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NonDeterministic Kleene Coalgebras
"... In this paper, we present a systematic way of deriving (1) languages of (generalised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of systems. This generalizes both the results of Kleene (on regular languages and deterministic finite automata) and Miln ..."
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Cited by 15 (4 self)
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In this paper, we present a systematic way of deriving (1) languages of (generalised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of systems. This generalizes both the results of Kleene (on regular languages and deterministic finite automata) and Milner (on regular behaviours and finite labelled transition systems), and includes many other systems such as Mealy and Moore machines.
Observational Ultraproducts of Polynomial Coalgebras
, 2002
"... Coalgebras of polynomial functors constructed from sets of observable elements have been found useful in modelling various kinds of data types and statetransition systems. ..."
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Cited by 6 (3 self)
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Coalgebras of polynomial functors constructed from sets of observable elements have been found useful in modelling various kinds of data types and statetransition systems.
A modal proof theory for final polynomial coalgebras. Theoret
 Comput. Sci
"... An infinitary proof theory is developed for modal logics whose models are coalgebras of polynomial functors on the category of sets. The canonical model method from modal logic is adapted to construct a final coalgebra for any polynomial functor. The states of this final coalgebra are certain “maxim ..."
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Cited by 4 (1 self)
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An infinitary proof theory is developed for modal logics whose models are coalgebras of polynomial functors on the category of sets. The canonical model method from modal logic is adapted to construct a final coalgebra for any polynomial functor. The states of this final coalgebra are certain “maximal ” sets of formulas that have natural syntactic closure properties. The syntax of these logics extends that of previously developed modal languages for polynomial coalgebras by adding formulas that express the “termination ” of certain functions induced by transition paths. A completeness theorem is proven for the logic of functors which have the Lindenbaum property that every consistent set of formulas has a maximal extension. This property is shown to hold if if the deducibility relation is generated by countably many inference rules. A counterexample to completeness is also given. This is a polynomial functor that is not Lindenbaum: it has an uncountable set of formulas that is deductively consistent but has no maximal extension and is unsatisfiable, even though all of its countable subsets are satisfiable. 1
Enlargements of Polynomial Coalgebras
 Proceedings of the 7th and 8th Asian Logic Conferences
"... This paper continues a series [GolOlc,a,b] of articles on the equational logic and model theory of coalgebras for certain functors T: Set  Set on the category of sets. A Tcoalgebra is a pair (A, () comprising a set A, thought of as a set of "states", and a function : A > TA called the transition ..."
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Cited by 3 (3 self)
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This paper continues a series [GolOlc,a,b] of articles on the equational logic and model theory of coalgebras for certain functors T: Set  Set on the category of sets. A Tcoalgebra is a pair (A, () comprising a set A, thought of as a set of "states", and a function : A > TA called the transition structure. We study the case of functors T that are polynomial, i.e. constructed from constantvalued functors and the identity functor by forming products, coproducts, and exponential functors with constant exponent. Many data structures and systems of interest to computer science  such as lists, streams, trees, automata, and classes in objectoriented programming languages  can be modelled as coalgebras for polynomial functors [Rei95, Jac96, Rut95, Rut00]. This has motivated the development of a theory of "universal coalgebra" [Rut95, Rut00], by analogy with, and categorically dual to, the study of abstract algebras
Kleene Coalgebra – an overview
"... Abstract. Coalgebras provide a uniform framework for the study of dynamical systems, including several types of automata. The coalgebraic view on systems has recently been proved relevant by the development of a number of expression calculi which generalize classical results by Kleene, on regular ex ..."
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Abstract. Coalgebras provide a uniform framework for the study of dynamical systems, including several types of automata. The coalgebraic view on systems has recently been proved relevant by the development of a number of expression calculi which generalize classical results by Kleene, on regular expressions, and by Kozen, on Kleene algebra. This note contains an overview of the motivation and results of the generic framework we developed – Kleene Coalgebra – to uniformly derive the aforementioned calculi. We present an historical overview of work on regular expressions and axiomatizations, as well a discussion of related work. We show applications of the framework to three types of probabilistic systems: simple Segala, stratified and PnueliZuck. 1