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Observational Coalgebras and Complete Sets of Cooperations
, 2008
"... In this paper we introduce the notion of an observational coalgebra structure and of a complete set of cooperations. We demonstrate in various example the usefulness of these notions, in particular, we show how they give rise to coalgebraic proof and definition principles. ..."
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In this paper we introduce the notion of an observational coalgebra structure and of a complete set of cooperations. We demonstrate in various example the usefulness of these notions, in particular, we show how they give rise to coalgebraic proof and definition principles.
UNIVERSAL COALGEBRAS AND THEIR LOGICS
, 2009
"... ABSTRACT. We survey coalgebras as models of state based systems together with their global and local logics. We convey some useful intuition regarding Setfunctors which leads naturally to coalgebraic modal logic where modalities are validity patterns for the successor object of a state. 1. ..."
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ABSTRACT. We survey coalgebras as models of state based systems together with their global and local logics. We convey some useful intuition regarding Setfunctors which leads naturally to coalgebraic modal logic where modalities are validity patterns for the successor object of a state. 1.
London, United Kingdom
"... The structure map turning a set into the carrier of a final coalgebra is not unique. This fact is wellknown but commonly elided. In this paper we argue that any such concrete representation of a set as a final coalgebra is potentially interesting on its own. We discuss several examples, in particul ..."
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The structure map turning a set into the carrier of a final coalgebra is not unique. This fact is wellknown but commonly elided. In this paper we argue that any such concrete representation of a set as a final coalgebra is potentially interesting on its own. We discuss several examples, in particular, we consider different coalgebra structures that turn the set of infinite streams into the carrier of a final coalgebra. After that we focus on coalgebra structures that are made up using socalled cooperations. We say that a collection of cooperations is complete for a given set X if it gives rise to a coalgebra structure that turns X into the carrier set of a subcoalgebra of a final coalgebra. Any complete set of cooperations yields a coalgebraic proof and definition principle. We exploit this fact and devise a general definition scheme for constants and functions on a set X that is parametrically in the choice of the complete set of cooperations for X. Key words: Coalgebra, coinduction, infinite data structures, hidden algebra. 1
A Coalgebraic Perspective on Minimization and Determinization ⋆
"... Abstract. Coalgebra offers a unified theory of state based systems, including infinite streams, labelled transition systems and deterministic automata. In this procedures for checking behavioural equivalence in coalgebras, which perform (a combination of) minimization and determinization. First, we ..."
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Abstract. Coalgebra offers a unified theory of state based systems, including infinite streams, labelled transition systems and deterministic automata. In this procedures for checking behavioural equivalence in coalgebras, which perform (a combination of) minimization and determinization. First, we show that for coalgebras in categories equipped with factorization structures, there exists an abstract procedure for equivalence checking. Then, we consider coalgebras in categories without suitable factorization structures: under certain conditions, it is possible to apply the above procedure after transforming coalgebras with reflections. This transformation can be thought of as some kind of determinization. We will apply our theory to the following examples: conditional transition systems and (nondeterministic) automata. 1