Behavioural and abstractor specifications

AbstractIn the literature, one can distinguish two main approaches to the definition of observational semantics of algebraic specifications. On one hand, observational semantics is defined using a notion of observational satisfaction for the axioms of the specifications and, on the other hand, one can define observational semantics by abstraction with respect to an observational equivalence relation between algebras. In this paper, we present an analysis and a comparative study of the different approaches in a more general framework which subsumes the observational case. The distinction between the different observational concepts is reflected by our notions of behavioural specification and abstractor specification. We provide necessary and sufficient conditions for the semantical equivalence of both kinds of specifications and we show that behavioural specifications can be characterized by an abstractor construction and, vice versa, abstractor specifications can be characterized in terms of behavioural specifications. Hence, there exists a duality between both concepts which allows to express each one by the other. We also study the relationships to fully abstract algebras which can be used for a further characterization of behavioural semantics. Finally, we provide proof-theoretic results which show that behavioural theories of specifications can be reduced to standard theories of some classes of algebras

Compatibility properties of synchronously and asynchronously communicating components

We study interacting components and their compatibility with respect to synchronous and asynchronous composition. The behavior of components is formalized by I/O-transition systems. Synchronous composition is based on simultaneous execution of shared output and input actions of two components while asynchronous composition uses unbounded FIFO-buffers for message transfer. In both contexts we study compatibility notions based on the idea that any output issued by one component should be accepted as an input by the other. We distinguish between strong and weak versions of compatibility, the latter allowing the execution of internal actions before a message is accepted. We consider open systems and study conditions under which (strong/weak) synchronous compatibility is sufficient and necessary to get (strong/weak) asynchronous compatibility. We show that these conditions characterize half-duplex systems. Then we focus on the verification of weak asynchronous compatibility for possibly non half-duplex systems and provide a decidable criterion that ensures weak asynchronous compatibility. We investigate conditions under which this criterion is complete, i.e. if it is not satisfied then the asynchronous system is not weakly asynchronously compatible. Finally, we discuss deadlock-freeness and investigate relationships between deadlock-freeness in the synchronous and in the asynchronous case

Closure properties for the class of behavioral models

Hidden k-logics can be considered as the underlying logics of program specification. They constitute natural generalizations of k-deductive systems and encompass deductive systems as well as hidden equational logics and inequational logics. In our abstract algebraic approach, the data structures are sorted algebras endowed with a designated subset of their visible parts, called filter, which represents a set of truth values. We present a hierarchy of classes of hidden k-logics. The hidden k-logics in each class are characterized by three different kinds of conditions, namely, properties of their Leibniz operators, closure properties of the class of their behavioral models, and properties of their equivalence systems. Using equivalence systems, we obtain a new and more complete analysis of the axiomatization of the behavioral models. This is achieved by means of the Leibniz operator and its combinatorial properties. © 2007 Elsevier Ltd. All rights reserved.FCT via UIM

Testing data types implementations from algebraic specifications

Algebraic specifications of data types provide a natural basis for testing data types implementations. In this framework, the conformance relation is based on the satisfaction of axioms. This makes it possible to formally state the fundamental concepts of testing: exhaustive test set, testability hypotheses, oracle. Various criteria for selecting finite test sets have been proposed. They depend on the form of the axioms, and on the possibilities of observation of the implementation under test. This last point is related to the well-known oracle problem. As the main interest of algebraic specifications is data type abstraction, testing a concrete implementation raises the issue of the gap between the abstract description and the concrete representation. The observational semantics of algebraic specifications bring solutions on the basis of the so-called observable contexts. After a description of testing methods based on algebraic specifications, the chapter gives a brief presentation of some tools and case studies, and presents some applications to other formal methods involving datatypes

Proving Behavioural Theorems with Standard First-Order Logic

. Behavioural logic is a generalization of first-order logic where the equality predicate is interpreted by a behavioural equality of objects (and not by their identity). We establish simple and general sufficient conditions under which the behavioural validity of some first-order formula with respect to a given first-order specification is equivalent to the standard validity of the same formula in a suitably enriched specification. As a consequence any proof system for first-order logic can be used to prove the behavioural validity of first-order formulas. 1 Introduction Observability plays a prominent role in formal software development, since it provides a suitable basis for defining adequate correctness concepts. For instance, for proving the correctness of a program with respect to a given specification, many examples show that it is essential to abstract from internal implementation details and to rely only on the observable behaviour of the program. A similar situation is the n..

Behavioural Theories and The Proof of Behavioural Properties

Behavioural theories are a generalization of first-order theories where the equality predicate symbol is interpreted by a behavioural equality of objects (and not by their identity). In this paper we first consider arbitrary behavioural equalities determined by some (partial) congruence relation and we show how to reduce the behavioural theory of any class of \Sigma -algebras to (a subset of) the standard theory of some corresponding class of algebras. This reduction is the basis of a method for proving behavioural theorems whenever an axiomatization of the behavioural equality is provided. Then we focus on the important special case of (partial) observational equalities where two elements are observationally equal if they cannot be distinguished by observable computations over some set of input values. We provide general conditions under which an obvious infinite axiomatization of the observational equality can be replaced by a finitary one and we provide methodological guidelines for..

Observational interpretations of hybrid dynamic logic with binders and silent transitions

We extend hybrid dynamic logic with binders (for state variables) by distinguishing between observable and silent transitions. This differentiation gives rise to two kinds of observational interpretations: The first one relies on observational abstraction from the ordinary model class of a specification Sp by considering its closure under weak bisimulation. The second one uses an observational satisfaction relation for the axioms of the specification Sp, which relaxes the interpretation of state variables and the satisfaction of modal formulæ by abstracting from silent transitions. We establish a formal relationship between both approaches and show that they are equivalent under mild conditions. For the proof we instantiate the previously introduced concept of a behaviour-abstractor framework to the case of dynamic logic with binders and silent transitions. As a particular outcome we provide an invariance theorem and show the Hennessy-Milner property for weakly bisimilar labelled transition systems and observational satisfaction. In the second part of the paper we integrate our results in a development methodology for reactive systems leading to two versions of observational refinement. We provide conditions under which both kinds of refinement are semantically equivalent, involving implementation constructors for relabelling, hiding, and parallel composition.publishe