Terminal Coalgebra Research Articles

Terminal metric spaces of finitely branching and image finite linear processes

Well-known metric spaces for modelling finitely branching and image finite systems are shown to be (the carrier of) terminal coalgebras.

Modeling Process Calculi with PVS

In this work we show how process calculi can be embedded into the general-purpose verification system PVS. We make use of PVS's higher order predicate logic to define the abstract co-data type “process” as a terminal coalgebra of the finite power-set functor ℘fin(−)A. In this context we are able to define the (strong) bisimulation in different ways and to prove the equivalence of these definitions. Moreover, coinduction is used as an approach for the definition of process combinators and for the verification of their properties.Other calculi can be integrated in PVS or a like prove-assistant in similar way too.

An approach to object semantics based on terminal co-algebras

The categorical dual construction of initial abstract data types is studied. The resulting terminal co-algebras represent not only an implementation independent semantics of infinite objects such as streams, but also a suitable formal basis for object types in the sense of the object-oriented programming paradigm. Instances of object types may be interpreted as abstract automata with several state transition functions representing the methods of an object, and several output functions representing the attributes. By structuring the index set of the possibly infinite family of methods, and by structuring the output set, one can specify specific object types. For dealing simultaneously with complex data types and object types, it is not necessary to live within a cartesian closed category. In ccc's there are standard functional constructions for object types, but object types are not necessarily a higher-order construction. A world of data types and object types may be combined with the Rewriting Logic of Meseguer to obtain a formal basis for concurrent object systems.

Terminal coalgebras in well-founded set theory

This paper shows that, in order to obtain the theorem of Aczel and Mendler on the existence of terminal coalgebras for an endofunctor on the category of sets, it is entirely unnecessary to delve into such exotica as non-well-founded set theory. In addition, we discuss the canonical map from the initial algebra for an endofunctor on sets to the terminal coalgebra and show that in many cases it embeds the former as a dense subset of the latter in a certain natural topology. By way of example, we calculate the terminal coalgebra for various simple endofunctors.

Algebraically compact functors

In a previous paper, we investigated the relation between the initial algebra and terminal coalgebra for an endofunctor on the category of sets. In this one we study conditions on a functor to be algebraically compact, which means that the canonical comparison morphism between those objects is an isomorphism.