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Coalgebraic modal logic beyond Sets
 In MFPS XXIII
, 2007
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Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be
Algebras, Coalgebras, Monads and Comonads
, 2001
"... Whilst the relationship between initial algebras and monads is wellunderstood, the relationship between nal coalgebras and comonads is less well explored. This paper shows that the problem is more subtle and that final coalgebras can just as easily form monads as comonads and dually, that initial a ..."
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Cited by 8 (3 self)
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Whilst the relationship between initial algebras and monads is wellunderstood, the relationship between nal coalgebras and comonads is less well explored. This paper shows that the problem is more subtle and that final coalgebras can just as easily form monads as comonads and dually, that initial algebras form both monads and comonads. In developing these theories we strive to provide them with an associated notion of syntax. In the case of initial algebras and monads this corresponds to the standard notion of algebraic theories consisting of signatures and equations: models of such algebraic theories are precisely the algebras of the representing monad. We attempt to emulate this result for the coalgebraic case by defining a notion cosignature and coequation and then proving the models of this syntax are precisely the coalgebras of the representing comonad.
On coalgebras over algebras
 In ”Proceedings of the Tenth Workshop on Coalgebraic Methods in Computer Science (CMCS 2010)”, Electr. Notes
"... We extend Barr’s wellknown characterization of the final coalgebra of a Setendofunctor H as the completion of its initial algebra to the EilenbergMoore category Alg(M) of algebras associated to a Setmonad M, if H can be lifted to Alg(M). As further analysis, we introduce the notion of commuting ..."
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We extend Barr’s wellknown characterization of the final coalgebra of a Setendofunctor H as the completion of its initial algebra to the EilenbergMoore category Alg(M) of algebras associated to a Setmonad M, if H can be lifted to Alg(M). As further analysis, we introduce the notion of commuting pair of endofunctors (T,H) with respect to a monad M and show that under reasonable assumptions, the final Hcoalgebra can be obtained as the completion of the free Malgebra on the initial Talgebra.
An AlphaCorecursion Principle for the Infinitary Lambda Calculus
, 2012
"... Gabbay and Pitts proved that lambdaterms up to alphaequivalence constitute an initial algebra for a certain endofunctor on the category of nominal sets. We show that the terms of the infinitary lambdacalculus form the final coalgebra for the same functor. This allows us to give a corecursion pri ..."
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Gabbay and Pitts proved that lambdaterms up to alphaequivalence constitute an initial algebra for a certain endofunctor on the category of nominal sets. We show that the terms of the infinitary lambdacalculus form the final coalgebra for the same functor. This allows us to give a corecursion principle for alphaequivalence classes of finite and infinite terms. As an application, we give corecursive definitions of substitution and of infinite normal forms (Böhm, LévyLongo and Berarducci trees).
A Coalgebraic Calculus for Component Based Systems ∗
"... In this paper we describe the coalgebraic models for statebased software components and componentbased systems. The behaviour patterns of components are specified by strong monads. A family of operators for combining components based on the category of coalgebras are defined and a set of algebraic ..."
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In this paper we describe the coalgebraic models for statebased software components and componentbased systems. The behaviour patterns of components are specified by strong monads. A family of operators for combining components based on the category of coalgebras are defined and a set of algebraic laws are also presented to specify the properties being satisfied by these operators. 1
Nominal Coalgebraic Data Types . . .
"... We investigate final coalgebras in nominal sets. This allows us to define types of infinite data with binding for which all constructions automatically respect alpha equivalence. We give applications to the infinitary lambda calculus. ..."
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We investigate final coalgebras in nominal sets. This allows us to define types of infinite data with binding for which all constructions automatically respect alpha equivalence. We give applications to the infinitary lambda calculus.
Nominal Kleene Coalgebra
"... Abstract. We develop the coalgebraic theory of nominal Kleene algebra, including an alternative languagetheoretic semantics, a nominal extension of the Brzozowski derivative, and a bisimulationbased decision procedure for the equational theory. 1 ..."
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Abstract. We develop the coalgebraic theory of nominal Kleene algebra, including an alternative languagetheoretic semantics, a nominal extension of the Brzozowski derivative, and a bisimulationbased decision procedure for the equational theory. 1
Structural Operational Semantics for Continuous State Stochastic Transition Systems
"... In this paper we show how to model syntax and semantics of stochastic processes with continuous states, respectively as algebras and coalgebras of suitable endofunctors over the category of measurable spaces Meas. Moreover, we present an SOSlike rule format, called MGSOS, representing abstract GSO ..."
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In this paper we show how to model syntax and semantics of stochastic processes with continuous states, respectively as algebras and coalgebras of suitable endofunctors over the category of measurable spaces Meas. Moreover, we present an SOSlike rule format, called MGSOS, representing abstract GSOS over Meas, and yielding fully abstract universal semantics, for which behavioral equivalence is a congruence. An MGSOS specification defines how semantics of processes are composed by means of measure terms, which are expressions specifically designed for describing finite measures. The syntax of these measure terms, and their interpretation as measures, are part of the MGSOS specification. We give two example applications, with a simple and neat MGSOS specification: a “quantitative CCS”, and a calculus of processes living in the plane R2 whose communication rate depends on their distance. The approach we follow in these cases can be readily adapted to deal with other quantitative aspects.
On a Categorical Framework for Coalgebraic Modal Logic
, 2014
"... A category of onestep semantics is introduced to unify different approaches to coalgebraic logic parametric in a contravariant functor that assigns to the state space its collection of predicates with propositional connectives. Modular constructions of coalgebraic logic are identified as colimits, ..."
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A category of onestep semantics is introduced to unify different approaches to coalgebraic logic parametric in a contravariant functor that assigns to the state space its collection of predicates with propositional connectives. Modular constructions of coalgebraic logic are identified as colimits, limits, and tensor products, extending known results for predicate liftings. Generalised predicate liftings as modalities are introduced. Under common assumptions, the logic of all predicate liftings together with a complete axiomatisation exists for any type of coalgebras, and it is onestep expressive for finitary functors. Colimits and compositions of onestep expressive coalgebraic logics are shown to remain onestep expressive. 1