This paper unveils and motivates an ambitious programme of hidden algebraic research in software engineering, beginning with our general goals, continuing with an overview of results, and including some future plans. The main contribution is powerful hidden coinduction techniques for proving behavioral correctness of concurrent systems; several mechanical proofs are given using OBJ3. We also show how modularization, bisimulation, transition systems, concurrency and combinations of the functional, constraint, logic and object paradigms fit into hidden algebra. 1. Introduction

This paper is a comparative study of a number of (intensional-semantically distinct) least and greatest fixed point operators that natural-deduction proof systems for intuitionistic logics can be extended with in a proof-theoretically defendable way. Eight pairs of such operators are analysed. The exposition is centered around a cube-shaped classification where each node stands for an axiomatization of one pair of operators as logical constants by intended proof and reduction rules and each arc for a proof- and reduction-preserving encoding of one pair in terms of another. The three dimensions of the cube reflect three orthogonal binary options: conventional-style vs. Mendler-style, basic (``[co]iterative'') vs. enhanced (``primitive-[co]recursive''), simple vs. course-of-value [co]induction. Some of the axiomatizations and encodings are well-known; others, however, are novel; the classification into a cube is also new. The differences between the least fixed point operators considered are illustrated on the example of the corresponding natural number types.

Unfolds generate data structures, and folds consume them. A hylomorphism is a fold after an unfold, generating then consuming a virtual data structure. A metamorphism is the opposite composition, an unfold after a fold; typically, it will convert from one data representation to another. In general, metamorphisms are less interesting than hylomorphisms: there is no automatic fusion to deforest the intermediate virtual data structure. However, under certain conditions fusion is possible: some of the work of the unfold can be done before all of the work of the fold is complete. This permits streaming metamorphisms, and among other things allows conversion of infinite data representations. We present a theory of metamorphisms and outline some examples.

Elements from the technical works

Human cognitive capacity includes recursively definable concepts, which are prevalent in domains involving lists, numbers, and languages. Cognitive science currently lacks a satisfactory explanation for the systematic nature of such capacities (i.e., why the capacity for some recursive cognitive abilities–e.g., finding the smallest number in a list–implies the capacity for certain others–finding the largest number, given knowledge of number order). The category-theoretic constructs of initial F-algebra, catamorphism, and their duals, final coalgebra and anamorphism provide a formal, systematic treatment of recursion in computer science. Here, we use this formalism to explain the systematicity of recursive cognitive capacities without ad hoc assumptions (i.e., to the same explanatory standard used in our account of systematicity for non-recursive capacities). The presence of an initial algebra/final coalgebra explains systematicity because all recursive cognitive capacities, in the domain of interest, factor through (are composed of) the same component process. Moreover, this factorization is unique, hence no further (ad hoc) assumptions are required to establish the intrinsic connection between members of a group of systematically-related capacities. This formulation also provides a new perspective on the relationship between recursive cognitive capacities. In particular, the link between number and language does not depend on recursion, as such, but on the underlying functor on which the group of recursive capacities is based. Thus, many species