Algebras, Coalgebras and Processes

Uwe Wolter
TU Berlin

In the last years theoretical computer scientists are put more and more in a position where they have to defend themselves for still doing "abstract nonsens". To a certain extend this is not so bad since we are forced to think more thoroughly about what kind of theory we will develop and teach in future.

Obviously theory should help people to solve special practical problems. On the other side, however, theory should give a proper basis and a guidance for understanding, constructing and managing a very complex world. Theory should help us to structure and to condense our knowledge and our systems of concepts and ideas.

One principle very helpful in bringing some order into our systems of knowledge is to seek for dualities of concepts and constructions. Well-known dualities are, e.g., the duality between the logical connectives "and" and "or" or the duality between the cartesian product AxB and the disjoint union A+B of sets.

In the talk we will present the concept of coalgebra which arises by dualizing the familiar concept of algebra. We discuss the corresponding duality between the algebraic principle of induction and the coalgebraic principle of coinduction. We show that dualizing the concept of initial algebra leads to the concept of processes developed in the theory of concurrent systems. Moreover we point out the connection between automata, coalgebras, and processes.

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