J.N. Kok and J.J.M.M. Rutten. Contractions in comparing concurrency semantics. Theoretical Computer Science, 76:179--222, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....function space. By Banach s Theorem they have a unique fixed point, which is the object specified by the equations of the original definitions. The nice point of using metric spaces as underlying mathematical structure is the close connection between measure theory and metric topology. We refer to [16,4] for a further discussion on the use of complete metric spaces and Banach s Theo3 rem in the area of programming language semantics. The advantage of using metric spaces is also reflected in a number of semantical investigations related to probability. See [7,17,5] for example. The mathematical ....

....the proof is cumbersome and the details are omitted here. However, it is here that we benefit from using the metric approach to programming language semantics, where Banach s Fixed Point Theorem 10 is instrumental in the justification of circular definitions. We mention the references [16,4,11] for more details of the application of this technique. Lemma 3.6 (a) T 0 is finitely branching. b) The transition system T 0 has no internal divergence, i.e. there are no statements s 0 ; s 1 ; and 1 ; 2 ; in fg[ 0; 1) such that s 0 1 s 1 2 Delta Delta Delta is an ....

J.N. Kok and J.J.M.M. Rutten. Contractions in comparing concurrency semantics. Theoretical Computer Science, 76:179--222, 1990.

....the operational and denotational semantics, respectively. In the design of O, arrays of processes are modelled using the concept of (nested) resumptions. For D, continuations are an essential tool. In Section 3, we prove the equivalence of O and D using the unique fixed point proof principle from [KR90]. Let us mention one subtlety in the semantic models: in order to apply Banach s theorem, we require contractiveness at various instances. At appropriate points a version of Park s hiaton ( Par83] is used to enforce contractiveness if this would not arise naturally. We now present the syntax of ....

J.N. Kok and J.J.M.M. Rutten. Contractions in Comparing Concurrency Semantics. Theoretical Computer Science, 76(2/3):179--222, 1990.

....(x) fix (f) where f 0 (x) x and f n 1 (x) f (f n (x) Proof See Theorem II.6 of [Ban22] ut In this paper, several recursive definitions are presented (cf. Definition 4.1, 4.3, 4.4, 6.1, and 6. 3) Banach s theorem can be used to prove the well definedness of these definitions (cf. [KR90]) The embeddings to be introduced in Section 4 will be defined by means of nonexpansive functions. Definition 1.3 Let (X; dX ) and (X 0 ; dX 0 ) be metric spaces. A function f : X X 0 is called nonexpansive if, for all x and x 0 , dX 0 (f (x) f (x 0 ) dX (x; x 0 ) 2. Three ....

.... composition, trace set, and fairification on P 1 processes similar complications arise (cf. BK87, BBKM84, Bre94] These complications do not arise in the definitions of the operators on P 3 processes (cf. Bre94] Also process domain P 2 does not give rise to these complications (cf. [KR90]) However, unlike process domain P 3 , process domain P 2 does not allow an elementary modelling of image finite language constructions like random assignment (cf. Bre94] Concluding remarks 13 Concluding remarks In this concluding section, some related work is discussed and some points for ....

J.N. Kok and J.J.M.M. Rutten. Contractions in Comparing Concurrency Semantics. Theoretical Computer Science, 76(2/3):179--222, 1990.

....(oe : oefff=vg Delta O (r) oefff=vg) oe : oefff=vg Delta D (r) oefff=vg) 1 2 Delta d (O (r) D (r) 1 2 Delta sup f d (O (r 0 ) D (r 0 ) j r 0 2 Res 1 g: Consequently, for all r, d (O (r) D (r) 0. Hence O = D . ut The first proof follows [KR90] (cf. BM88] but with a substantial simplification thanks to our avoiding procedure environments. corollary 1.10 For all s, O (s) D (s) oe : f g) 2. A sequential language with second order assignment The central notion of this section is second order assignment, in the form of the statement ....

J.N. Kok and J.J.M.M. Rutten. Contractions in Comparing Concurrency Semantics. Theoretical Computer Science, 76(2/3):179--222, 1990.

....to occur at the same time. The more complex operational semantics O s and the denotational semantics Den have no counterpart in [Br90] FLP84] and [Mo81] Although they are of classical metric inspiration, these last two semantics still present some originality with related work ( BZ82] B91] [KR90], BKPR89] It arises essentially from the three following points : i) our concern with generalized Horn clauses, which has not been done before and which requires new solutions; in particular, it should be noticed that the form of communication provided by the Pi and ....

Kok J.N., Rutten J.J.M.M., Contractions in Comparing Concurrency Semantics, Theoretical Computer Science, 76, 1990, pp. 179-222.

....exploited to handle recursion. This denotational model uses a complete metric space as its mathematical domain, which has been initiated by Nivat [40] and de Bakker and Zucker [16] To compare these models a method based on higher order transformations, which has been described by Kok and Rutten [36], is used. This general method for comparing different semantic models has already been applied successfully to several programming paradigms varying from notions related to concurrency [7, 13, 14] to notions related to logic programming [6, 10, 20] and object oriented programming [15, 44] The ....

....captured by the operational model. In this section we will show that we can relate the operational model O and the denotational model D. We will prove that these models are equivalent. To prove this we will use a general method for comparing different semantic models as described by Kok and Rutten [36]: if two models are both a fixed point of a higher order transformation and this higher order transformation is a contraction on a complete metric space, we can conclude that those models are equivalent. We will introduce an intermediate operational semantic model O d and relate this model to ....

J.N. Kok and J.J.M.M. Rutten. Contractions in Comparing Concurrency Semantics. Theoretical Computer Science 76 (1990), 179-222.

....operational semantics 3 . If an operational semantics has one of the above mentioned topological properties, then we can possibly use the unique fixed point proof principle in order to relate the semantics to another semantics. This proof principle has been introduced by Kok and Rutten in [KR90]. It is based on Banach s fixed point theorem ( Ban22] a contractive mapping from a complete metric space to itself has a unique fixed point. The proof principle has been applied successfully to relate semantic models for various programming language notions (see, e.g. the theses [AR89a, Eli91, ....

....loe j c l Gamma c 0 oe 2 O (c 0 ) g otherwise Theorem 4. 5 generalizes the already known result that the operational semantics induced by a finitely branching or an image finite labelled transition system is the unique fixed point of the induced higher order transformation (see, e.g. [KR90]) Second, we discuss the so called branching higher order transformations. Property 4.6 1. A compactly branching and nonexpansive metric labelled transition system (C; L; induces a branching higher order transformation Phi : C 1 B 1 [L] C 1 B 1 [L] defined by Phi (OE) c) f ....

J.N. Kok and J.J.M.M. Rutten. Contractions in Comparing Concurrency Semantics. Theoretical Computer Science, 76(2/3):179--222, 1990.

....variant of the considered program P that allows several independent reductions to occur at the same time. The denotational semantics Den has no counterpart in [3] 10] and [16] Although it is of classical metric inspiration, it still presents some originality with related work ( 6] 5] [14], which arises essentially from the two following points : i) our concern with extended Horn clauses, which has not been done before and which requires new solutions; in particular, it should be noticed that the form of communication provided by the Pi and operators is different ....

....relation , and of the declarative semantics Decl m and Decl f , based on model theory and fixed point theory, respectively. The other semantics, namely the denotational semantics Den, is issued from the imperative tradition, and, more particularly, from its metric semantic branch ( 6] 5] [14], It describes computations, in a compositional way, via histories, possibly including hypotheses. All these semantics have been related throughout the paper, thanks to propositions, 15, 16 and 34. The minimal relations have only been stated. From them, it is possible to deduce other ....

J.N. Kok and J.J.M.M. Rutten. Contractions in Comparing Concurrency Semantics. Theoretical Computer Science, 76:179--222, 1990.

....will have a premise that is syntactically more complex than the conclusion. This rule is allowed because the limitation to guarded recursion prevents problems. The proofs can be given by using induction on the weight function introduced below. Induction on a weight function was introduced in [20] and the systematic use is due to Van Breugel [12] Definition 3.2 wgt: Res N is given by wgt(E) 0 wgt(a) 1 wgt(x) wgt(D(x) 1 wgt(s 1 ;s 2 ) wgt(s 1 ) 1; and similarly for k Gamma wgt(s 1 ks 2 ) 1 wgt(s 1 k Gamma s 2 ) wgt(s 2 k Gamma s 1 ) wgt(s 1 js 2 ) ....

J.N. Kok and J.J.M.M. Rutten. Contractions in comparing concurrency semantics. Theoretical Computer Science, 76:179--222, 1990.

....contribution can be found in the equivalence proof itself. The equivalence proof does not split as usual into O D and D O. Rather, both the operational and denotational semantics are represented as least upperbounds of chains and we prove equality of the approximating elements. See also [KR], BM] where although not made explicit in the setting of complete metric spaces operational and denotational semantics can be represented as limits of Cauchy sequences. The denotational semantics makes use of a fixed point construction with respect to environments. The environment ....

J.N. Kok and J.J.M.M. Rutten, "Contractions in Comparing Concurrency Semantics," pp. 317-332 in Proc. ICALP'88, T. Lepistö & A. Salomaa (eds.), LNCS 317 (1988).

....function space. By Banach s Theorem they have a unique fixed point, which is the object specified by the equations of the original definitions. The nice point of using metric spaces as underlying mathematical structure is the close connection between measure theory and metric topology. We refer to [15, 3] for a further discussion on the use of complete metric spaces and Banach s Theorem in the area of programming language semantics. The advantage of using metric spaces is also reflected in a number of semantical investigations related to probability. See [6, 16, 4] for example. The mathematical ....

....used. Though straightforward, the proof is cumbersome and omitted here. However, it is here that we benefit from using the metric approach to programming language semantics, where Banach s Fixed Point Theorem is instrumental in the justification of circular definitions. We mention the references [15, 3, 10] for more details of the application of this technique. Lemma 3.6 (a) T 0 is finitely branching. b) The transition system T 0 has no internal divergence, i.e. there are no statements s 0 ; s 1 ; and 1 ; 2 ; in fg [ 0; 1) such that s 0 1 s 1 2 Delta Delta Delta is an ....

J.N. Kok and J.J.M.M. Rutten. Contractions in comparing concurrency semantics. Theoretical Computer Science, 76:179--222, 1990.

....is guaranteed for instance when one does not use cpo s and a continuity argument to ensure the existence of the fixed points, but when complete metric spaces are used instead and the operators are contracting functions on these spaces. For in that case Banach s theorem can be applied. In [KR] unicity of the fixed point of the operational higher order operator F has been exploited successfully to derive compact equivalence proofs for operational and denotational meanings along the lines sketched above. A similar line of reasoning has been used in [BM] The fact that our operator F ....

....described language B: Section 4 describes the operational and denotational semantics of B. Section 5 will be devoted to the actual equivalence proof. Acknowledgements. From the above it will be clear that our work relies heavily on that of others. The immediate starting point of this paper is [KR]a where for the first time the observation was made that compact equivalence proofs could be realized using higher order transformations. To a great extent, we also benefit from the work on metric semantics of concurrency performed by De Bakker e.a. e.g. BZ] BKMOZ] BM]a, Ba2] It is a ....

[Article contains additional citation context not shown here]

J.N. Kok and J.J.M.M. Rutten, "Contractions in Comparing Concurrency Semantics," 21. pp. 317-332 in Proc. ICALP'88, T. Lepistö & A. Salomaa (eds.), LNCS 317 (1988).

....metric spaces as a tool over the allegedly simpler cpo s. The main advantage of the metric approach that they mention is the usefulness of Banach s unique fixed point theorem for complete metric spaces over Kleene s least fixed point theorem for cpo s. This is illustrated by Kok and Rutten [8], who prove that the denotational and operational semantics of certain languages coincide by showing that both are fixed points of the same contractive functional on a complete metric space. We apply this technique in Section 8 to establish computational adequacy of a metric model of PCF. They ....

J.N. Kok and J.J.M.M. Rutten. Contractions in comparing concurrency semantics. Theoretical Computer Science, 76:179--222, 1990. Escard' o

....e.g. in Section 2.4 of [Pra86] of the preorders corresponding to g X and g Y . We conjecture that in this setting Property 5.4 and 5.5 are still valid. We are interested to see whether constructions like the sequential composition of branching processes (as defined in, e.g. Definition 2. 14 of [KR90]) are universal. ....

J.N. Kok and J.J.M.M. Rutten. Contractions in Comparing Concurrency Semantics. Theoretical Computer Science, 76(2/3):179--222, November 1990.

....both semantic operators assign to each statement a meaning that is a maximal element in the powerdomain based on the Egli Milner ordering described above. Equivalence can then, for instance, be proved using a characterization of O as the unique fixed point of a higher order transformation. cf. [KR] for this approach in a metrical setting. However to deal with the present situation first we will introduce in section 5 the variants O and D of O and D, which differ from the latter semantics only in the fact that for each execution of a procedure call a silent step t is recorded in the ....

....of retracts. We will introduce less abstract semantical definitions O d and D d (for d Decl) in which the silent t steps can be observed. We relate the abstract and concrete models for the operational and denotational case. The picture is then completed by establishing O d =D d in a [KR]a style, using a higher order transformation with a unique fixed point. We start with defining a t version of the operational semantics and relate it to the original one. 5.1) DEFINITION Let d Decl. Define the semantical mapping O d : Stat E P (A t st ) by O d ( s 3 ) min ....

[Article contains additional citation context not shown here]

J.N. Kok and J.J.M.M. Rutten, "Contractions in Comparing Concurrency Semantics," pp. 317-332 in Proc. ICALP'88, T. Lepistö & A. Salomaa (eds.), LNCS 317, Springer (1988).

....metric spaces they used were defined as solutions of recursive domain equations. By means of Banach s theorem America and Rutten [AR89] proved that a particular class of domain equations has unique solutions. Banach s theorem has also been used to prove semantic models to be equal. Kok and Rutten [KR90] applied a proof principle which we baptize the unique fixed point proof principle. By means of this proof principle elements of a metric space can be proved to be equal. First, one introduces a function from the metric space to itself. Second, one shows that the function is a contraction. And ....

J.N. Kok and J.J.M.M. Rutten. Contractions in Comparing Concurrency Semantics. Theoretical Computer Science, 76(2/3):179--222, November 1990.

....language B that will serve as a vehicle for an examination of the asynchronous rendez vous as present in DLP. The strategy of analyzing the control flow of a skeletal language first, before given an account of the full programming language under consideration has been shown profitable in, e.g. [Kok, 1988] and [de Bruin and de Vink, 1989] Since the semantics for (the core of) Prolog including the cut operator, and therefore of individual DLP objects, has been extensively studied in [Jones and Mycroft, 1984] de Vink, 1990] and [de Bakker, 1991] attention is given primarily to the creation of ....

J. Kok and J. Rutten, Contractions in comparing concurrency semantics, in: Proc. Automata, Languages and Programming, T. Lepisto and A. Salomaa (eds.), Lecture Notes in Computer Science 317, Springer (1988) pp. 317-332

....parallel composition are defined as the unique fixed point (Banach s theorem) of a contractive function (see Definition A.12) from the nonempty complete space IP Theta IP 1 IP to itself. We only give the equations characterizing the semantic operators. For the details we refer the reader to [ABKR89, KR90] where various related semantic operators are defined in this way. 7 An occurrence of IP is negative if it is to the left, hereditarily, of an odd number of 1 s. 8 This terminology is taken from [RT92] In [AR89] hom contractive is used instead. Definition 3.6 The operator ; is the ....

....D (s) D [ s] D (p) p D (r ; s) D (r) D (s) D (r 1 k r 2 ) D (r 1 ) k D (r 2 ) The semantic models I and D are shown to be equivalent by uniqueness of fixed point, viz we show that D is a fixed point of Phi. This proof technique is due to Kok and Rutten [KR90]. Property 4.11 Phi (D ) D . Proof First, we should check that D is nonexpansive, which can be verified by structural induction. Second, we should show that D is a fixed point of Phi, which can also be proved by structural induction. 2 By uniqueness of fixed point we can ....

J.N. Kok and J.J.M.M. Rutten. Contractions in Comparing Concurrency Semantics. Theoretical Computer Science, 76(2/3):179--222, November 1990.

....function space. By Banach s Theorem they have a unique fixed point, which is the object specified by the equations of the original definitions. The nice point of using metric spaces as underlying mathematical structure is the close connection between measure theory and metric topology. We refer to [14, 3] for a further discussion on the use of complete metric spaces and Banach s Theorem in the area of programming language semantics. The advantage of using metric spaces is also reflected in a number of semantical investigations related to probability. See [5, 15, 4] for example. The mathematical ....

....used. Though straightforward, the proof is cumbersome and omitted here. However, it is here that we benefit from using the metric approach to programming language semantics, where Banach s Fixed Point Theorem is instrumental in the justification of circular definitions. We mention the references [14, 3, 8] for more details of the application of this technique. Lemma 3.6 (a) T 0 is finitely branching. b) The transition system T 0 has no internal divergence, i.e. there are no statements s 0 ; s 1 ; s 2 ; and 1 ; 2 ; in fg [ 0; 1) such that s 0 1 s 1 2 s 2 Delta Delta ....

J.N. Kok and J.J.M.M. Rutten. Contractions in comparing concurrency semantics. TCS, 76:179-- 222, 1990.

....L 0 is well known (see [Dij76] including the separate treatment of guards b ( Hes88a] Probably less familiar is the justification of the above definition of wp 0 . The semantics can be obtained as least fixed point of a higher order transformation, as promoted originally in a metric setting in [KR90], thus avoiding employment of environments. We need at first a map which assigns a natural number to every statement in L 0 to be used in proofs based on induction. 5.4. Definition. The map wgt 0 : L 0 N is given by wgt 0 (hd; v : ei) 1; wgt 0 (hd; b i) 1; wgt 0 (hd; xi) wgt 0 (hd; ....

J.N. Kok and J.J.M.M. Rutten. Contractions in Comparing Concurrency Semantics. Theoretical Computer Science, 76 (2/3):179--222, 1990.

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J.N. Kok and J.J.M.M. Rutten. Contractions in Comparing Concurrency Semantics. Theoretical Computer Science, 76:179--222, 1990.

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