in Section 3 te sequentialization property is proved; Section 4 contains te main result, i.e., standardization; in Section 5 some conservativity results wit respect to Plotkin s λ v -calculus are proved. Section 6 concludes te paper, wit some s for future work. 2 Te call-by-value lambda calculus wit sigma-rules In tis section we present λ σ v, a call-by-value λ-calculus roduced in [3] tat adds two σ-reduction rules to pure (i.e. witout constants) call-by-value λ-calculus defined by Plotkin in [15]. Te syntax of terms of λ σ v [3] is te same as te one of ordinary λ-calculus and Plotkin s

3 G. Guerrieri, L. Paolini and S. Ronci Della Rocca 3 call-by-value λ-calculus λ v [15] (witout constants). Given a countable set V of variables (denoted by x, y, z,... ), te sets Λ of terms and Λ v of values are defined by mutual induction: (Λ v ) V, U ::= x λx.m values (Λ) M, N, L ::= V MN terms Clearly, Λ v Λ. All terms are considered up to α-conversion. Te set of free variables of a term M is denoted by fv(m). Given V 1,..., V n Λ v and pairwise distinct variables x 1,..., x n, M{V 1 /x 1,..., V n /x n } denotes te term obtained by te capture-avoiding simultaneous substitution of V i for eac free occurrence of x i in te term M (for all 1 i n). Note tat, for all V, V 1,..., V n Λ v and pairwise distinct variables x 1,..., x n, V {V 1 /x 1,..., V n /x n } Λ v. Contexts (wit exactly one ole ), denoted by C, are defined as usual via te grammar: C ::= λx.c CM MC. We use C M for te term obtained by te capture-allowing substitution of te term M for te ole in te context C. Notation. From now on, we set I = λx.x and = λx.xx. Te reduction rules of λ σ v consist of Plotkin s β v -reduction rule, roduced in [15], and two simple commutation rules called σ 1 and σ 3, studied in [3]. Definition 1 (Reduction rules). We define te following binary relations on Λ (for any M, N, L Λ and any V Λ v ): (λx.m)v βv M{V/x} (λx.m)nl σ1 (λx.ml)n wit x / fv(l) V ((λx.l)n) σ3 (λx.v L)N wit x / fv(v ). For any r {β v, σ 1, σ 3 }, if M r M ten M is a r-redex and M is its r-contractum. In tis sense, a term of te sape (λx.m)n (for any M, N Λ) is a β-redex. We set σ = σ1 σ3 and v = βv σ. Te side conditions on σ1 and σ3 in Definition 1 can be always fulfilled by α-renaming. Obviously, any β v -redex is a β-redex but te converse does not old: (λx.z)(yi) is a β-redex but not a β v -redex. Redexes of different kind may overlap: for example, te term I is a σ 1 -redex and it contains te β v -redex I; te term (I )(xi) is a σ 1 -redex and it contains te σ 3 -redex (I ), wic contains in turn te β v -redex I. According to te Girard s call-by-value boring translation ( ) v of terms o Intuitionistic Multiplicative Exponential Linear Logic proof-nets, defined by (A B) v =!A v!b v (see [6]), te images under ( ) v of a σ 1 -redex (resp. σ 3 -redex) and its contractum are equal modulo some bureaucratic steps of cut-elimination. Notation. Let R be a binary relation on Λ. We denote by R (resp. R + ; R = ) te reflexive-transitive (resp. transitive; reflexive) closure of R. Definition 2 (Reductions). Let r {β v, σ 1, σ 3, σ, v}. Te r-reduction r is te contextual closure of r, i.e. M r M iff tere is a context C and N, N Λ suc tat M = C N, M = C N and N r N. Te r-equivalence = r is te reflexive-transitive and symmetric closure of r. Let M be a term: M is r-normal if tere is no term N suc tat M r N; M is r-normalizable if tere is a r-normal term N suc tat M r N; M is strongly r-normalizing if tere is no sequence (N i ) i N suc tat M = N 0 and N i r N i+1 for any i N. Finally, r is strongly normalizing if every N Λ is strongly r-normalizing.

4 4 Standardization of a call-by-value lambda-calculus Patently, σ v and βv v. Remark 3. For any r {β v, σ 1, σ 3, σ, v} (resp. r {σ 1, σ 3, σ}), values are closed under r- reduction (resp. r-expansion): for any V Λ v, if V r M (resp. M r V ) ten M Λ v ; more precisely, V = λx.n and M = λx.n for some N, N Λ wit N r N (resp. N r N). Proposition 4 (See [3]). Te σ-reduction is confluent and strongly normalizing. Te v-reduction is confluent. Te λ σ v -calculus, λ σ v for sort, is te set Λ of terms endowed wit te v-reduction v. Te set Λ endowed wit te β v -reduction βv is te λ v -calculus (λ v for sort), i.e. te Plotkin s call-by-value λ-calculus [15] (witout constants), wic is tus a sub-calculus of λ σ v. Example 5. M = (λy. )(xi) σ1 (λy. )(xi) βv (λy. )(xi) βv... and N = ((λy. )(xi)) σ3 (λy. )(xi) βv (λy. )(xi) βv... are te only possible v-reduction pats from M and N respectively: M and N are not v-normalizable, and M = v N. Meanwile, M and N are β v -normal and different, ence M βv N (by confluence of βv, see [15]). Informally, σ-rules unblock β v -redexes wic are idden by te yper-sequential structure of terms. Tis approac is alternative to te one in [1] were idden β v -redexes are reduced using rules acting at a distance (troug explicit substitutions). It can be sown tat te call-by-value λ-calculus wit explicit substitution roduced in [1] can be embedded in λ σ v. 3 Sequentialization In tis section we aim to prove a sequentialization teorem (Teorem 22) for te λ σ v -calculus by adapting Takaasi s metod [17, 4] based on parallel reductions. Notation. From now on, we always assume tat V, V Λ v. Note tat te generic form of a term is V M 1... M m for some m N (in particular, values are obtained wen m = 0). Te sequentialization result is based on a partitioning of v-reduction between ead and ernal reduction. Definition 6 (Head β v -reduction). We define inductively te ead β v -reduction βv te following rules (m N in bot rules): by β v N βv N rigt (λx.m)v M 1... M m βv M{V/x}M 1... M m V NM 1... M m βv V N M 1... M m Te ead β v -reduction βv reduces exactly te same redexes (see also [13]) as te left reduction defined in [15, p. 136] for λ v and called evaluation in [16, 4]. If N βv N ten N is obtained from N by reducing te leftmost-outermost β v -redex, not in te scope of a λ: tus, te ead β v -reduction is deterministic (i.e., it is a partial function from Λ to Λ) and does not reduce values. Definition 7 (Head σ-reduction). We define inductively te ead σ-reduction σ by te following rules (m N in all te rules, x / fv(l) in te rule σ 1, x / fv(v ) in te rule σ 3 ): N σ N rigt (λx.m)nlm 1... M m σ (λx.ml)nm 1... M m V NM 1... M m σ V N M 1... M m V ((λx.l)n)m 1... M m σ 1 σ (λx.v L)NM 1... M m σ 3

5 G. Guerrieri, L. Paolini and S. Ronci Della Rocca 5 Te ead (v-)reduction is v = βv σ. Te ernal (v-)reduction is v = v v. Notice tat βv βv βv and σ σ σ and v v v. Values are normal forms for te ead reduction, but te converse does not old: xi / Λ v is ead-normal. Informally, if N σ N ten N is obtained from N by reducing one of te leftmost σ 1 - or σ 3 -redexes, not in te scope of a λ: in general, a term may contain several ead σ 1 - and σ 3 -redexes. Indeed, differently from βv, te ead σ-reduction σ is not deterministic, for example te leftmost-outermost σ 1 - and σ 3 -redexes may overlap: if M = (λy.y )( (xi))i ten M σ (λy.y I)( (xi)) = N 1 by applying te rule σ 1 and M σ (λz.(λy.y )(zz))(xi)i =N 2 by applying te rule σ 3. Note tat N 1 contains only a ead σ 3 -redex and N 1 σ (λz.(λy.y I)(zz))(xI) = N wic is normal for v ; meanwile N 2 contains only a ead σ 1 -redex and N 2 σ (λz.(λy.y )(zz)i)(xi) = N wic is normal for v : N N, ence te ead reduction v is not confluent and a term may ave several ead-normal forms (tis example does not contradict te confluence of σ-reduction because N σ N but by performing an ernal reduction step). Later, in Corollary 26.2 we sow tat if a term M as a ead normal form N Λ v ten N is te unique ead normal form of M. Definition 8 (Parallel reduction). We define inductively te parallel reduction by te following rules (x / fv(l) in te rule σ 1, x / fv(v ) in te rule σ 3 ): V V M i M i (m N, 0 i m) βv (λx.m 0)VM 1... M m M 0{V /x}m 1... M m N N L L M i M i (m N, 0 i m) σ1 (λx.m 0)NLM 1... M m (λx.m 0L )N M 1... M m V V N N L L M i M i (m N, 1 i m) σ3 V ((λx.l)n)m 1... M m (λx.v L )N M 1... M m M i M i (m N, 0 i m) λ (λx.m 0)M 1... M m (λx.m 0)M 1... M m M i M i (m N, 1 i m) var x M 1... M m x M 1... M m In Definition 8 te rule var as no premises wen m = 0: tis is te base case of te inductive definition of. Te rules σ 1 and σ 3 ave exactly tree premises wen m = 0. Intuitively, M M means tat M is obtained from M by reducing a number of β v -, σ 1 - and σ 3 -redexes (existing in M) simultaneously. Definition 9 (Internal and strong parallel reduction). We define inductively te ernal parallel reduction by te following rules: N N λx.n λx.n λ x x var V V N N M i M i (m N, 1 i m) rigt V NM 1... M m V N M 1... M m Te strong parallel reduction is defined by: M N iff M N and tere exist M, M Λ suc tat M β v M σ M N. Notice tat te rule rigt for as exactly two premises wen m = 0. Remark 10. Te relations, and are reflexive. Te reflexivity of follows immediately from te reflexivity of and. Te proofs of reflexivity of and are bot by structural induction on a term: in te case of, recall tat every term is of te form (λx.n)m 1... M m or x M 1... M m for some m N and ten apply te rule λ or var respectively, togeter wit te inductive ypotesis; in te case of, recall tat every term is of te form λx.m or x or V NM 1... M m for some m N and ten apply te rule λ (togeter wit te reflexivity of ) or var or rigt (togeter wit te reflexivity of and te inductive ypotesis) respectively.

6 6 Standardization of a call-by-value lambda-calculus One as (first, prove tat by induction on te derivation of M M, te oter inclusions follow from te definition of ) and, since is reflexive (Remark 10), βv and σ. Observe tat R for any R { βv, βv,,, }, even if for different reasons: for example, by reflexivity of (Remark 10), wereas βv by reducing te (leftmost-outermost) β v -redex. Next two furter remarks collect many minor properties tat can be easily proved. Remark Te ead β v -reduction βv does not reduce a value (in particular, does not reduce under λ s), i.e., for any M Λ and any V Λ v, one as V βv M. 2. Te ead σ-reduction σ does neiter reduce a value nor reduce to a value, i.e., for any M Λ and any V Λ v, one as V σ M and M σ V. 3. Variables and abstractions are preserved under ( -expansion), i.e., if M x (resp. M λx.n ) ten M = x (resp. M = λx.n for some N Λ suc tat N N ). 4. If M M ten λx.m R λx.m for any R {,, }. Indeed, for R {, } apply te rule λ to conclude, ten λx.m λx.m according to te definition of. 5. For any V, V Λ v, V V iff V V. Te left-to-rigt direction olds because ; conversely, assume V V : if V is a variable ten necessarily V = V and ence V V by applying te rule var for ; oterwise V = λx.n for some N Λ, and ten necessarily V = λx.n wit N N, so V V by applying te rule λ for. Remark If M M and N N ten MN M N. For te proof, it is sufficient to consider te last rule of te derivation of M M. 2. If R { βv, σ } and M R M, ten MN R M N for any N Λ. For te proof, it is sufficient to consider te last rule of te derivation of M R M, for any R { βv, σ }. 3. If M M and N N were M / Λ v, ten MN M N : indeed, te last rule in te derivation of M M can be neiter λ nor var because M / Λ v. Te ypotesis M / Λ v is crucial: for example, x x and I but I and tus x(i ) x. 4. v v. As a consequence, = v and (by Proposition 4) is confluent. 5. v v, so = v. Tus, by Remark 11.3, variables and abstractions are preserved under v -expansion, i.e., if M v x (resp. M v λx.n ) ten M = x (resp. M = λx.n wit N v N ). 6. For any R { βv, σ }, if M R M ten M{V/x} R M {V/x} for any V Λ v. Te proof is by straigtforward induction on te derivation of M R M for any R { βv, σ }. As expected, a basic property of parallel reduction is te following: Lemma 13 (Substitution lemma for ). If M M and V V ten M{V/x} M {V /x}. Proof. By straigtforward induction on te derivation of M M. Te following lemma will play a crucial role in te proof of Lemmas and sows tat te ead σ-reduction σ can be postponed after te ead β v -reduction βv. Lemma 14 (Commutation of ead reductions). 1. If M σ L βv N ten tere exists L Λ suc tat M βv L = σ N. 2. If M σ L β v N ten tere exists L Λ suc tat M β v L σ N. 3. If M v M ten tere exists N Λ suc tat M β v N σ M. Proof. 1. By induction on te derivation of M σ L. Let us consider its last rule r. If r = σ 1 ten M = (λx.m 0 )N 0 L 0 M 1... M m and L = (λx.m 0 L 0 )N 0 M 1... M m were m N and x / fv(l 0 ). Since L βv N, tere are only two cases:

7 G. Guerrieri, L. Paolini and S. Ronci Della Rocca 7 eiter N 0 βv N 0 and N = (λx.m 0 L 0 )N 0M 1... M m (according to te rule rigt for βv ), ten M βv (λx.m 0 )N 0L 0 M 1... M m σ N; or N 0 Λ v and N =M 0 {N 0 /x}l 0 M 1... M m (by te rule β v, as x / fv(l 0 )), so M βv N. If r = σ 3 ten M = V ((λx.l 0 )N 0 )M 1... M m and L = (λx.v L 0 )N 0 M 1... M m wit m N and x / fv(v ). Since L βv N, tere are only two cases: eiter N 0 βv N 0 and N = (λx.v L 0 )N 0M 1... M m (according to te rule rigt for βv ), ten M βv V ((λx.l 0 )N 0)M 1... M m σ N; or N 0 Λ v and N =VL 0 {N 0 /x}m 1... M m (by te rule β v, as x / fv(v )), so M βv N. Finally, if r = rigt ten M = V N 0 M 1... M m and L = V N 0M 1... M m wit m N and N 0 σ N 0. By Remark 11.2, N 0 / Λ v and tus, since L βv N, te only possibility is tat N 0 βv N 0 and N = V N 0 M 1... M m (according to te rule rigt for βv ). By induction ypotesis, tere exists N 0 Λ suc tat N 0 βv N 0 = σ N 0. Terefore, M βv V N 0 M 1... M m = σ N. 2. Immediate consequence of Lemma 14.1, using standard tecniques of rewriting teory. 3. Immediate consequence of Lemma 14.2, using standard tecniques of rewriting teory. We are now able to travel over again Takaasi s metod [17, 4] in our setting wit β v - and σ-reduction. Te next four lemmas govern te strong parallel reduction and will be used to prove Lemma 19. Lemma 15. If M M and N N and M / Λ v, ten MN M N. Proof. One as MN M N by Remark 12.1 and since M M. By ypotesis, tere exist m, n N and M 0,..., M m, N 0,..., N n suc tat M = M 0, M m = N 0, N n M, M i βv M i+1 for any 0 i < m and N j σ N j+1 for any 0 j < n; by Remark 12.2, M i N βv M i+1 N for any 0 i < m and N j N σ N j+1 N for any 0 j < n. As M / Λ v, one as N n N M N by Remark Terefore, MN M N. Lemma 16. If M M and N N ten MN M N. Proof. If M / Λ v ten MN M N by Lemma 15 and since N N. Assume M Λ v : MN M N by Remark 12.1, as M M and N N. By ypotesis, tere are m, m, n, n N and M 0,..., M m, M 0,..., M m, N 0,..., N n, N 0,..., N n suc tat: M = M 0, M m = M 0, M m M, M i βv M i+1 for any 0 i < m, and M i σ M i +1 for any 0 i < m, N = N 0, N n = N 0, N n N, N j βv N j+1 for any 0 j < n and N j σ N j +1 for any 0 j < n. By Remark 11.3, M m Λ v since M Λ v, terefore m = 0 by Remark 11.2, and tus M m = M 0 M (and M m M since ) and M m Λ v. Using te rules rigt for βv and σ, one as M m N j βv M m N j+1 for any 0 j < n, and M m N j σ M m N j +1 for any 0 j < n. By Remark 12.2, M i N 0 βv M i+1 N 0 for any 0 i < m. By applying te rule rigt for, one as M m N n M N. Terefore, MN = M 0 N 0 β v M m N 0 β v M m N n = M m N 0 σ M m N n M N and ence MN M N. Lemma 17. If M M and V V, ten M{V/x} M {V /x}. Proof. By Lemma 13, one as M{V/x} M {V /x} since M M and V V. We proceed by induction on M Λ. Let us consider te last rule r of te derivation of M M. If r = var ten tere are two cases: eiter M = x and ten M{V/x}=V V =M {V /x}; or M = y x and ten M{V/x} = y = M {V /x}, so M{V/x} M {V /x} by Remark 10.

8 8 Standardization of a call-by-value lambda-calculus If r = λ ten M = λy.n and M = λy.n wit N N ; we can suppose witout loss of generality tat y / fv(v ) {x}. One as N{V/x} N {V /x} according to Lemma 13. By applying te rule λ for, one as M{V/x} = λy.n{v/x} λy.n {V /x} = M {V /x} and tus M{V/x} M {V /x}. Finally, if r = rigt ten M = UNM 1... M m and M = U N M 1... M m for some m N wit U, U Λ v, U U, N N and M i M i for any 1 i m. By induction ypotesis, U{V/x} U {V /x} (indeed U U according to Remark 11.5) and N{V/x} N {V /x}. By Lemma 13, M i {V/x} M i {V /x} for any 1 i m. By Lemma 16, U{V/x}N{V/x} U {V /x}n {V/x} and ence, by applying Lemma 15 m times since U {V /x}n {V/x} / Λ v, one as M{V/x} = U{V/x}N{V/x}M 1 {V/x}... M m {V/x} U {V /x}n {V /x}m 1{V /x}... M m{v /x} = M {V /x}. In te proof of te two next lemmas, as well as in te proof of Corollary 21 and Teorem 22, our Lemma 14 plays a crucial role: indeed, since te ead σ-reduction well eract wit te ead β v -reduction, Takaasi s metod [17, 4] is still working wen adding te reduction rules σ 1 and σ 3 to Plotkin s β v -reduction. Lemma 18 (Substitution lemma for ). If M M and V V ten M{V/x} M {V /x}. Proof. By Lemma 13, one as M{V/x} M {V /x} since M M and V V. By ypotesis, tere exist m, n N and M 0,..., M m, N 0,..., N n suc tat M = M 0, M m = N 0, N n M, M i βv M i+1 for any 0 i < m and N j σ N j+1 for any 0 j < n; by Remark 12.6, M i {V/x} βv M i+1 {V/x} for any 0 i < m, and N j {V/x} σ N j+1 {V/x} for any 0 j < n. By Lemma 17, one as N n {V/x} M {V /x}, tus tere exist L, N Λ suc tat M{V/x} β v N 0 {V/x} σ N n {V/x} β v N σ L M {V /x}. By Lemma 14.2, tere exists N Λ suc tat M{V/x} β v N 0 {V/x} β v N σ N σ L M {V /x}, terefore M{V/x} M {V /x}. Now we are ready to prove a key lemma, wic states tat parallel reduction coincides wit strong parallel reduction (te inclusion is trivial). Lemma 19 (Key Lemma). If M M ten M M. Proof. By induction on te derivation of M M. Let us consider its last rule r. If r = var ten M = x M 1... M m and M = x M 1... M m were m N and M i M i for any 1 i m. By reflexivity of (Remark 10), x x. By induction ypotesis, M i M i for any 1 i m. Terefore, M M by applying Lemma 16 m times. If r = λ ten M = (λx.m 0 )M 1... M m and M = (λx.m 0)M 1... M m were m N and M i M i for any 0 i m. By induction ypotesis, M i M i for any 1 i m. According to Remark 11.4, λx.m 0 λx.m 0. So M M by applying Lemma 16 m times. If r = β v ten M = (λx.m 0 )V M 1... M m and M = M 0{V /x}m 1... M m were m N, V V and M i M i for any 0 i m. By induction ypotesis, V V and M i M i for any 0 i m. By applying te rule β v for βv, one as M βv M 0 {V/x}M 1... M m ; moreover M 0 {V/x}M 1... M m M by Lemma 18 and by applying Lemma 16 m times, tus tere are L, N Λ suc tat M βv M 0 {V/x}M 1... M m β v L σ N M. So M M. If r = σ 1 ten M = (λx.m 0 )N 0 L 0 M 1... M m and M = (λx.m 0L 0)N 0M 1... M m were m N, L 0 L 0, N 0 N 0 and M i M i for any 0 i m. By induction ypotesis, N 0 N 0 and M i M i for any 1 i m. By applying te rule σ 1 for σ, one as M σ (λx.m 0 L 0 )N 0 M 1... M m. By Remark 12.1, M 0 L 0 M 0L 0 and tus λx.m 0 L 0 λx.m 0L 0 according to Remark So (λx.m 0 L 0 )N 0 M 1... M m M by applying Lemma 16 m + 1

9 G. Guerrieri, L. Paolini and S. Ronci Della Rocca 9 times, ence tere are L, N Λ suc tat M σ (λx.m 0 L 0 )N 0 M 1... M m β v L σ N M. By Lemma 14.2, tere is L Λ suc tat M β v L σ L σ N M and tus M M. Finally, if r = σ 3 ten M = V ((λx.l 0 )N 0 )M 1... M m and M = (λx.v L 0)N 0M 1... M m wit m N, V V, L 0 L 0, N 0 N 0 and M i M i for any 1 i m. By induction ypotesis, N 0 N 0 and M i M i for any 1 i m. By te rule σ 3 for σ, one as M σ (λx.v L 0 )N 0 M 1... M m. By Remark 12.1, V L 0 V L 0 and tus λx.v L 0 λx.v L 0 according to Remark So (λx.v L 0 )N 0 M 1... M m M by applying Lemma 16 m + 1 times, ence tere are L, N Λ suc tat M σ (λx.v L 0 )N 0 M 1... M m β v L σ N M. By Lemma 14.2, tere is L Λ suc tat M β v L σ L σ N M, so M M. Next Lemma 20 and Corollary 21 sow tat ernal parallel reduction can be sifted after ead reductions. Lemma 20 (Postponement). If M L and L βv N (resp. L σ N) ten tere exists L Λ suc tat M βv L (resp. M σ L ) and L N. Proof. By induction on te derivation of M L. Let us consider its last rule r. If r = var, ten M = x = L wic contradicts L βv N and L σ N by Remarks If r = λ ten L = λx.l for some L Λ, wic contradicts L βv N and L σ N by Remarks Finally, if r = rigt ten M = V M 0 M 1... M m and L = V L 0 L 1... L m were m N, V V (so V V by Remark 11.5), M 0 L 0 (tus M 0 L 0 since ) and M i L i for any 1 i m. If L βv N ten tere are two cases, depending on te last rule r of te derivation of L βv N. If r = β v ten V = λx.n 0, L 0 Λ v and N = N 0{L 0 /x}l 1... L m, tus M 0 Λ v and V = λx.n 0 wit N 0 N 0 by Remark By Lemma 13, one as N 0 {M 0 /x} N 0{L 0 /x}. Let L = N 0 {M 0 /x}m 1... M m : so M = (λx.n 0 )M 0 M 1... M m βv L (apply te rule β v for βv ) and L N by applying Remark 12.1 m times. If r = rigt ten N = V N 0 L 1... L m wit L 0 βv N 0. By induction ypotesis, tere exists L 0 Λ suc tat M 0 βv L 0 N 0. Let L = V L 0M 1... M m : so M βv L (apply te rule rigt for βv ) and L N by applying Remark 12.1 m + 1 times. If L σ N ten tere are tree cases, depending on te last rule r of te derivation of L σ N. If r = σ 1 ten m > 0, V = λx.n 0 and N = (λx.n 0L 1 )L 0 L 2... L m, tus V = λx.n 0 wit N 0 N 0 by Remark Using Remarks 12.1 and 11.4, one as λx.n 0 M 1 λx.n 0L 1. Let L = (λx.n 0 M 1 )M 0 M 2... M m : so M = (λx.n 0 )M 0 M 1... M m σ L (apply te rule σ 1 for σ ) and L N by applying Remark 12.1 m times. If r = σ 3 ten L 0 = (λx.l 01 )L 02 and N = (λx.v L 01 )L 02 L 1... L m. Since M 0 (λx.l 01 )L 02, necessarily M 0 = (λx.m 01 )M 02 wit M 01 L 01 and M 02 L 02 (so M 02 L 02 ). Using Remarks 12.1 and 11.4, one as λx.v M 01 λx.v L 01. Let L = (λx.v M 01 )M 02 M 1... M m : terefore M = V ((λx.m 01 )M 02 )M 1... M m σ L (apply te rule σ 3 for σ ) and L N by applying Remark 12.1 m + 1 times. If r = rigt ten N = V N 0 L 1... L m wit L 0 σ N 0. By induction ypotesis, tere exists L 0 Λ suc tat M 0 σ L 0 N 0. Let L = V L 0M 1... M m : so M σ L (apply te rule rigt for σ ) and L N by applying Remark 12.1 m + 1 times. Corollary 21. If M L and L βv N (resp. L σ N), ten tere exist L, L Λ suc tat M + β v L σ L N (resp. M β v L σ L N).

10 10 Standardization of a call-by-value lambda-calculus Proof. Immediate by Lemma 20 and Lemma 19, applying Lemma 14.2 if L σ N. Now we obtain our first main result (Teorem 22): any v-reduction sequence can be sequentialized o a ead β v -reduction sequence followed by a ead σ-reduction sequence, followed by an ernal reduction sequence. In ordinary λ-calculus, te well-known result corresponding to our Teorem 22 says tat a β-reduction sequence can be factorized in a ead reduction sequence followed by an ernal reduction sequence (see for example [17, Corollary 2.6]). Teorem 22 (Sequentialization). If M v M ten tere exist L, N Λ suc tat M β v L σ N v M. Proof. By Remark 12.4, M M and tus tere are m N and M 0,..., M m Λ suc tat M = M 0, M m = M and M i M i+1 for any 0 i < m. We prove by induction on m N tat tere are L, N Λ suc tat M β v L σ N M, so N v M by Remark If m = 0 ten M = M 0 = M and ence we conclude by taking L = M = N. Finally, suppose m > 0. By induction ypotesis applied to M 1 M, tere exist L, N Λ suc tat M 1 β v L σ N M. By applying Lemma 19 to M, tere exist L 0, N 0 Λ suc tat M β v L 0 σ N 0 M 1. By applying Corollary 21 repeatedly, tere exists N Λ suc tat N 0 Lemma 14.3, tere exists L Λ suc tat M β v v N N and ence M v N L σ N M. M. According to It is wort noticing tat in Definition 7 tere is no distinction between ead σ 1 - and ead σ 3 -reduction steps, and, according to it, te sequentialization of Teorem 22 imposes no order between ead σ-reductions. We denote by σ1 and σ3 respectively te reduction relations σ1 σ and σ3 σ. So, a natural question arises: is it possible to sequentialize tem? Te answer is negative, as proved by te next two counterexamples. Let M = x((λy.z )(zi)) and N = (λy.xz )(zi): M σ3 (λy.xz )(zi) σ1 N, but tere exists no L suc tat M σ 1 L σ 3 N. In fact M contains only a ead σ 3 -redex and (λy.xz )(zi) contains only a ead σ 1 -redex Let M = x((λy.z )(zi) ) and N = (λy.x(z ))(zi): M σ1 x((λy.z )(zi)) σ3 N but tere is no L suc tat M σ 3 L σ 1 N. In fact M contains only a ead σ 1 -redex and x((λy.z )(zi)) contains only a ead σ 3 -redex. So, te impossibility of sequentializing a ead σ-reduction sequence is due to te fact tat a ead σ 1 -reduction step can create a ead σ 3 -redex, and viceversa. Tis is not a problem, since ead σ-reduction is strongly normalizing (by Proposition 4 and since σ σ ). Our approac does not force a strict order of ead σ-reductions. 4 Standardization Now we are able to prove te main result of tis paper, i.e., a standardization teorem for λ σ v (Teorem 25). In particular we provide a notion of standard reduction sequence tat avoids any auxiliary notion of residual redexes, by closely following te definition given in [15]. Notation. For any k, m N wit k m, we denote by M 0,, M k, M m ead a sequence of terms suc tat M i βv M i+1 wen 0 i < k, and M i σ M i+1 wen k i < m. It is easy to ceck tat M ead for any M Λ. Te notion of standard sequence of terms is defined by using te previous notion of ead-sequence. Our notion of standard reduction sequence is mutually defined togeter wit te notion of inner-sequence of terms (Definition 23). Tis definition allows us to avoid non-deterministic cases remarked in [7]

11 G. Guerrieri, L. Paolini and S. Ronci Della Rocca 11 (we provide more details at te end of tis section). We denote by M 0,, M m std (resp. M 0,, M m in ) a standard (resp. inner) sequence of terms. Definition 23 (Standard and inner sequences). Standard and inner sequences of terms are defined by mutual induction as follows: 1. if M 0,, M m ead and M m,, M m+n in ten M 0,, M m+n std, were m, n N; 2. M in, for any M Λ; 3. if M 0,, M m std ten λz.m 0,, λz.m m in, were m N; 4. if V 0,, V std and N 0,, N n in ten V 0 N 0,, V 0 N n,, V N n in, were, n N; 5. if N 0,, N n in, M 0,, M m std and N 0 Λ v, ten N 0 M 0,, N n M 0,, N n M m in, were m, n N. For instance, let M = (λy.ix)(z( I))(II): M v (λy.ix)(z( I))I v (λy.x)(z( I))I and M v (λy.ix(ii))(z( I)) v (λy.ix(ii))(z(ii)) are not standard sequences; M v (λy.ix)(z( I))I and M v (λy.ix)(z(ii))(ii) v (λy.ix)(zi)(ii) v (λy.ix(ii))(zi) v (λy.x(ii))(zi) v (λy.xi)(zi) are standard sequences. Remark 24. For any n N, if N 0,, N n in (resp. N 0,, N n ead ) ten N 0,, N n std. Indeed, N 0 ead (resp. N n in by Definition 23.2), so N 0,, N n std by Definition In particular, N std for any N Λ: apply Definition 23.2 and Remark 24 for n = 0. Teorem 25 (Standardization). 1. If M v M ten tere is a sequence M,, M std. 2. If M v M ten tere is a sequence M,, M in. Proof. Bot statements are proved simultaneously by induction on M Λ. 1. If M = z ten, by Teorem 22, M β v L σ N z for some L, N Λ. By Remarks 12.5 and 11.2, L = N = z; terefore M β v z and ence tere is a sequence M,, z ead. Tus, M,, z std by Remark 24. If M = λz.n ten, by Teorem 22, M β v L σ L v λz.n for some L, L Λ. By Remarks 12.5 and 11.2, L = L = λz.n wit N v N. So M β v λz.n and ence tere is a sequence M,, λz.n ead. By induction on (1), tere is a sequence N,, N std, tus λz.n,, λz.n in by Definition Terefore M,, λz.n,, λz.n std by Definition If M = N L ten, by Teorem 22, M β v M σ M 0 v N L for some M, M 0 Λ. By Remark 3, M 0 = NL for some N, L Λ, since v v and M / Λ v. Tus tere is a sequence M,, M,, NL ead. By Remark 12.5, NL N L ; clearly, eac step of is an instance of te rule rigt of Definition 9. Tere are two sub-cases. If N Λ v ten N N and L L, so N v N and L v L by Remarks By induction respectively on (1) and (2), tere are sequences N,, N std and L,, L in, tus NL,, NL,, N L in by Definition Terefore M,, M,, NL,, NL,, N L std by Definition If N / Λ v (i.e., N =VM 1... M m wit m>0) ten N N and L L, so N v N and L v L by Remarks By induction respectively on (2) and (1), tere are sequences N,, N in and L,, L std. Hence NL,, N L,, N L in by Definition Tus M,, M,, NL,, N L,, N L std by Definition If M = z ten M = z by Remark 12.5, ence z in by Definition If M = λz.l ten M = λz.l and L v L by Remark Hence tere is a sequence L,, L std by induction on (1). By Definition 23.3, λz.l,, λz.l in. v

12 12 Standardization of a call-by-value lambda-calculus If M = N L ten M = NL for some N, L Λ by Remark 3, since v v and M / Λ v. By Remark 12.5, NL N L ; clearly, eac step of is an instance of te rule rigt of Definition 9. Tere are two sub-cases. If N Λ v ten N N and L L, so N v N and L v L by Remarks Tus tere are sequences N,, N std and L,, L in by induction respectively on (1) and (2). Terefore, by Definition 23.4, NL,, NL,, N L in. If N / Λ v (i.e. N = VM 1... M m wit m > 0) ten N N and L L, tus N v N and L v L by Remarks By induction respectively on (2) and (1), tere are sequences N,, N in and L,, L std. So NL,, N L,, N L in by Definition Due to non-sequentialization of ead σ 1 - and ead σ 3 -reductions, several standard sequences may ave te same starting term and ending term: for instance, if M = I( I)I and N = (λz.(λx.xi)(zz))i ten M v (λx.xi)( I) v N and M v (λz.i(zz))ii v (λz.i(zz)i)i v N are bot standard sequences from M to N. Finally, we can compare our notion of standardization wit tat given in [15]. To make te comparison possible we avoid σ-reductions and we recall tat βv is exactly te Plotkin s left-reduction [15, p. 136]. As remarked in [7, 1.5 p. 149], bot sequences (λz.ii)(ii) v (λz.i)(ii) v (λz.i)i and (λz.ii)(ii) v (λz.ii)i v (λz.i)i are standard according to [15]. On te oter and, only te second sequence is standard in our sense. It is easy to ceck tat collapsing te two notions of inner and standard sequence given in Definition 23, we get a notion of standard sequence tat accept bot te above sequences. 5 Some conservativity results Te sequentialization result (Teorem 22) as some eresting semantic consequences. It allows us to prove tat (Corollary 29) te λ σ v -calculus is sound wit respect to te call-byvalue observational equivalence roduced by Plotkin in [15] for λ v. Moreover we can prove tat some notions, like tat of potential valuability and solvability, roduced in [13] for λ v, coincide wit te respective notions for λ σ v (Teorem 31). Tis justifies te idea tat λ σ v is a useful tool for studying te properties of λ v. Our starting po is te following corollary. Corollary If M v V Λ v ten tere exists V Λ v suc tat M β v V v V. 2. For every V Λ v, M β v V if and only if M v V. Proof. Te first po is proved by observing tat, by Teorem 22, tere are N, L Λ suc tat M β v L σ N v V. By Remark 12.5, N Λ v and tus L = N according to Remark Concerning te second po, te rigt-to-left direction is a consequence of Lemma 14.3 and Remark 11.2; te left-to-rigt direction follows from βv v. Let us recall te notion of observational equivalence defined by Plotkin [15] for λ v. Definition 27 (Halting, observational equivalence). Let M Λ. We say tat ( te evaluation of) M alts if tere exists V Λ v suc tat M β v V. Te ( call-by-value) observational equivalence is an equivalence relation = on Λ defined by: M = N if, for every context C, one as tat C M alts iff C N alts. 1 1 Original Plotkin s definition of call-by-value observational equivalence (see [15]) also requires tat C M and C N are closed terms, according to te tradition identifying programs wit closed terms.

13 G. Guerrieri, L. Paolini and S. Ronci Della Rocca 13 Clearly, similar notions can be defined for λ σ v using v instead of βv. Head σ-reduction plays no role neiter in deciding te alting problem for evaluation (Corollary 26.1), nor in reacing a particular value (Corollary 26.2). So, we can conclude tat te notions of alting and observational equivalence in λ σ v coincide wit te ones in λ v, respectively. Now we compare te equational teory of λ σ v wit Plotkin s observational equivalence. Teorem 28 (Adequacy of v-reduction). If M v M ten: M alts iff M alts. Proof. If M β v V Λ v ten M v M v V since βv v. By Corollary 26.1, tere exists V Λ v suc tat M β v V. Tus M alts. Conversely, if M β v V Λ v ten M v V since βv v. By confluence of v (Proposition 4, since M v M ) and Remark 3 (since V Λ v ), tere is V Λ v suc tat V v V and M v V. By Corollary 26.1, tere is V Λ v suc tat M β v V. So M alts. Corollary 29 (Soundness). If M = v N ten M = N. Proof. Let C be a context. By confluence of v (Proposition 4), M = v N implies tat tere exists L Λ suc tat M v L and N v L, ence C M v C L and C N v C L. By Teorem 28, C M alts iff C L alts iff C N alts. Terefore, M = N. Plotkin [15, Teorem 5] as already proved tat M = βv N implies M = N, but our Corollary 29 is not obvious since our λ σ v -calculus equates more tan Plotkin s λ v -calculus (= βv = v since βv v, and Example 5 sows tat tis inclusion is strict). Te converse of Corollary 29 does not old since λx.x(λy.xy) = but λx.x(λy.xy) and are different v-normal forms and ence λx.x(λy.xy) v by confluence of v (Proposition 4). A furter remarkable consequence of Corollary 26.1 is tat te notions of potential valuability and solvability for λ σ v -calculus (studied in [3]) can be sown to coincide wit te ones for Plotkin s λ v -calculus (studied in [13, 14]), respectively. Let us recall teir definition. Definition 30 (Potential valuability, solvability). Let M be a term: M is v-potentially valuable (resp. β v -potentially valuable) if tere are m N, pairwise distinct variables x 1,..., x m and V, V 1,..., V m Λ v suc tat M{V 1 /x 1,..., V m /x m } v V (resp. M{V 1 /x 1,..., V m /x m } β v V ); M is v-solvable (resp.β v -solvable) if tere are n,m N, variables x 1,..., x m and N 1,...,N n Λ suc tat (λx 1... x m.m)n 1 N n v I (resp. (λx 1... x m.m)n 1 N n β v I). Teorem 31. Let M be a term: 1. M is v-potentially valuable if and only if M is β v -potentially valuable; 2. M is v-solvable if and only if M is β v -solvable. Proof. In bot pos, te implication from rigt to left is trivial since βv v. Let us prove te oter direction. 1. Since M is v-potentially valuable, tere are variables x 1,..., x m and V, V 1,..., V m Λ v (wit m 0) suc tat M{V 1 /x 1,..., V m /x m } v V ; ten, tere exists V Λ v suc tat M{V 1 /x 1,..., V m /x m } β v V by Corollary 26.1 and because βv βv, terefore M is β v -potentially valuable. 2. Since M is v-solvable, tere exist variables x 1,..., x m and terms N 1,..., N n (for some n, m 0) suc tat (λx 1... x m.m)n 1 N n v I; ten, tere exists V Λ v suc tat (λx 1... x m.m)n 1 N n β v V v I by Corollary 26.1 and because βv βv. According to Remark 12.5, V = λx.n for some N Λ suc tat N v x. By Corollary 26.1, tere is V Λ v suc tat N β v V v x, ence V = x by Remark 12.5 again. Since βv βv, N β v x and tus V = λx.n β v I, terefore M is β v -solvable.

14 14 Standardization of a call-by-value lambda-calculus So, due to Teorem 31, te semantic (via a relational model) and operational (via two sub-reductions of v ) caracterization of v-potential valuability and v-solvability given in [3, Teorems 24-25] is also a semantic and operational caracterization of β v -potential valuability and β v -solvability. Te difference is tat in λ σ v tese notions can be studied operationally inside te calculus, wile it as been proved in [13, 14] tat te β v -reduction is too weak to caracterize tem: an operational caracterization of β v -potential valuability and β v -solvability cannot be given inside λ v. Hence, λ σ v is a useful, conservative and complete tool for studying semantic properties of λ v. 6 Conclusions In tis paper we ave proved a standardization teorem for te λ σ v -calculus roduced in [3]. Te used tecnique is a notion of parallel reduction. Let us recall tat parallel reduction in λ-calculus as been defined by Tait and Martin-Löf in order to prove confluence of te β-reduction, witout referring to te difficult notion of residuals. Takaasi in [17] as simplified tis tecnique and sowed tat it can be successfully applied to standardization. We would like to remark tat our parallel reduction cannot be used to prove confluence of v. Indeed, take M = (λx.l) ( (λy.n)((λz.n )N ) ) L, M 1 = (λx.ll ) ( (λy.n)((λz.n )N ) ) and M 2 = ( (λy.(λx.l)n)((λz.n )N ) ) L : ten M M 1 and M M 2 but tere is no term M suc tat M 1 M and M 2 M. To sum up, does not enjoy te Diamond Property. Te standardization result allows us to formally verify te correctness of λ σ v wit respect to te semantics of λ v, so we can use λ σ v as a tool for studying properties of λ v. Tis is a remarkable result: in fact some properties, like potential valuability and solvability, cannot be caracterized in λ v by means of β v -reduction (as proved in [13, 14]), but tey ave a natural operational caracterization in λ σ v (via two sub-reductions of v ). We plan to continue to explore te call-by-value computation, using λ σ v. As a first step, we would like to revisit and improve te Separability Teorem given in [11] for λ v. Still te issue is more complex tan in te call-by-name, indeed in ordinary λ-calculus different βη-normal forms can be separated (by te Böm Teorem), wile in λ v tere are different normal forms tat cannot be separated, but wic are only semi-separable (e.g. I and λz.(λu.z)(zz)). We ope to completely caracterize separable and semi-separable normal forms in λ σ v. Tis sould be a first step aimed to define a semantically meaningful notion of approximants. Ten, we sould be able to provide a new insigt on te denotational analysis of te call-by-value, maybe overcoming limitations as tat of te absence of fully abstract filter models [14, Teorem ]. Last but not least, an unexplored but callenging researc direction is te use of commutation rules to improve te call-by-value evaluation. We do not ave concrete evidence supporting suc possibility, but since λ σ v is strongly related to te calculi presented in [7, 1], wic are endowed wit explicit substitutions, we are confident tat a sarp use of commutations can ave a relevant impact in te evaluation. References 1 Beniamino Accattoli and Luca Paolini. Call-by-Value Solvability, Revisited. In Tom Scrijvers and Peter Tiemann, editors, Functional and Logic Programming, volume 7294 of Lecture Notes in Computer Science, pages Springer-Verlag, Henk Barendregt. Te Lambda Calculus: Its Syntax and Semantics, volume 103 of Studies in logic and te foundation of matematics. Nort Holland, 1984.

15 G. Guerrieri, L. Paolini and S. Ronci Della Rocca 15 3 Alberto Carraro and Giulio Guerrieri. A Semantical and Operational Account of Call-by- Value Solvability. In Anca Muscoll, editor, Foundations of Software Science and Computation Structures, volume 8412 of Lecture Notes in Computer Science, pages Springer-Verlag, Karl Crary. A Simple Proof of Call-by-Value Standardization. Tecnical Report CMU-CS , Carnegie Mellon University, Haskell B. Curry and Robert Feys. Combinatory Logic, volume 1. Nort Holland, Jean-Yves Girard. Linear logic. Teoretical Computer Science, 50(1):1 102, Hugo Herbelin and Stépane Zimmermann. An Operational Account of Call-by-Value Minimal and Classical lambda-calculus in "Natural Deduction" Form. In Pierre-Louis Curien, editor, Typed Lambda Calculi and Applications, volume 5608 of Lecture Notes in Computer Science, pages Springer-Verlag, Roger Hindley. Standard and normal reductions. Transactions of te American Matematical Society, pages , Jon Maraist, Martin Odersky, David N. Turner, and Pilip Wadler. Call-by-name, call-byvalue, call-by-need and te linear lambda calculus. Teoretical Computer Science, 228(1 2): , Eugenio Moggi. Computational Lambda-Calculus and Monads. In Logic in Computer Science, pages IEEE Computer Society, Luca Paolini. Call-by-Value Separability and Computability. In Antonio Restivo, Simona Ronci Della Rocca, and Luca Roversi, editors, Italian Conference in Teoretical Computer Science, volume 2202 of Lecture Notes in Computer Science, pages Springer-Verlag, Luca Paolini and Simona Ronci Della Rocca. Parametric parameter passing lambdacalculus. Information and Computation, 189(1):87 106, Luca Paolini and Simona Ronci Della Rocca. Call-by-value Solvability. Teoretical Informatics and Applications, 33(6): , RAIRO Series, EDP-Sciences. 14 Luca Paolini and Simona Ronci Della Rocca. Te Parametric λ-calculus: a Metamodel for Computation. Texts in Teoretical Computer Science: An EATCS Series. Springer-Verlag, Gordon D. Plotkin. Call-by-name, call-by-value and te lambda-calculus. Teoretical Computer Science, 1(2): , Amr Sabry and Mattias Felleisen. Reasoning about programs in continuation-passing style. Lisp and symbolic computation, 6(3-4): , Masako Takaasi. Parallel reductions in lambda-calculus. Information and Computation, 118(1): , 1995.