M. Bidoit, R. Hennicker, and M. Wirsing. Behavioural and abstractor specifications. Science of Computer Programming, 25(2-3):149--186, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... and minimal realisations of automata, to bisimulations in process algebras [21] and in coalgebras, particularly with respect to the object paradigm [1, 13, 25, 20] while notions of observational or behavioural equivalence have formed a continuous thread through the area of algebraic speci cation [24, 26, 23, 3, 10]. In this paper we present a categorical logic in which the notion of behavioural satisfaction (or truth up to observability) can be abstracted as a modality. The logic in question is the internal logical language of a topos of presheaves; modal operators in such languages correspond to ....

Michel Bidoit, Rolf Hennicker, and Martin Wirsing. Behavioural and abstractor speci cations. Science of Computer Programming, 25(2{ 3):149-186, 1995. 8

....sorts. Indistinguishability relations are necessarily partial congruences, from which we derive an equivalence of algebras: two algebras are observationally equivalent if their quotients by these indistinguishability relations (understood as corresponding fully abstract models) are isomorphic [7,2]. The following characterization will be helpful in the sequel: Lemma 6.1 ( 2] Two algebras A and B are observationally equivalent i for some set X Set OBS of variables of observable sorts and surjective valuations v A and v B of X in the carriers of observable sorts of A and B, ....

....which we derive an equivalence of algebras: two algebras are observationally equivalent if their quotients by these indistinguishability relations (understood as corresponding fully abstract models) are isomorphic [7,2] The following characterization will be helpful in the sequel: Lemma 6. 1 ([2]) Two algebras A and B are observationally equivalent i for some set X Set OBS of variables of observable sorts and surjective valuations v A and v B of X in the carriers of observable sorts of A and B, respectively, the sets of equalities of observable sorts satis ed in A by v A and in B by ....

Bidoit, M., Hennicker, R., Wirsing, M. Behavioural and abstractor speci cations. Science of Computer Programming 25(2-3), 1995.

....here. De nition 1 ( 4] Observational equivalence OBS of regular algebras is A OBS B i AOBS = BOBS = AOBS = A will be written shortly A= A or A= in the sequel. All de nitions in this subsection are still valid when standard algebras are taken into account (cf. [3, 2]) the only modi cation required is to restrict contexts to only nitary terms T (X) in de nition of indistinguishability. Behavioural implementation A concept of observational equivalence plays a crucial role in the process of step wise re nement. It allows one to consider possibly large ....

....AOBS . Formally, A OBS ( A= Behavioural semantics is closely related to the observational equivalence of algebras. When only equational speci cations are considered, the class of behavioural models coincides with the closure of classical models under observational equivalence (see [3, 5]) The most straightforward formalization of the concept of implementation refers to the inclusion of model classes: speci cation SP = h ; Ax i implements (re nes) SP , if each model of SP is a model of SP : Mod(SP ) Mod(SP ) both SP and SP are over the same signatures ....

Bidoit, M., Hennicker, R., Wirsing, M. Behavioural and abstractor speci cations. Science of Computer Programming 25(2-3), 1995.

....the KBN grant 8 T11C 046 14. representing abstractly a bisimulation. A dual approach was used for de ning the behavioural equivalence for algebras. Roughly speaking, two algebras A and B are behaviourally equivalent if their quotients by indistinguishability relations A and B are isomorphic [BHW95], that is if they are related by a co span of quotient projections: A= A = 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 ....

....as noticed there, in the way sketched above one can uniformly introduce observational equivalence in other algebraic settings, e.g. for partial and regular algebras. There exist an explicit characterization of A as the relation of indistinguishability by means of observable contexts ([BHW95]) Similar characterizations for partial and regular algebras can be found in [BT96] and [BT95] respectively. 5 Bisimulation from open maps be a category of models of computation, in which we choose a subcategory (not necessarily full) of observation objects. Any morphism p : O A from ....

M. Bidoit, R. Hennicker, M. Wirsing, Behavioural and abstractor speci cations. Science of Computer Programming 25(2-3) (1995) 149-186.

....an 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, R. Hennicker and M. Wirsing. Behavioural and abstractor specifications. Science of Computer Programming 25:149--186 (1995).

....use in writing specifications (see e.g. SW96] But in a recent development, Bidoit et al. have generalized Reichel s result to the case of specifications with infinitary firstorder equational formulae as axioms, and to arbitrary relations of behavioural equivalence and indistinguishability. In [BHW95] they show that the coincidence of classes described above holds in this context as well, whenever the class of models of h Sigma; Phii (under ordinary satisfaction) is closed under quotienting with respect to indistinguishability of values, provided that indistinguishability is weakly regular ....

....Subsequently, BH95] and [BH96] use this characterization as the basis for reasoning methods. In this paper we examine these issues for the case of (flat) specifications with higher order logical formulae as axioms. Our first main contribution is a generalization of the framework and results of [BHW95]. Although it is not made explicit there, the main results in [BHW95] including the characterization theorem do not strongly depend on the form of axioms. The same result holds for any logical system for which behavioural satisfaction of a formula in A coincides with ordinary satisfaction of ....

[Article contains additional citation context not shown here]

M. Bidoit, R. Hennicker and M. Wirsing. Behavioural and abstractor specifications. Science of Computer Programming 25:149--186 (1995).

....but to stick to algebraic techniques for the specification of the involved data. This is consistent with the purely algebraic approaches which use initial semantics for the specification of data structures, and final semantics (based on behavioural equivalences) for state spaces (see for example [ONS93, BHW95] and the references therein) The results in this paper are based instead on a purely coalgebraic approach: we leave as a topic of future research to investigate how far they can be generalized to a hybrid coalgebraic algebraic framework. We start introducing in Section 2 the class of coalgebras ....

M. Bidoit, R. Hennicker, and M. Wirsing. Behavioural and Abstractor Specifications. Science of Computer Programming, 25:146--186, 1995.

....of behavioral equivalence in work of E.F. Moore [80] from 1956. 6 [48] which also rst recognized the connection between tree automata and software engineering. Reichel s notion of behavioral theory further developed in various directions, again mainly using partial algebras, e.g. see [6,21,22] and the survey [82] In order to get the powerful module and type system of parameterized programming [54,27,29,28,31,60] it is necessary that the signatures (with their morphisms) models and axioms form what is called an institution [39] What we call half institutions are used in [22] ....

....di erent purposes, and show that the one that should form an institution (the morphisms) does in fact do so (see also Section 3. 5) The rst e ective algebraic proof technique for behavioral properties was context induction, introduced by Hennicker [68] and further developed with Bidoit (e.g. [6]) Their research programme is similar to ours in several ways; however, their approach is more concerned with semantics than with proofs, and their context induction can be very awkward to apply in practice (e.g. see [24] for a discussion of some of the diculties) We have found that hidden ....

Michel Bidoit, Rolf Hennicker, and Martin Wirsing. Behavioural and abstractor specications. Science of Computer Programming, vol. 25, no. 2-3, 1995.

No context found.

M. Bidoit, R. Hennicker and M. Wirsing. Behavioural and abstractor specifications. Science of Computer Programming 25:149--186 (1995).

No context found.

M. Bidoit, R. Hennicker and M. Wirsing. Behavioural and abstractor specifications. Science of Computer Programming 25:149--186 (1995).

No context found.

Michel Bidoit, Rolf Hennicker, and Martin Wirsing. Behavioural and abstractor specifications. Science of Computer Programming, 25(2--3):149--186, 1995.

....of full abstractness leads to proof principles for verifying observational consequences of a specification. In the context of automata theory, a similar duality was already investigated by Arbib and Manes in [3] The relationships between the two approaches have been intensively studied in [10]. 1.2 Reachability Reachability concepts provide means to specify generation principles for datatypes. The standard approach to reachability is to introduce a set of datatype constructors and to admit as models of a specification only those algebras which are reachable w.r.t. the given ....

....with the observational and standard satisfaction relations. Theorem 2.12. Let # Obs be an observational signature with underlying standard signature #,let # be a # formula and let A be a # Obs algebra. Then #Obs # if and only if # #. This theorem is a generalization of Theorem 3. 11 in [10]. The proof is done by induction on the form of the formula # (along the lines of the proof of Theorem 3.11 in [10] Similar results are provided in [23] and in [33] Definition 2.13 (Basic observational specification) A basic observational specification Obs , Ax# consists of an observational ....

[Article contains additional citation context not shown here]

M. Bidoit, R. Hennicker, and M. Wirsing. Behavioural and abstractor specifications. Science of Computer Programming, 25(2--3):149--186, 1995.

.... introduce 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 ....

....easiest to verify. However, i) is strictly stronger in general than the perhaps more expected Mod (SP G ) Abs ( SP ) This weaker condition turns out to be sucient (and in fact, equivalent to (i) if we additionally assume that the two speci cations involved are behaviourally consistent [BHW95] that is, closed under observational quotients. When this is not the case, then the use of this weaker condition must be paid for by a stronger version of (ii) Abs ( G ) which seems even less convenient to use than (i) Overall, we need a way to pass information on the global ....

M. Bidoit, R. Hennicker and M. Wirsing. Behavioural and abstractor speci cations. Science of Computer Programming 25:149-186 (1995).

.... The rst one is based on an observational equivalence relation between algebras which is used In the context of automata theory, a similar duality was already investigated by Arbib and Manes in [3] The relationships between the two approaches have been intensively studied in [10]. to abstract from the (standard) model class of a speci cation, see, e.g. 36] The second approach relaxes the (standard) satisfaction relation so that the observational models of a speci cation are all algebras which satisfy the given set of axioms up to observational equality of the ....

....and standard satisfaction relations. Theorem 12 Let Obs be an observational signature with underlying standard signature , let be a formula and let A be a Obs algebra. Then: A j= Obs if and only if BB Obs (A) j= This theorem is a generalization of Theorem 3. 11 in [10]. The proof is done by induction on the form of the formula (along the lines of the proof of Theorem 3.11 in [10] Similar results are provided in [23] and in [33] De nition 13 (Basic observational speci cation) A basic observational speci cation SP Obs = h Obs ; Axi consists of an ....

[Article contains additional citation context not shown here]

M. Bidoit, R. Hennicker, and M. Wirsing. Behavioural and abstractor speci cations. Science of Computer Programming, 25(2-3):149-186, 1995.

.... introduce 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 ....

....easiest to verify. However, i) is strictly stronger in general than the perhaps more expected Mod (SP G ) #(SP ) This weaker condition turns out to be su#cient (and in fact, equivalent to (i) if we additionally assume that the two specifications involved are behaviourally consistent [BHW95] that is, closed under observational quotients. When this is not the case, then the use of this weaker condition must be paid for by a stronger version of (ii) # # (SP # ) Mod (# # (SP G ) SP # G ) which seems even less convenient to use than (i) Overall, we need a way to pass ....

M. Bidoit, R. Hennicker and M. Wirsing. Behavioural and abstractor specifications. Science of Computer Programming 25:149--186 (1995).

....are admitted. Thus standard implementations which simply contain junk (like the realization of the natural numbers by the integers) are ruled out from the models of a speci cation. The ultra loose approach of [20] the notion of behavioral speci cation w.r.t. a partial observational equality in [5, 12] and the hidden algebra approach are closely related to our framework. The main di erence to [20] is that no explicit notion of observer or constructor operation is used there while in our approach they are the basic ingredients of a signature which lead to a speci cation methodology and to an ....

....to [20] is that no explicit notion of observer or constructor operation is used there while in our approach they are the basic ingredients of a signature which lead to a speci cation methodology and to an institution tailored to observability and reachability. The partial observational equality of [5] does not take into account a distinguished set of observer and constructor operations which in our case facilitates proofs and leads to a powerful notion of signature morphism. The main di erence to the presentation of hidden algebra in [9] is that there the reachable values are given by a xed ....

[Article contains additional citation context not shown here]

M. Bidoit, R. Hennicker, and M. Wirsing. Behavioural and abstractor specications. Science of Computer Programming, 25:149-186, 1995.

....specification with signature Sigma Obs . 1. A j= Sigma Obs if and only if FA Sigma Obs (A) j= 2. SP Obs j= Sigma Obs if and only if [ SP Obs ] j= This theorem shows the adequacy of the black box semantics in the observational case. The theorem is a variant of Theorem 3. 11 in [4] and it is related to similar results in [10] 5.2 Black Box Semantics of Constructor Based Specifications Let Sigma Cons be a constructor based signature. Since for any Sigma Cons algebra A, the reachable part hAi Cons of A is a Sigma algebra, which by definition contains only 11 those ....

....of the two concepts and not on their integration. The combination of the two concepts offers a promising perspective of future reserach. We believe that such an integration will be strongly related to (a generalization of) the notion of partial observational equivalence considered e.g. in [4] and [10] 15 ....

M. Bidoit, R. Hennicker, and M. Wirsing. Behavioural and abstractor specifications. Science of Computer Programming, 25:149--186, 1995.

....types. Formally, we introduce an institution of observational logic and we study proof methods for first order observational properties of structured specifications built over this institution. Although our approach is novel, it is influenced by previous behavioural approaches, in particular of [BHW 95] P 96] GM 97] R 95] and [JR 97] The important difference to [BHW 95] is that in the present approach we use a built in observational semantics which previously led to problems w.r.t. the encapsulation of observational properties of parts of a system specification (cf. BB 91] HN 93] In ....

....study proof methods for first order observational properties of structured specifications built over this institution. Although our approach is novel, it is influenced by previous behavioural approaches, in particular of [BHW 95] P 96] GM 97] R 95] and [JR 97] The important difference to [BHW 95] is that in the present approach we use a built in observational semantics which previously led to problems w.r.t. the encapsulation of observational properties of parts of a system specification (cf. BB 91] HN 93] In the hidden sorted algebra approach (cf. e.g. GM 97] encapsulation is ....

[Article contains additional citation context not shown here]

M. Bidoit, R. Hennicker, M. Wirsing: Behavioural and abstractor specifications. Science of Computer Programming 25, 149-186, 1995.

No context found.

M. Bidoit, R. Hennicker, and M. Wirsing. Behavioural and abstractor specifications. Science of Computer Programming, 25(2-3):149--186, 1995.

No context found.

M. Bidoit, R. Hennicker, and M. Wirsing. Behavioural and abstractor specifications. Science of Computer Programming, 25(2--3):149--186, 1995.

No context found.

Michel Bidoit, Rolf Hennicker, and Martin Wirsing. Behavioral and abstractor speci cations. Science of Computer Programming, 25:149-186, 1995.

No context found.

Michel Bidoit, Rolf Hennicker, and Martin Wirsing. Behavioral and abstractor speci cations. Science of Computer Programming, 25:149-186, 1995.

No context found.

Bidoit,M., R.Hennicker, and M.Wirsing: Behavioural and abstractor specifications. Science of Computer Programming 25 (1995) 149--186.

No context found.

Michel Bidoit, Rolf Hennicker, and Martin Wirsing. Behavioral and abstractor speci cations. Science of Computer Programming, 25:149-186, 1995.

No context found.

M. Bidoit, R. Hennicker, M. Wirsing, Behavioural and Abstractor Specifications, Science of Computer Programming 25 (1995) 149-186

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC