This paper gives a short introduction to the algebraic specification language Spectrum. Using simple, well-known examples, the objectives and concepts of Spectrum are explained. The Spectrum language is based on axiomatic specification techniques and is oriented towards functional programs. Spectrum includes the following features: ffl partial functions, definedness logic and fixed point theory ffl higher-order elements and typed -abstraction ffl non-strict functions and infinite objects ffl full first-order predicate logic with induction principles ffl predicative polymorphism with sort classes ffl parameterization and modularization Spectrum is based on the concept of loose semantics.

Equational presentations with ordered sorts encompass partially defined functions and subtyping information in an algebraic framework. In this work we address the problem of computing in order-sorted algebras, with very few restrictions on the allowed presentations. We adopt an algebraic framework where equational, membership and existence formulas can be expressed. A complete deduction calculus is provided to incorporate the interaction between all these formulas. The notion of decorated terms is proposed to memorize local sort information, dynamically changed by a rewriting process. A completion procedure for equational presentations with ordered sorts computes a set of rewrite rules with which not only equational theorems of the form (t = t 0 ), but also typing theorems of the for...

A precise and perspicuous specification of mathematical domains of computation and their inherently related type inference mechanisms is a prerequisite for the design and systematic development of a system for symbolic computing. This paper describes Formal, a language for giving modular and well-structured specifications of such domains and particularly of "mathematical objects". A novel framework for algebraic specification involving so-called "unified algebras" has been adopted, where sorts are treated as values. The adoption of this framework aims also at being capable of specifying polymorphism, unifying the notions of "parametric" and "inclusion" polymorphisms. Furthermore, the operational nature of the specification formalisms allows a straightforward transformation into an executable form.

Narrowing is a way to integrate function evaluation and equality definition into logic programming. Here we show how this can be combined with the constraint paradigm. We propose a solver for goals with constraints in theories defined by unconstrained equalities and rewrite rules with constraints expressed in an algebraic built-in structure. The narrowing method reduces the goal solving problem in the whole theory to rewriting and constraint solving in an adequate combined theory. The combined solver is obtained through the combination of a solver in the built-in structure and a solver for the unconstrained equalities. Sufficient syntactic conditions are proposed to get a process that enumerates a complete set of solutions. 1 Introduction Narrowing provides integration of function evaluation and equality definition into logic programming [6, 12, 8, 18, 10]. In this work, we show how this can be connected with the constraint paradigm to get a constraint solver on combined algebraic dom...

. We introduce a rewriting implementation of the reduction relation of ß-calculus and prove its correctness. The implementation is based on terms with De Bruijn indices and an explicit substitution operator. The resulting rewrite rules need to be applied modulo a large and complex equational theory, and are only of theoretical interest. Applying the coherence techniques introduced in a previous paper, we transform this specification into an equivalent one that only requires rewriting modulo associativity and commutativity. This latter rewrite system can then be straightforwardly encoded in currently available rewriting-based programming languages. Finally, we sketch a possible application of this implementation as the basis for adding input-output capabilities to such languages. 1 Introduction Rewriting interpreters are very nice toys for the theoretically minded. You can feed them equations and rules, and they will show you the possible derivations, or compute normal forms, many vari...

This paper gives a short introduction to the algebraic specification language Spectrum.

This paper points at the retract problem that arises, when using subsorting in a not necessarily executable algebraic specification language. We showhow this problem can be circumvented in many important cases by the use of conditionally polymorphic functions. Further, we will see that this sorting facility is well-suited to model inheritance in a smart way. Wegive a simple example which demonstrates this approach in the speci cation language Spectrum.

Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.5.4 Special Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.6 Semantics of Programming Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6.1 Semantics of Ada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6.2 Action Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.7 Specification Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.7.1 Early Algebraic Specification Languages . . . . . . . . . . . . . . . . . . . . . . . . 53 4.7.2 Recent Algebraic Specification Languages . . . . . . . . . . . . . . . . . . . . . . . 55 4.7.3 The Common Framework Initiative. . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5 Methodology 57 5.1 Development Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.1.1 Applica...

This paper first discusses the advantages of sorts in algebraic specification languages. It gives a brief introduction into polymorphism approaches used in functional programming languages to weaken the strength of traditional sort systems. We then sketch a universally polymorphic specification language which combines all polymorphism concepts within one universal approach. The approach is based on qualified sorts which generalize the Haskell sort classes to n-ary sort predicates. To describe the properties of arbitrary sort predicates we use a separate Horn-Clause specification. In an introductory example we show how to model sort classes with our approach. A further example...