## Milius, Stefan - Proper Functors and Fixed Points for Finite Behaviour

lmcs:4822 - Logical Methods in Computer Science, September 24, 2018, Volume 14, Issue 3
Proper Functors and Fixed Points for Finite Behaviour

Authors: Milius, Stefan

The rational fixed point of a set functor is well-known to capture the behaviour of finite coalgebras. In this paper we consider functors on algebraic categories. For them the rational fixed point may no longer be fully abstract, i.e. a subcoalgebra of the final coalgebra. Inspired by \'Esik and Maletti's notion of a proper semiring, we introduce the notion of a proper functor. We show that for proper functors the rational fixed point is determined as the colimit of all coalgebras with a free finitely generated algebra as carrier and it is a subcoalgebra of the final coalgebra. Moreover, we prove that a functor is proper if and only if that colimit is a subcoalgebra of the final coalgebra. These results serve as technical tools for soundness and completeness proofs for coalgebraic regular expression calculi, e.g. for weighted automata.

Source : oai:arXiv.org:1705.09198
DOI : 10.23638/LMCS-14(3:22)2018
Volume: Volume 14, Issue 3
Published on: September 24, 2018
Submitted on: September 13, 2018
Keywords: Computer Science - Logic in Computer Science