Summary. The article deals with parameterized families of sets. When treated in a similar way as sets (due to systematic overloading notation used for sets) they are called many sorted sets. For instance, if x and X are two many-sorted sets (with the same set of indices I) then relation x ∈ X is defined as ∀i∈Ixi ∈ Xi. I was prompted by a remark in a paper by Tarlecki and Wirsing: “Throughout the paper we deal with many-sorted sets, functions, relations etc.... We feel free to use any standard set-theoretic notation without explicit use of indices ” [6, p. 97]. The aim of this work was to check the feasibility of such approach in Mizar. It works. Let us observe some peculiarities:- empty set (i.e. the many sorted set with empty set of indices) belongs to itself (theorem 133),- we get two different inclusions X ⊆ Y iff ∀i∈IXi ⊆ Yi and X ⊑ Y iff ∀xx ∈ X ⇒ x ∈ Y equivalent only for sets that yield non empty values. Therefore the care is advised.

: This paper presents a number of concepts of a mapping between logical systems modelled as institutions, discusses their mutual merits and demerits, and sketches their role in the process of system specification and development. Some simple properties of the resulting categories of institutions are given. 1 Introduction We have to live with a multitude of logical systems used in various approaches to software specification and development. The proliferation of logical systems in the area is not just researchers' fancy, but results from the very practical needs to capture various aspects of software systems and to cater for various programming paradigms. Each of them leads to a different notion of a semantic model capturing the semantic essence of the adopted view of software systems. For instance, standard (many-sorted) algebras [BL70], [GTW78] provide a satisfactory framework for modelling data types where all operations always yield well-defined results. However, if general recursi...

This paper presents a particular approach to the design and verification of large sequential systems. It is based on structured algebraic specifications and stepwise refinement by program modules. The approach is implemented in Kiv (Karlsruhe Interactive Verifier), and supports the entire design process starting from formal specifications and ending with verified code. Its main characteristics are a strict decompositional design discipline for modular systems, a powerful proof component, and an evolutionary verification model supporting incremental error correction and verification. We present the design methodology for modular systems, a feasible verification method for single modules, and an evolutionary verification technique based on reuse of proofs. We report on the current performance of the system, compare it to others in the field, and discuss future perspectives.

An overview of past, present and future work on the Extended ML formal program development framework is given, with emphasis on two topics of current active research: the semantics of the Extended ML specification language, and tools to support formal program development.

The focus of this chapter is the incremental presentation of partial firstorder logic, seen as a powerful framework where the specification of most data types can be directly represented in the most natural way. Both model theory and logical deduction are described in full detail. Alternatives to partiality, like (variants of) error algebras and order-sortedness are also discussed, showing their uses and limitations. Moreover, both the total and the partial (positive) conditional fragment are investigated in detail, and in particular the existence of initial (free) models for such restricted logical paradigms is proved. Some more powerful algebraic frameworks are sketched at the end. Equational specifications introduced in last chapter, are a powerful tool to represent the most common data types used in programming languages and their semantics. Indeed, Bergstra and Tucker have shown in a series of papers (see [BT87] for a complete exposition of results) that a data type is semicompu...

this paper we introduce the entity framework (entity algebras and entity specifications) and show, also with the help of several examples, how they can be used for formally modelling and specifying dynamic systems.

. A single pushout approach to the transformation of attributed partial graphs based on categories of partial algebras and partial morphisms is introduced. A sufficient condition for pushouts in these categories is presented. As the synchronization mechanism we use amalgamation of rules and show how synchronization can be minimized. We point out how the results obtained can be employed in order to define an operational semantics for object specification languages. 1 Introduction Graphs and graph grammars usually yield intuitive descriptions of complex phenomena in computer science. Therefore, numerous approaches to graph grammars have been put forward, among them the logical approach [6], the set theoretic approach [29], and the algebraic approach [9]. Graph-based techniques have for instance been successfully applied in the realm of software engineering development environments [13, 14], for object-oriented languages based on asynchronous communication [22, 24, 20, 21] and in logic p...

There are two widely accepted notions of behavioural equivalence, formalizing the idea of observational indistinguishability: observational equivalence for algebras (which are models for sequential systems) and bisimulation equivalence (bisimilarity) for concurrent processes. In this paper we show that the observational equivalences for standard, partial and regular algebras are bisimulation equivalences. This is done in the setting of open maps, proposed in [JNW93] as an abstract approach to behavioural equivalences of processes. The main advantage of the results is capturing the models for sequential and concurrent systems in a uniform framework. In such an abstract setting we formulate the property of determinism, shared by all the algebras considered in this paper, and identify some interesting facts about bisimilarity in the deterministic case. All the results for standard, regular and partial algebras are obtained by the applications of a general machinery developed in the pape...

This paper presents an algebraic compositional semantics for a schema of an object-oriented syntax which models many existing features as class hierarchies, polymorphism and concurrency, using a pattern which could be applied to different concrete languages (in what follows O-O stands for "object-oriented"). The semantics is defined in a classical denotational style, ie giving an abstract syntax, the semantic domains and the interpretation of the syntactic operators. From this point of view the paper yields a formalization of the model which underlies an O-O notation as done for example in [Wo] for Smalltalk-80. However our approach is different from the traditional one for at least two reasons. First, the given semantics is algebraic in the sense that the value denoted by a class (the basic language unit in an O-O notation) is in general a class of algebras described by an algebraic specification. In particular, we use the algebraic specification language ASL ([Wi]). This approach allows to model in a natural way some typical O-O features related to combining classes. A class combinator (eg inheritance) is semantically interpreted in this framework as a function which handles classes of algebras or, in an equivalent way, since we use an algebraic specification language, as a specification combinator (see [Wi] for foundations). Moreover, our schema of semantic definition allows to model also concurrent features of the O-O language if any, by underlying an approach to concurrency based on algebraic transition systems (see for example [AR] for foundations and [AGRZ] for a recent survey with already some hints on the treatment of objects). In this respect our aim is not to present a particular language but to give a general framework in which different concurrent features,...

In the literature a host of different kinds of algebras have been suggested as models of algebraic specifications, among them ordinary total algebras, partial algebras, ordered/monotonic algebras, continuous algebras, and topological algebras. In order to unify as may as possible of these algebra concepts into one, we make a general construction based on some parameters: a Cartesian category with functors to and from the category Set (of sets and total functions) satisfying some assumptions. From these parameters we construct a category of algebras of a given signature and the subcategory of those satisfying a given set of equations. We show that the two categories have initial objects strongly related to the ordinary term and quotient algebras, respectively. Finally, we show how to apply this general construction to obtain total and ordered algebras. 1 Introduction This paper reports work in progress. It is concerned with specifications of abstract data types 1 and in particular w...