Results 1  10
of
21
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 ..."
Abstract

Cited by 48 (10 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 16 (8 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
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
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. ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
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
A category of compositional domainmodels for separable Stone spaces
, 2000
"... In this paper we introduce SFP M , a category of SFP domains which provides very satisfactory domainmodels, i.e. "partializations", of separable Stone spaces (2Stone spaces). More specifically, SFP M is a subcategory of SFP ep , closed under direct limits as well as many construc ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
In this paper we introduce SFP M , a category of SFP domains which provides very satisfactory domainmodels, i.e. "partializations", of separable Stone spaces (2Stone spaces). More specifically, SFP M is a subcategory of SFP ep , closed under direct limits as well as many constructors, such as lifting, sum, product and Plotkin powerdomain. SFP M is "structurally well behaved", in the sense that the functor MAX, which associates to each object of SFP M the Stone space of its maximal elements, is compositional with respect to the constructors above, and wcontinuous. A correspondence can be established between these constructors over SFP M and appropriate constructors on Stone spaces, whereby SFP domainmodels of Stone spaces defined as solutions of a vast class of recursive equations in 2Stone, can be obtained simply by solving the corresponding equations in SFP M . Moreover any continuous function between two 2Stone spaces can be extended to a continuous function b...
Axiomatic Characterizations of Hyperuniverses and Applications
 University of Southern
, 1996
"... Hyperuniverses are topological structures exhibiting strong closure properties under formation of subsets. They have been used both in Computer Science, for giving denotational semantics `a la Scottde Bakker, and in Mathematical Logic, in order to show the consistency of set theories which do not a ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
Hyperuniverses are topological structures exhibiting strong closure properties under formation of subsets. They have been used both in Computer Science, for giving denotational semantics `a la Scottde Bakker, and in Mathematical Logic, in order to show the consistency of set theories which do not abide by the "limitation of size" principle. We present correspondences between settheoretic properties and topological properties of hyperuniverses. We give existence theorems and discuss applications and generalizations to the non compact case. Work partially supported by 40% and 60% MURST grants, CNR grants, and EEC Science MASK, and BRA Types 6453 contracts. y Member of GNSAGA of CNR. z The main results of this paper have been communicated by this author at the "11 th Summer Conference on General Topology and Applications" August 1995, Portland, Maine. Introduction Natural frameworks for dicussing Selfreference and other circular phenomena are extremely useful in areas such ...
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 ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
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.
Partializing Stone Spaces using SFP domains (Extended Abstract)
 CAAP ’97, volume 1158 of LNCS
, 1997
"... ) F. Alessi, P. Baldan, F. Honsell Dipartimento di Matematica ed Informatica via delle Scienze 208, 33100 Udine, Italy falessi, baldan, honsellg@dimi.uniud.it Abstract. In this paper we investigate the problem of "partializing" Stone spaces by "Sequence of Finite Posets" (SFP) ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
) F. Alessi, P. Baldan, F. Honsell Dipartimento di Matematica ed Informatica via delle Scienze 208, 33100 Udine, Italy falessi, baldan, honsellg@dimi.uniud.it Abstract. In this paper we investigate the problem of "partializing" Stone spaces by "Sequence of Finite Posets" (SFP) domains. More specifically, we introduce a suitable subcategory SFP m of SFP which is naturally related to the special category of Stone spaces 2Stone by the functor MAX, which associates to each object of SFP m the space of its maximal elements. The category SFP m is closed under limits as well as many domain constructors, such as lifting, sum, product and Plotkin powerdomain. The functor MAX preserves limits and commutes with these constructors. Thus, SFP domains which "partialize" solutions of a vast class of domain equations in 2Stone, can be obtained by solving the corresponding equations in SFP m . Furthermore, we compare two classical partializations of the space of Milner's Synchronization Tre...