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Coalgebraic semantics for timed processes
 Inf. & Comp
, 2006
"... We give a coalgebraic formulation of timed processes and their operational semantics. We model time by a monoid called a “time domain”, and we model processes by “timed transition systems”, which amount to partial monoid actions of the time domain or, equivalently, coalgebras for an “evolution comon ..."
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We give a coalgebraic formulation of timed processes and their operational semantics. We model time by a monoid called a “time domain”, and we model processes by “timed transition systems”, which amount to partial monoid actions of the time domain or, equivalently, coalgebras for an “evolution comonad ” generated by the time domain. All our examples of time domains satisfy a partial closure property, yielding a distributive law of a monad for total monoid actions over the evolution comonad, and hence a distributive law of the evolution comonad over a dual comonad for total monoid actions. We show that the induced coalgebras are exactly timed transition systems with delay operators. We then integrate our coalgebraic formulation of time qua timed transition systems into Turi and Plotkin’s formulation of structural operational semantics in terms of distributive laws. We combine timing with action via the more general study of the combination of two arbitrary sorts of behaviour whose operational semantics may interact. We give a modular account of the operational semantics for a combination induced by that of each of its components. Our study necessitates the investigation of products of comonads. In particular, we characterise when a monad lifts to the category of coalgebras for a product comonad, providing constructions with which one can readily calculate. Key words: time domains, timed transition systems, evolution comonads, delay operators, structural operational semantics, modularity, distributive laws 1
Logical Construction of Final Coalgebras
, 2003
"... We prove that every finitary polynomial endofunctor of a category C has a final coalgebra, provided that C is locally Cartesian closed, it has finite coproducts and is an extensive category, it has a natural number object. ..."
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We prove that every finitary polynomial endofunctor of a category C has a final coalgebra, provided that C is locally Cartesian closed, it has finite coproducts and is an extensive category, it has a natural number object.
Lane’s Categories for the Working Mathematician? Simply put: because
"... Why write a new textbook on Category Theory, when we already have Mac ..."
Abstract CMCS’03 Preliminary Version Logical Construction of Final Coalgebras
"... We prove that every finitary polynomial endofunctor of a category C has a final coalgebra, provided that C is locally Cartesian closed, it has finite coproducts and ..."
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We prove that every finitary polynomial endofunctor of a category C has a final coalgebra, provided that C is locally Cartesian closed, it has finite coproducts and
CMCS'03 Preliminary Version Logical Construction of Final Coalgebras
"... Abstract We prove that every finitary polynomial endofunctor of a category C has a final coalgebra, provided that C is locally Cartesian closed, it has finite coproducts and is an extensive category, it has a natural number object. 1 Introduction In this note we prove that every polynomial endofunct ..."
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Abstract We prove that every finitary polynomial endofunctor of a category C has a final coalgebra, provided that C is locally Cartesian closed, it has finite coproducts and is an extensive category, it has a natural number object. 1 Introduction In this note we prove that every polynomial endofunctor
Final Coalgebras And a Solution Theorem for Arbitrary Endofunctors
"... Every endofunctor F of Set has an initial algebra and a final coalgebra, but they are classes in general. Consequently, the endofunctor F ∞ of the category of classes that F induces generates a completely iterative monad T. And solutions of arbitrary guarded systems of iterative equations w.r.t. F e ..."
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Every endofunctor F of Set has an initial algebra and a final coalgebra, but they are classes in general. Consequently, the endofunctor F ∞ of the category of classes that F induces generates a completely iterative monad T. And solutions of arbitrary guarded systems of iterative equations w.r.t. F exist, and can be found in naturally defined subsets of the classes T Y.