In this short note we study the semantics of two basic computational effects, exceptions and states, from a new point of view. In the handling of exceptions we dissociate the control from the elementary operation that recovers from the exception. In this way it becomes apparent that there is a duality, in the categorical sense, between exceptions and states.

This paper provides an abstract definition of a class of logics, called diagrammatic logics, together with a definition of morphisms and 2-morphisms between them. The definition of the 2-category of diagrammatic logics relies on category theory, mainly on adjunction, categories of fractions and limit sketches. This framework is applied to the formalisation of a parameterisation process. This process, which consists of adding a formal parameter to some operations in a given specification, is presented as a morphism of logics. Then the parameter passing process for recovering a model of the given specification from a model of the parameterised specification and an actual parameter is shown to be a 2-morphism of logics.

This paper presents a simple and powerful diagrammatic framework for dealing with specifications in computer science. Following a classical line, we define diagrammatic specifications as a kind of generalised sketch. In addition, the specifications themselves are defined as the realisations of projective sketches. This meta level provides adjunction properties: this is due to a well-known result of Ehresmann. Moreover, we prove in this paper that this meta level also provides an efficient definition of deduction. This work results from a collaboration with Christian Lair.

We define a categorical framework, based on the notion of sketch, for specification and evaluation in the senses of algebraic specifications and algebraic programming. This framework goes far beyond our initial motivation, which was to specify computation with algebraic numbers. We begin by redefining sketches in order to deal explicitly with programs. Expressions and terms are carefully defined and studied, then quasi-projective sketches are introduced. We describe static evaluation in these sketches: we propose a rigorous basis for evalution in the corresponding structures. These structures admit an initial model, but are not necessarily equational. In Part II (Duval and Reynaud 1994), we study a more general process, called dynamic evaluation, for structures that may have no initial model.

In the first part of this paper (Duval and Reynaud 1994), we defined a categorical framework, based on the notion of sketch, for specification and evaluation in the senses of algebraic specifications and algebraic programming. Static evaluation in quasi-projective sketches was defined in Part I; in this paper, dynamic evaluation is introduced. It deals with more general structures, which may have no initial model. Until now, this process has not been used in algebraic specification systems, but computer algebra systems are beginning to use it as a basic tool. Finally, we give some applications of dynamic evaluation to computation in field extensions.

In this paper a new algorithm is presented for factoring bivariate polynomials over algebraically closed fields. Or, equivalently, for determining the irreducible components of a plane curve. This algorithm is based on properties of some geometric invariants of the curve, and is similar to Berlekamp’s algorithm for factorization of univariate polynomials over finite fields.