M. Bidoit and A. Tarlecki. Behavioural satisfaction and equivalence in concrete model categories. In Proc. 21st Int. Coll. on Trees in Algebra and Programming (CAAP '96), volume 1059 of LNCS, pages 241--256. Springer, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....contain properly continuous algebras [15] intended usually to model in nite datatypes. Unfortunately continuous algebras are not well suited for behavioural semantics of speci cation, as they lack some crucial algebraic properties, e.g. quotients of continuous algebras do not compose (cf. [5]) The subject of this paper is to analyze the applicability of regular algebras as models of behavioural speci cations in the process of stepwise development of software systems. A general methodology is proposed for regular algebras, as an adaptation of the constructor behavioural ....

....comments during this work. 1 Preliminaries Regular algebras Let , be xed many sorted algebraic signatures throughout this paper. We omit introducing classical notions of standard algebra, homomorphism (the category of those is denoted Alg ) subalgebra, congruence, quotient (cf. e.g. [5]) By t A[v] we denote the value of term t in algebra A under valuation v. For A being a standard or regular algebra, by jAj we denote the many sorted carrier set of A; by jAj s the carrier of sort s; by jAj S , for a subset S of sorts of , the carrier sets of sorts from S. All sets are ....

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Bidoit, M., Tarlecki A. Behavioural satisfaction and equivalence in concrete model categories. Manuscript. A short version appeared in Proc. 20th Coll. on Trees in Algebra and Computing CAAP'96, Linkoping, LNCS 1059, 241-256, SpringerVerlag, 1996.

....= B= B R We say that behavioural equivalence is factorized by the family of indistinguishability relations. This scheme is appropriate for various categories of algebras, and more generally for an arbitrary concrete category supporting algebraic constructions of subobject and quotient [BT96]. The undeniable conceptual analogies (despite technical di erences) between both approaches are well recognized and much work was done in linking the two notions, mainly in applying methods based on bisimulation to sequential programs [AO93, Gor95] In this paper, roughly speaking, we ....

.... has been rst introduced in [Bed91] called there hereditary history preserving bisimilarity) and rediscovered independently in [JNW93] similar ideas can be found already in [DNM89] Behavioural equivalence in a concrete category We introduce here a categorical setting, borrowed from [BT96], in which we are going to work in the following sections. All the de nitions below, being very abstract, follow usual algebraic intuitions. We aim at identifying abstractly some constructions on algebras, like quotients and subobjects, sucient for dealing with behavioural equivalence. be an ....

[Article contains additional citation context not shown here]

Bidoit, M., Tarlecki A. Behavioural satisfaction and equivalence in concrete model categories. Proc. 20th Coll. on Trees in Algebra and Computing CAAP'96, Linkoping, LNCS 1059 (Springer-Verlag, 1996) 241-256.

....bisimilar if they are related by a span of open maps, representing abstractly a bisimulation. C # # # # # ### # # # # # A B (1) In [3] one can nd an overview of di erent equivalence de nable by means of open maps. Open maps were also successfully applied to behavioural equivalences [1] of algebras, see [10] In this paper we focus on weak bisimilarity of transition systems, proved already in [3] to coincide with open maps bisimilarity in the category of transition systems. The method was similar as in the case of strong bisimilarity, that is open maps were characterized as ....

Bidoit, M., Tarlecki A. Behavioural satisfaction and equivalence in concrete model categories. Proc. 20th Coll. on Trees in Algebra and Computing CAAP'96, Linkoping, 241-256, LNCS 1059, 1996.

....external observational equivalence relation on algebras, and re interpret specifications by closing their class of models under such equivalence. It turns out that under some acceptable technical conditions, the two approaches are closely related and coincide for most basic specifications [BHW95,BT96] From now on, for the sake of simplicity, we will assume that all signatures contain a distinguished Boolean part: a sort bool with two constants true and false preserved by all signature morphisms. Similarly, we restrict attention to algebras with a fixed, standard interpretation of the ....

M. Bidoit and A. Tarlecki. Behavioural satisfaction and equivalence in concrete model categories. Proc. 20th Coll. on Trees in Algebra and Computing CAAP'96, Linkoping, Springer LNCS 1059, 241--256 (1996).

....: v n ) b [ c] A 0 (v n ) and such that b is a bijection for b 2 OBS ) This characterization is useful for proving that specific algebras are In fact, the definition of A j OBS A in [Sch90] requires [ b] A = b] A 0 for all b 2 24 behaviourally equivalent. Very recently, [BT96] has generalized this result. First, they consider an arbitrary concrete category of models, rather than that of ordinary algebras, and study the concepts of behavioural satisfaction and behavioural equivalence in this context. They then generalize the characterization theorem of [Sch90] to the ....

....dec(dec(inc(inc(zero) In [Sch94] an expressibility result analogous to Theorem 5.4 for the indistinguishability relation used in [Rei85] is given for a language of second order logic. Detailed comparisons are rendered difficult by the fact that the logic used there is untyped. Very recently, [BT96] has shown that regularity (Proposition 4.7 above) among other properties of OBS , follows from the characterization of OBS as the greatest partial congruence that is equality on OBS and is defined only for OBS reachable values. Let = hA i A2Alg ( Sigma) be a family of partial congruences ....

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M. Bidoit and A. Tarlecki. Behavioural satisfaction and equivalence in concrete model categories. Proc. 21st Colloq. on Trees in Algebra and Programming, Linkoping. Springer LNCS 1059, 241--256 (1996). 47

.... each signature Sigma, its sorts S; correspondingly, there is a natural transformation j Gammaj:Mod ffl SSet ffi Sorts op giving, for each model A over Sigma with sorts S, its carrier which is an S sorted set; in other words we consider model parts of concrete institutions in the sense of [12]. This is technically needed since methods manipulate elements in states, and not restrictive since mostly commonly used formalisms are in this class [12] Gamma There is a notion of extension E allowing to define an observational equivalence between object premodels. In the first subsection we ....

.... over Sigma with sorts S, its carrier which is an S sorted set; in other words we consider model parts of concrete institutions in the sense of [12] This is technically needed since methods manipulate elements in states, and not restrictive since mostly commonly used formalisms are in this class [12]. Gamma There is a notion of extension E allowing to define an observational equivalence between object premodels. In the first subsection we state the requirements over Sig and define on top of it the category OSig of object signatures. In the second subsection we state the requirements over ....

Bidoit, M. and A. Tarlecki: 1996, `Behavioural Satisfaction and Equivalence in Concrete Model Categories'. In: H. Kirchner (ed.): CAAP '96 - 20th Coll. on Trees in Algebra and Computing. Berlin, pp. 241--256.

....above. However, it is easy to see that the approach is (rather) institution independent concerning the data state models. The only requirement used in this paper has been that data state models have carrier sets from which the parameters can be chosen, i.e. the model theory is concrete (see [BT96]) and that the model categories have limits and colimits. In this way institution independent transformation systems can be defined. What is left open in this paper are the development of a syntax to represent or specify the transition graphs, and axioms for the description of transformation ....

M. Bidoit and A. Tarlecki. Behavioural satisfaction and equivalence in concrete model categories. In Proc. CAAP'96, Springer LNCS 1059. 1996.

....A 0 (v 0 1 ; v 0 n ) and such that b is a bijection for b 2 OBS ) 4 This characterization is useful for proving that specific algebras are 4 In fact, the definition of A j OBS A 0 in [Sch90] requires [ b] A = b] A 0 for all b 2 24 behaviourally equivalent. Very recently, [BT96] has generalized this result. First, they consider an arbitrary concrete category of models, rather than that of ordinary algebras, and study the concepts of behavioural satisfaction and behavioural equivalence in this context. They then generalize the characterization theorem of [Sch90] to the ....

....dec(dec(inc(inc(zero) In [Sch94] an expressibility result analogous to Theorem 5.4 for the indistinguishability relation used in [Rei85] is given for a language of second order logic. Detailed comparisons are rendered difficult by the fact that the logic used there is untyped. Very recently, [BT96] has shown that regularity (Proposition 4.7 above) among other properties of OBS , follows from the characterization of OBS as the greatest partial congruence that is equality on OBS and is defined only for OBS reachable values. Let = hA i A2Alg ( Sigma) be a family of partial congruences ....

[Article contains additional citation context not shown here]

M. Bidoit and A. Tarlecki. Behavioural satisfaction and equivalence in concrete model categories. Proc. 21st Colloq. on Trees in Algebra and Programming, Linkoping. Springer LNCS 1059, 241--256 (1996). 47

....for the semantical equivalence of both kinds of specifications and shows that behavioral specifications can be characterized by an abstractor construction and vice versa. These results are generalized to higher order logic in [501] and to an even more general setting based on institutions in [115]. Constraints In some approaches, restrictions to the class of models of a given specification are given explicitly using socalled constraints. A constraint is a semantic construction fixing or restricting the interpretation of a specification contained within a larger one. As a ....

M. Bidoit and A. Tarlecki. Behavioural satisfaction and equivalence in concrete model categories. In Proc. CAAP '96, 1996. To appear. \Phi.

....we mean a behavioural equivalence of algebras whereas term behavioural equivalence will be used more generally, for observational equivalence as well as for bisimilarity. of standard algebras [Rei81] ST87] but also in some other frameworks, like regular algebras [BT95] and partial algebras [BT96]. Another approach to behavioural equivalence is the notion of bisimilarity of concurrent processes [Mil89] Par81] Two processes are bisimilar if there exists a relation of bisimulation between them, meaning that every action of one process can be simulated in the other one and vice versa. ....

....observational equivalence. In Section 4.2 the observational equivalence of partial algebras is proved to be a bisimilarity induced by union of all families of free objects of some multiadjunction. This is done in the category of partial algebras with strong partial homomorphisms, similarly as in [BT96]. Such a result is not quite satisfactory, as the choice of strong homomorphisms can not be easily justified. To do the same for partial algebras with all (weak) homomorphisms, we introduce the notion of pre adjunction, being a further relaxation of requirements of multiadjunction. It turns out ....

[Article contains additional citation context not shown here]

Bidoit, M., Tarlecki A. Behavioural satisfaction and equivalence in concrete model categories. Manuscript. The short version appeared in Proc. 20th Coll. on Trees in Algebra and Computing CAAP'96, Linkoping, Springer-Verlag.

....model parts in the sense that models are (sorted) sets enriched by some structure, like operations in the case of standard algebras; correspondingly, signatures have a set of sorts. In other words, the category of Sigma models, for any signature Sigma , is a concrete model category (see [5]) i.e. a particular case of concrete category (see [1] the same notion has been called static framework in a context where the aim was to enrich such a framework by dynamic features (see [3, 15] The choice of working with concrete model parts corresponds to consider modular languages in ....

M. Bidoit and A. Tarlecki. Behavioural satisfaction and equivalence in concrete model categories. In CAAP '96, LNCS, 1059, pages 241--256, 1996. Springer Verlag.

....method can (cannot) be applied if the data state satisfies a certain condition, like being stable for instance. Furthermore a diagram language should be developed that allows to specify diagrams of connected components. Ideally such a language should also 6 i.e. the model theory is concrete, see [BT96] 7 Note that in general more colimits in the data state categories are needed than can be constructed in the category of transformation systems. e.g. if finite limits of data states exist this does not imply that all finite limits of transformation systems exist. support dynamic evolution of ....

M. Bidoit and A. Tarlecki. Behavioural satisfaction and equivalence in concrete model categories. In H. Kirchner, editor, Proc. CAAP'96, Springer LNCS 1059, pages 241--256, 1996.

.... expression ex and each Sigma 0 model M 0 [ Exp(oe) ex ) M 0 Sigma 0 = ex ] Mod(oe) M) Sigma : In order to give a natural description of programming languages, we endow generalized institutions with a notion of typing. A similar notion is that of concrete institutions (see e.g. [1]) i.e. standard institutions with boolean sentences, where signatures have an underlying set of sorts or types and models over a signature Sigma with types S have an underlying carrier which is an S sorted set. Definition2. A typed institution T GI consists of a generalized institution GI = ....

M. Bidoit and A. Tarlecki. Behavioural satisfaction and equivalence in concrete model categories. In H. Kirchner, editor, CAAP '96 - 20th Coll. on Trees in Algebra and Computing, number 1059 in LNCS, pages 241--256, Berlin, 1996. Springer Verlag.

....that signatures have a set of sorts, we consider model parts where models are (sorted) sets enriched by some structure, like operations in the case of standard algebras. We call them concrete model parts, since the category of Sigma models, for any signature Sigma, is a concrete model category [11], i.e. a particular case of concrete category [1] the same notion has been called static framework in a context where the aim was to enrich such a framework by dynamic features [4, 45] Def. 4.3 The functor SSet : Set op Set is defined as follows: ffl for any set S, the elements of SSet(S) ....

M. Bidoit and A. Tarlecki. Behavioural satisfaction and equivalence in concrete model categories. In H. Kirchner, editor, CAAP '96 - 20th Coll. on Trees in Algebra and Computing, number 1059 in Lecture Notes in Computer Science, pages 241--256, Berlin, 1996. Springer Verlag.

....S, the category of S sorted sets. The assumption that j Gammaj Sigma is faithful (i.e. all the hom set restrictions are injective) models the fact that Sigma morphisms are basically maps; in other words, for each signature Sigma over S, Alg( Sigma) is an S concrete category in the sense of [5, 1]. In particular an inclusion from A into B, denoted i A;B or A , B, is the unique Sigma morphism, if any, s.t. ji A;B j = i jAj;jBj . If Sigma , Sigma 0 and A is a Sigma 0 algebra, then we write A j Sigma instead of A ji Sigma; Sigma 0 . Assume in what follows a fixed static ....

M. Bidoit and A. Tarlecki. Behavioural satisfaction and equivalence in concrete model categories. In Proc. CAAP '96, LNCS. Springer Verlag, 1996. To appear.

No context found.

M. Bidoit and A. Tarlecki. Behavioural satisfaction and equivalence in concrete model categories. Proc. 20th Coll. on Trees in Algebra and Computing CAAP'96, Linkoping, Springer LNCS 1059, 241--256 (1996).

No context found.

M. Bidoit and A. Tarlecki. Behavioural satisfaction and equivalence in concrete model categories. Proc. 20th Coll. on Trees in Algebra and Computing CAAP'96, Linkoping, Springer LNCS 1059, 241--256 (1996).

....an external observational equivalence on algebras, and re interpret speci cations by closing their class of models under such equivalence. It turns out that under some acceptable technical conditions, the two approaches are closely related and coincide for most basic speci cations [BHW95,BT96] However, the former approach seems more dicult to extend to structured speci cations and parametrization. Hence, we follow here the latter possibility. De nition 4.1. Consider a signature with observable sorts OBS sorts( We always assume that bool 2 OBS . A correspondence between two ....

....cation may be provided via links with indistinguishability relations (via factorization properties, like Thm. 4. 3, which in turn may require a richer context of concrete institutions, with model categories equipped with concretization structure subject to a number of technical requirements as in [BT96] On the other hand, to transfer the present work to the speci c framework of Casl we need a precise and convincing de nition of observational equivalence between Casl models (many sorted algebras with predicates, partial operations and subsorting) In terms of the institutional structure ....

M. Bidoit and A. Tarlecki. Behavioural satisfaction and equivalence in concrete model categories. Proc. 20th Coll. on Trees in Algebra and Computing CAAP'96, Linkoping, Springer LNCS 1059, 241-256 (1996).

....an external observational equivalence on algebras, and re interpret specifications by closing their class of models under such equivalence. It turns out that under some acceptable technical conditions, the two approaches are closely related and coincide for most basic specifications [BHW95,BT96] However, the former approach seems more di#cult to extend to structured specifications and parametrization. Hence, we follow here the latter possibility. Definition 4.1. Consider a signature # with observable sorts OBS sorts(#) We always assume that bool OBS . A correspondence between ....

....may be provided via links with indistinguishability relations (via factorization properties, like Thm. 4. 3, which in turn may require a richer context of concrete institutions, with model categories equipped with concretization structure subject to a number of technical requirements as in [BT96] On the other hand, to transfer the present work to the specific framework of Casl we need a precise and convincing definition of observational equivalence between Casl models (many sorted algebras with predicates, partial operations and subsorting) In terms of the institutional structure ....

M. Bidoit and A. Tarlecki. Behavioural satisfaction and equivalence in concrete model categories. Proc. 20th Coll. on Trees in Algebra and Computing CAAP'96, Linkoping, Springer LNCS 1059, 241--256 (1996).

.... to de ne implementation relations where implementations may relax (some of) the properties of the given requirement speci cation (cf. e.g. abstractor implementations in [24] or behavioural implementations in [13,6] and similarly in [17] for a survey on implementation concepts see [20] 7] and [8] provide an in depth study of the relationships between behavioural and abstractor speci cations. The underlying idea of the behavioural approach is to consider the behaviour A= A of any algebra A w.r.t. a given behavioural equality , while the abstractor approach is based on an equivalence ....

....abstract algebra, since two distinct sequences may be observationally equal. The algebra of nite sets of elements, which is also a model of CONTAINER, is fully abstract. We will show at the end of this section that observational equalities are always regular (see also [7] for a direct proof and [8] for more general properties of observational equalities) 48 We will now focus on behavioural theories of behavioural and abstractor speci cations. Let us rst recall a result established in [7] Theorem 10.8 (Behavioural theories of behavioural and abstractor speci cations [7, Section 7] Let ....

M. Bidoit and A. Tarlecki. Behavioural satisfaction and equivalence in concrete model categories. In Proc. of CAAP'96. Springer-Verlag L.N.C.S., 1996. To appear.

....This work is very much in the spirit of the theory of institutions, as introduced by Goguen and Burstall [GB92] and used in a number of similar attempts to generalise and clarify some fundamental ideas and concepts of algebraic speci cation and program development. We follow a similar pattern in [BT96], assuming that model categories are concrete, which results in a concept of concrete institution. For example, a rather intuitive treatment of variables is possible in an arbitrary concrete institution, yielding a more explicit justi cation for the treatment of variables in institutions proposed ....

Bidoit, M., Tarlecki, A. Behavioural satisfaction and equivalence in concrete institutions. Technical report, Warsaw University and Ecole Normale Superieure, in preparation.

....between the two approaches to behavioural semantics of speci cations. We recast here the main results of [BHW95] throwing away the ballast of unnecessary speci c properties of standard algebras and relying only on some abstract properties of A full version of this paper is available as [BT95b]. model categories. This yields a more general and, we believe, more clear view of this work. The basic results are presented for an arbitrary behavioural indistinguishability relation and the abstract behavioural equivalence it determines, and hinge on the crucial assumption that the ....

....frameworks we study. By providing a more abstract view of behavioural semantics we indirectly propose in this paper a way of introducing related concepts for logical systems and speci cation formalisms based on various, not necessarily quite standard de nitions of a model. This is illustrated in [BT95b], where we present the concepts and results introduced throughout the paper in the frameworks based on, respectively, standard many sorted algebras, partial algebras, and regular algebras. To our knowledge, behavioural semantics for speci cations of partial and of regular algebras has never been ....

[Article contains additional citation context not shown here]

Bidoit, M., Tarlecki, A. Behavioural satisfaction and equivalence in concrete model categories. Full version. Technical report, Warsaw University and Ecole Normale Superieure, September 1995. Available by WWW: http://zls.mimuw.edu.pl/~tarlecki/drafts/behabs.{ps,dvi}.

.... to define implementation relations where implementations may relax (some of) the properties of the given requirement specification (cf. e.g. abstractor implementations in [24] or behavioural implementations in [13,6] and similarly in [17] for a survey on implementation concepts see [20] 7] and [8] provide an in depth study of the relationships between behavioural and abstractor specifications. The underlying idea of the behavioural approach is to consider the behaviour A= A of any Sigma algebra A w.r.t. a given Sigma behavioural equality , while the abstractor approach is based on an ....

....abstract algebra, since two distinct sequences may be observationally equal. The algebra of finite sets of elements, which is also a model of CONTAINER, is fully abstract. We will show at the end of this section that observational equalities are always regular (see also [7] for a direct proof and [8] for more general properties of observational equalities) We will now focus on behavioural theories of behavioural and abstractor specifications. Let us first recall a result established in [7] Theorem 10.8 (Behavioural theories of behavioural and abstractor specifications [7, Section 7] Let ....

M. Bidoit and A. Tarlecki. Behavioural satisfaction and equivalence in concrete model categories. In Proc. of CAAP'96. Springer-Verlag L.N.C.S., 1996. To appear.

....This work is very much in the spirit of the theory of institutions, as introduced by Goguen and Burstall [GB92] and used in a number of similar attempts to generalise and clarify some fundamental ideas and concepts of algebraic specification and program development. We follow a similar pattern in [BT96], assuming that model categories are concrete, which results in a concept of concrete institution. For example, a rather intuitive treatment of variables is possible in an arbitrary concrete institution, yielding a more explicit justification for the treatment of variables in institutions ....

Bidoit, M., Tarlecki, A. Behavioural satisfaction and equivalence in concrete institutions. Technical report, Warsaw University and Ecole Normale Sup'erieure, in preparation.

....between the two approaches to behavioural semantics of specifications. We recast here the main results of [BHW95] throwing away the ballast of unnecessary specific properties of standard algebras and relying only on some abstract properties of A full version of this paper is available as [BT95b]. model categories. This yields a more general and, we believe, more clear view of this work. The basic results are presented for an arbitrary behavioural indistinguishability relation and the abstract behavioural equivalence it determines, and hinge on the crucial assumption that the ....

....frameworks we study. By providing a more abstract view of behavioural semantics we indirectly propose in this paper a way of introducing related concepts for logical systems and specification formalisms based on various, not necessarily quite standard definitions of a model. This is illustrated in [BT95b], where we present the concepts and results introduced throughout the paper in the frameworks based on, respectively, standard many sorted algebras, partial algebras, and regular algebras. To our knowledge, behavioural semantics for specifications of partial and of regular algebras has never been ....

[Article contains additional citation context not shown here]

Bidoit, M., Tarlecki, A. Behavioural satisfaction and equivalence in concrete model categories. Full version. Technical report, Warsaw University and Ecole Normale Sup'erieure, September 1995. Available by WWW: http://zls.mimuw.edu.pl/~tarlecki/drafts/behabs.--ps,dvi.

No context found.

M. Bidoit and A. Tarlecki. Behavioural satisfaction and equivalence in concrete model categories. In Proc. 21st Int. Coll. on Trees in Algebra and Programming (CAAP '96), volume 1059 of LNCS, pages 241--256. Springer, 1996.

No context found.

M. Bidoit and A. Tarlecki. Behavioural satisfaction and equivalence in concrete model categories. In H. Kirchner, editor, CAAP '96 - 20th Coll. on Trees in Algebra and Computing, number 1059 in Lecture Notes in Computer Science, pages 241--256, Berlin, 1996. Springer Verlag.

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