On the foundations of final coalgebra semantics: non-well-founded sets, partial orders, metric spaces

Abstract

This paper, a revised version of Rutten and Turi (1993), is part of a programme aiming at formulating a mathematical theory of structural operational semantics to complement the established theory of domains and denotational semantics to form a coherent whole (Turi 1996; Turi and Plotkin 1997). The programme is based on a suitable interplay between the induction principle, which pervades modern mathematics, and a dual, non-standard ‘coinduction principle’, which underlies many of the recursive phenomena occurring in computer science.The aim of the present survey is to show that the elementary categorical notion of a final coalgebra is a suitable foundation for such a coinduction principle. The properties of coalgebraic coinduction are studied both at an abstract categorical level and in some specific categories used in semantics, namely categories of non-well-founded sets, partial orders and metric spaces.

Non-well-founded trees in categories

“…For this reason, they have been used in the theory of concurrency and specification, as well as in the study of semantics for programming languages with coinductive types [10,11,18,29,4].…”

“…Those of finite length form the corresponding W-type. More examples of importance in computer science can be found in [18] and [29]. But there are examples that are not toposes.…”

A Perspective on Component Refinement

“…Being final means that there exists a unique morphism to out T from each other coalgebra U, p . This is called the coinductive extension of p [48] or the anamorphism generated by p [28], and written as [(p)] T or, simply, [(p)], if the functor is clear from context. In other words, an anamorphism is defined as the unique function making the following diagram to commute:…”

Some Properties and Some Problems on Set Functors

“…categories where objects are sets (classes) of a possible non-wellfounded universe and morphisms are set (class) functions, have been used as convenient settings for studying the foundations of coalgebraic semantics, see e.g. [1,2,7,8,10,18,15,12,3,4].…”