It has recently become evident that categories of representations of Hopf algebras provide fundamental examples of monoidal categories. In this expository paper, we examine such categories as models of (multiplicative) linear logic. By varying the Hopf algebra, it is possible to model several variants of linear logic. We present models of the original commutative logic, the noncommutative logic of Lambek and Abrusci, the braided variant due to the author, and the cyclic logic of Yetter. Hopf algebras provide a unifying framework for the analysis of these variants. While these categories are monoidal closed, they lack sufficient structure to model the involutive negation of classical linear logic. We recall work of Lefschetz and Barr in which vector spaces are endowed with an additional topological structure, called linear topology. The resulting category has a large class of reflexive objects, which form a *-autonomous category, and so model the involutive negation. We show that the monoidal closed structure of the category of representations of a Hopf algebra can be extended to this topological category in a natural and simple manner. The models we obtain have the advantage of being nondegenerate in the sense that the two multiplicative connectives, tensor and par, are not equated. It has been recently shown by Barr that this category of topological vector spaces can be viewed as a subcategory of a certain Chu category. In an Appendix, Barr uses this equivalence to analyze the structure of its tensor product.

In this paper we provide a categorical interpretation of the first-order Hoare logic of a small programming language by giving a weakest precondition semantics for the language. To this end, we extend the well-known notion of a (first-order) hyperdoctrine to include partial maps. The most important new aspect of the resulting partial (first-order) hyperdoctrine is a different notion of morphism between the fibres. We also use this partial hyperdoctrine to give a model for Beeson's Partial Function Logic such that (a version of) his axiomatization is complete with respect to this model. This shows the usefulness of the notion, independent of its intended use as a model for Hoare logic.

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.

Every accessible category is proved to be sketchable by a sketch with finite colimits. In contrast, a finitely accessible category is presented that cannot be sketched by a finitary sketch, i.e., a sketch with finite limits and finite colimits. Also, a category sketchable by a finitary sketch is found that is not finitely accessible.

The study of type theory may offer a uniform language for modular programming, structured specification and logical reasoning. We develop an approach to program specification and data refinement in a type theory with a strong logical power and nice structural mechanisms to show that it provides an adequate formalism for modular development of programs and specifications. Specification of abstract data types is considered, and a notion of abstract implementation between specifications is defined in the type theory and studied as a basis for correct and modular development of programs by stepwise refinement. The higher-order structural mechanisms in the type theory provide useful and flexible tools (specification operations and parameterized specifications) for modular design and structured specification. Refinement maps (programs and design decisions) and proofs of implementation correctness can be developed by means of the existing proof development systems based on type theories.

Within a categorical framework for primitive recursion, equality between p.r. maps is shown to be definable by suitable p.r. equality predicates. Equivalence is shown between a direct categorical formalization of classical p.r. functions and p.r. maps in the sense of Lawvere and Freyd. An extension of the theory is shown to admit a ‘universal set’ containing all objects of the extended theory of subobjects.

Non-strict don't care functions, whose foremost representative is the ubiquitous if_then_else, play an essential role in computer science. As far as the semantics is concerned, they can be modelled by their totalizations with the appropriate use of elements representing undefinedness, as D. Scott has shown in his denotational approach. The situation is not so straightforward when we consider non-strict functions in the context of an algebraic framework; this point is discussed in the last section, where we explore the relationship between non-strict don't care and total algebras. The central part of this paper, after presenting the basic properties of the category of non-strict algebras, is an investigation of conditional algebraic specifications. It is shown that non-strict conditional specifications are equivalent to disjunctive specifications, and necessary and sufficient conditions for the existence of initial models are given. Since non-strict don't care specifications generalize both the total and the partial case, it is shown how the results for initiality can be obtained as specializations.

A subtyping relation ≤ between types is often accompanied by a typing rule, called subsumption: if a term a has type T and T≤U, then a has type U. In presence of subsumption, a well-typed term does not codify its proof of well typing. Since a semantic interpretation is most naturally defined by induction on the structure of typing proofs, a problem of coherence arises: different typing proofs of the same term must have related meanings. We propose a proof-theoretical, rewriting approach to this problem. We focus on F≤, a second-order lambda calculus with bounded quantification, which is rich enough to make the problem interesting. We define a normalizing rewriting system on proofs, which transforms different proofs of the same typing judgement into a unique normal proof, with the further property that all the normal proofs assigning different types to a given term in a given environment differ only by a final application of the subsumption rule. This rewriting system is not defined on the proofs themselves but on the terms of an auxiliary type system, in which the terms carry complete information about their typing proof. This technique gives us three different results:

— Any semantic interpretation is coherent if and only if our rewriting rules are satisfied as equations.

— We obtain a proof of the existence of a minimum type for each term in a given environment.

— From an analysis of the shape of normal form proofs, we obtain a deterministic typechecking algorithm, which is sound and complete by construction.

Since Pnueli’s seminal paper in 1977, Temporal Logic has been used as a formalism for specifying and verifying the correctness of reactive systems. In this paper, we show that, besides its expressive power, Temporal Logic enjoys a very strong structural property: it is categorical on processes. That is, we show how temporal specifications (as theories) can be embedded in categories of process behaviour, and out of this adjunction we build an institution that is categorical in the sense of Meseguer. This characterisation means that temporal logic is, in a sense, ‘sound and complete’ with respect to process specification and interconnection techniques.