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Coalgebraic modal logic beyond Sets
 In MFPS XXIII
, 2007
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Cited by 11 (3 self)
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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 7 (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.
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).
Expressiveness of Positive Coalgebraic Logic
"... From the point of view of modal logic, coalgebraic logic over posets is the natural coalgebraic generalisation of positive modal logic. From the point of view of coalgebra, posets arise if one is interested in simulations as opposed to bisimulations. From a categorical point of view, one moves from ..."
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From the point of view of modal logic, coalgebraic logic over posets is the natural coalgebraic generalisation of positive modal logic. From the point of view of coalgebra, posets arise if one is interested in simulations as opposed to bisimulations. From a categorical point of view, one moves from ordinary categories to enriched categories. We show that the basic setup of coalgebraic logic extends to this more general setting and that every finitary functor on posets has a logic that is expressive, that is, has the HennessyMilner property. Keywords: Coalgebra, Modal Logic, Poset
DOI: 10.1017/S0960129502003912 Printed in the United Kingdom Dualising initial algebras
, 2001
"... Whilst the relationship between initial algebras and monads is well understood, the relationship between final coalgebras and comonads is less well explored. This paper shows that the problem is more subtle than might appear at first glance: final coalgebras can form monads just as easily as comonad ..."
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Whilst the relationship between initial algebras and monads is well understood, the relationship between final coalgebras and comonads is less well explored. This paper shows that the problem is more subtle than might appear at first glance: final coalgebras can form monads just as easily as comonads, and, dually, 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 first defining a notion of cosignature and coequation and then proving that the models of such coalgebraic presentations are precisely the coalgebras of the representing comonad. 1.
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
Formal Software Development: From Foundations to Tools
"... This exposé gives an overview of the author’s contributions to the area of formal software development. These range from foundational issues dealing with abstract models of computation to practical engineering issues concerned with tool integration and user interface design. We can distinguish three ..."
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This exposé gives an overview of the author’s contributions to the area of formal software development. These range from foundational issues dealing with abstract models of computation to practical engineering issues concerned with tool integration and user interface design. We can distinguish three lines of work: Firstly, there is foundational work, centred around categorical models of rewriting. A new semantics for rewriting is developed, which abstracts over the concrete term structure while still being able to express key concepts such as variable, layer and substitution. It is based on the concept of a monad, which is wellknown in category theory to model algebraic theories. We generalise this treatment to term rewriting systems, infinitary terms, term graphs, and other forms of rewriting. The semantics finds applications in functional programming, where monads are used to model computational features such as state, exceptions and I/O, and modularity proofs, where
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.