. Algebraic structures which are generated by a collection of constructors--- like natural numbers (generated by a zero and a successor) or finite lists and trees--- are of well-established importance in computer science. Formally, they are initial algebras. Induction is used both as a definition principle, and as a proof principle for such structures. But there are also important dual "coalgebraic" structures, which do not come equipped with constructor operations but with what are sometimes called "destructor" operations (also called observers, accessors, transition maps, or mutators). Spaces of infinite data (including, for example, infinite lists, and non-well-founded sets) are generally of this kind. In general, dynamical systems with a hidden, black-box state space, to which a user only has limited access via specified (observer or mutator) operations, are coalgebras of various kinds. Such coalgebraic systems are common in computer science. And "coinduction" is the appropriate te...

This paper generalizes many-sorted algebra (hereafter, MSA) to order-sorted algebra (hereafter, OSA) by allowing a partial ordering relation on the set of sorts. This supports abstract data types with multiple inheritance (in roughly the sense of object-oriented programming), several forms of polymorphism and overloading, partial operations (as total on equationally defined subsorts), exception handling, and an operational semantics based on term rewriting. We give the basic algebraic constructions for OSA, including quotient, image, product and term algebra, and we prove their basic properties, including Quotient, Homomorphism, and Initiality Theorems. The paper's major mathematical results include a notion of OSA deduction, a Completeness Theorem for it, and an OSA Birkhoff Variety Theorem. We also develop conditional OSA, including Initiality, Completeness, and McKinsey-Malcev Quasivariety Theorems, and we reduce OSA to (conditional) MSA, which allows lifting many known MSA results to OSA. Retracts, which intuitively are left inverses to subsort inclusions, provide relatively inexpensive run-time error handling. We show that it is safe to add retracts to any OSA signature, in the sense that it gives rise to a conservative extension. A final section compares and contrasts many different approaches to OSA. This paper also includes several examples demonstrating the flexibility and applicability of OSA, including some standard benchmarks like STACK and LIST, as well as a much more substantial example, the number hierarchy from the naturals up to the quaternions.

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This is an introduction to the philosophy and use of OBJ, emphasizing its operational semantics, with aspects of its history and its logical semantics. Release 2 of OBJ3 is described in detail, with many examples. OBJ is a wide spectrum first-order functional language that is rigorously based on (order sorted) equational logic and parameterized programming, supporting a declarative style that facilitates verification and allows OBJ to be used as a theorem prover.

A formalism for constructing and using axiomatic specifications in an arbitrary logical system is presented. This builds on the framework provided by Goguen and Burstall’s work on the notion of an institution as a formalisation of the concept of a logical system for writing specifications. We show how to introduce free variables into the sentences of an arbitrary institution and how to add quantitiers which bind them. We use this foundation to define a set of primitive operations for building specifications in an arbitrary institution based loosely on those in the ASL kernel specification language. We examine the set of operations which results when the definitions are instantiated in institutions of total and partial tirst-order logic and compare these with the operations found in existing specification languages. We present proof rules which allow proofs to be conducted in specifications built using the operations we define. Finally, we introduce a simple mechanism for defining and applying parameterised specifications and briefly discuss the program development process. 1 1988 Academic Press. Inc. 1.

The properties of a simple and natural notion of observational equivalence of algebras and the corresponding specification-building operation are studied. We begin with a defmition of observational equivalence which is adequate to handle reachable algebras only, and show how to extend it to cope with unreachable algebras and also how it may be generalised to make sense under an arbitrary institution. Behavioural equivalence is treated as an important special case of observational equivalence, and its central role in program development is shown by means of an example.

The main ideas underlying work on the model-theoretic foundations of algebraic specification and formal program development are presented in an informal way. An attempt is made to offer an overall view, rather than new results, and to focus on the basic motivation behind the technicalities presented elsewhere.

A formal methodology is presented for the systematic evolution of modular Standard ML programs from specifications by means of verified refinement steps, in the framework of the Extended ML specification language. Program development proceeds via a sequence of design (modular decomposition), coding and refinement steps. For each of these three kinds of steps, conditions are given which ensure the correctness of the result. These conditions seem to be as weak as possible under the constraint of being expressible as "local" interface matching requirements. Interfaces are only required to match up to behavioural equivalence, which is seen as vital to the use of data abstraction in program development. Copyright c fl 1989 by D. Sannella and A. Tarlecki. All rights reserved. An extended abstract of this paper will appear in Proc. Colloq. on Current Issues in Programming Languages, Joint Conf. on Theory and Practice of Software Development (TAPSOFT), Barcelona, Springer LNCS (1989)....

Extended ML (EML) is a framework for the formal development of modular Standard ML (SML) software systems. Development commences with a specification of the behaviour required and proceeds via a sequence of partial solutions until a complete solution, an executable SML program, is obtained. All stages in this development process are expressed in the EML language, an extension of SML with axioms for describing properties of module components. This is an overview of the formal definition of the EML language. To complement the full technical details presented elsewhere, it provides an informal explanation of the main ideas, gives the rationale for certain design decisions, and outlines some of the technical issues involved. EML is unusual in being built around a "real" programming language having a formally-defined syntax and semantics. Interesting and complex problems arise both from the nature of this relationship and from interactions between the features of the language.

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 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 finding such...