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On the Foundations of Final Semantics: NonStandard Sets, Metric Spaces, Partial Orders
 PROCEEDINGS OF THE REX WORKSHOP ON SEMANTICS: FOUNDATIONS AND APPLICATIONS, VOLUME 666 OF LECTURE NOTES IN COMPUTER SCIENCE
, 1998
"... Canonical solutions of domain equations are shown to be final coalgebras, not only in a category of nonstandard sets (as already known), but also in categories of metric spaces and partial orders. Coalgebras are simple categorical structures generalizing the notion of postfixed point. They are ..."
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Cited by 48 (9 self)
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Canonical solutions of domain equations are shown to be final coalgebras, not only in a category of nonstandard sets (as already known), but also in categories of metric spaces and partial orders. Coalgebras are simple categorical structures generalizing the notion of postfixed point. They are also used here for giving a new comprehensive presentation of the (still) nonstandard theory of nonwellfounded sets (as nonstandard sets are usually called). This paper is meant to provide a basis to a more general project aiming at a full exploitation of the finality of the domains in the semantics of programming languages  concurrent ones among them. Such a final semantics enjoys uniformity and generality. For instance, semantic observational equivalences like bisimulation can be derived as instances of a single `coalgebraic' definition (introduced elsewhere), which is parametric of the functor appearing in the domain equation. Some properties of this general form of equivalence are also studied in this paper.
Automata and fixed point logics: a coalgebraic perspective
 Electronic Notes in Theoretical Computer Science
, 2004
"... This paper generalizes existing connections between automata and logic to a coalgebraic level. Let F: Set → Set be a standard functor that preserves weak pullbacks. We introduce various notions of Fautomata, devices that operate on pointed Fcoalgebras. The criterion under which such an automaton a ..."
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Cited by 26 (11 self)
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This paper generalizes existing connections between automata and logic to a coalgebraic level. Let F: Set → Set be a standard functor that preserves weak pullbacks. We introduce various notions of Fautomata, devices that operate on pointed Fcoalgebras. The criterion under which such an automaton accepts or rejects a pointed coalgebra is formulated in terms of an infinite twoplayer graph game. We also introduce a language of coalgebraic fixed point logic for Fcoalgebras, and we provide a game semantics for this language. Finally we show that any formula p of the language can be transformed into an Fautomaton Ap which is equivalent to p in the sense that Ap accepts precisely those pointed Fcoalgebras in which p holds.
Algebras and Coalgebras
 Handbook of Modal Logic
, 2007
"... This chapter 1 sketches some of the mathematical surroundings of modal logic. First, we discuss the algebraic perspective on the field, showing how the theory of universal algebra, and more specifically, that of Boolean algebras with operators, can be used to prove significant results in modal logic ..."
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Cited by 25 (3 self)
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This chapter 1 sketches some of the mathematical surroundings of modal logic. First, we discuss the algebraic perspective on the field, showing how the theory of universal algebra, and more specifically, that of Boolean algebras with operators, can be used to prove significant results in modal logic. In the second and last part of the chapter we describe how modal logic, and its model theory, provides many natural manifestations of the more general theory of universal coalgebra.
Relating coalgebraic notions of bisimulation
 Logical Methods in Computer Science 7 (1:13) (2011
"... Vol. 7 (1:13) 2011, pp. 1–21 www.lmcsonline.org ..."
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Applications of metric coinduction
 Proc. 2nd Conf. Algebra and Coalgebra in Computer Science (CALCO 2007), volume 4624 of Lecture Notes in Computer Science
, 2007
"... Abstract. Metric coinduction is a form of coinduction that can be used to establish properties of objects constructed as a limit of finite approximations. One proves a coinduction step showing that some property is preserved by one step of the approximation process, then automatically infers by the ..."
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Cited by 9 (3 self)
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Abstract. Metric coinduction is a form of coinduction that can be used to establish properties of objects constructed as a limit of finite approximations. One proves a coinduction step showing that some property is preserved by one step of the approximation process, then automatically infers by the coinduction principle that the property holds of the limit object. This can often be used to avoid complicated analytic arguments involving limits and convergence, replacing them with simpler algebraic arguments. This paper examines the application of this principle in a variety of areas, including infinite streams, Markov chains, Markov decision processes, and nonwellfounded sets. These results point to the usefulness of coinduction as a general proof technique. 1
A Structural CoInduction Theorem
 PROC. MFPS '93, SPRINGER LNCS 802
, 1993
"... The Structural Induction Theorem (Lehmann and Smyth, 1981; Plotkin, 1981) characterizes initial Falgebras of locally continuous functors F on the category of cpo's with strict and continuous maps. Here a dual of that theorem is presented, giving a number of equivalent characterizations of fi ..."
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Cited by 7 (1 self)
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The Structural Induction Theorem (Lehmann and Smyth, 1981; Plotkin, 1981) characterizes initial Falgebras of locally continuous functors F on the category of cpo's with strict and continuous maps. Here a dual of that theorem is presented, giving a number of equivalent characterizations of final coalgebras of such functors. In particular, final coalgebras are order stronglyextensional (sometimes called internal full abstractness): the order is the union of all (ordered) Fbisimulations. (Since the initial fixed point for locally continuous functors is also final, both theorems apply.) Further a similar coinduction theorem is given for a category of complete metric spaces and locally contracting functors.
Free modal algebras: a coalgebraic perspective
"... Abstract. In this paper we discuss a uniform method for constructing free modal and distributive modal algebras. This method draws on works by (Abramsky 2005) and (Ghilardi 1995). We revisit the theory of normal forms for modal logic and derive a normal form representation for positive modal logic. ..."
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Abstract. In this paper we discuss a uniform method for constructing free modal and distributive modal algebras. This method draws on works by (Abramsky 2005) and (Ghilardi 1995). We revisit the theory of normal forms for modal logic and derive a normal form representation for positive modal logic. We also show that every finitely generated free modal and distributive modal algebra axiomatised by equations of rank 1 is a reduct of a temporal algebra. 1
Finitely generated free Heyting algebras via Birkhoff duality and coalgebra
 Logical Methods in Computer Science 7
, 2011
"... Vol. 7 (2:9) 2011, pp. 1–24 www.lmcsonline.org ..."
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Nabla Algebras and Chu Spaces
"... Abstract. This paper is a study into some properties and applications of Moss ’ coalgebraic or ‘cover ’ modality ∇. First we present two axiomatizations of this operator, and we prove these axiomatizations to be sound and complete with respect to basic modal and positive modal logic, respectively. M ..."
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Cited by 6 (4 self)
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Abstract. This paper is a study into some properties and applications of Moss ’ coalgebraic or ‘cover ’ modality ∇. First we present two axiomatizations of this operator, and we prove these axiomatizations to be sound and complete with respect to basic modal and positive modal logic, respectively. More precisely, we introduce the notions of a modal ∇algebra and of a positive modal ∇algebra. We establish a categorical isomorphism between the category of modal ∇algebra and that of modal algebras, and similarly for positive modal ∇algebras and positive modal algebras. We then turn to a presentation, in terms of relation lifting, of the Vietoris hyperspace in topology. The key ingredient is an Flifting construction, for an arbitrary set functor F, on the category Chu of twovalued Chu spaces and Chu transforms, based on relation lifting. As a case study, we show how to realize the Vietoris construction on Stone spaces as a special instance of this Chu construction for the (finite) power set functor. Finally, we establish a tight connection with the axiomatization of the modal ∇algebras.