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15
Universal coalgebra: a theory of systems
, 2000
"... In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certa ..."
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Cited by 298 (31 self)
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In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certain types of automata and more generally, for (transition and dynamical) systems. An important property of initial algebras is that they satisfy the familiar principle of induction. Such a principle was missing for coalgebras until the work of Aczel (NonWellFounded sets, CSLI Leethre Notes, Vol. 14, center for the study of Languages and information, Stanford, 1988) on a theory of nonwellfounded sets, in which he introduced a proof principle nowadays called coinduction. It was formulated in terms of bisimulation, a notion originally stemming from the world of concurrent programming languages. Using the notion of coalgebra homomorphism, the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras. Thus, the three basic notions of universal algebra: algebra, homomorphism of algebras, and congruence, turn out to correspond to coalgebra, homomorphism of coalgebras, and bisimulation, respectively. In this paper, the latter are taken
Towards a Mathematical Operational Semantics
 In Proc. 12 th LICS Conf
, 1997
"... We present a categorical theory of `wellbehaved' operational semantics which aims at complementing the established theory of domains and denotational semantics to form a coherent whole. It is shown that, if the operational rules of a programming language can be modelled as a natural transformation ..."
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Cited by 134 (9 self)
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We present a categorical theory of `wellbehaved' operational semantics which aims at complementing the established theory of domains and denotational semantics to form a coherent whole. It is shown that, if the operational rules of a programming language can be modelled as a natural transformation of a suitable general form, depending on functorial notions of syntax and behaviour, then one gets both an operational model and a canonical, internally fully abstract denotational model for free; moreover, both models satisfy the operational rules. The theory is based on distributive laws and bialgebras; it specialises to the known classes of wellbehaved rules for structural operational semantics, such as GSOS.
Initial Algebra and Final Coalgebra Semantics for Concurrency
, 1994
"... The aim of this paper is to relate initial algebra semantics and final coalgebra semantics. It is shown how these two approaches to the semantics of programming languages are each others dual, and some conditions are given under which they coincide. More precisely, it is shown how to derive initial ..."
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Cited by 55 (9 self)
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The aim of this paper is to relate initial algebra semantics and final coalgebra semantics. It is shown how these two approaches to the semantics of programming languages are each others dual, and some conditions are given under which they coincide. More precisely, it is shown how to derive initial semantics from final semantics, using the initiality and finality to ensure their equality. Moreover, many facts about congruences (on algebras) and (generalized) bisimulations (on coalgebras) are shown to be dual as well.
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 (10 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.
Final Semantics for a Higher Order Concurrent Language
 CAAP'96 Conference Proceedings, H.Kirchner ed., Springer LNCS
, 1995
"... We show that adequate semantics can be provided for imperative higher order concurrent languages simply using syntactical final coalgebras. In particular we investigate and compare various behavioural equivalences on higher order processes defined by finality using hypersets and c.m.s.'s. Correspond ..."
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Cited by 15 (11 self)
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We show that adequate semantics can be provided for imperative higher order concurrent languages simply using syntactical final coalgebras. In particular we investigate and compare various behavioural equivalences on higher order processes defined by finality using hypersets and c.m.s.'s. Correspondingly, we derive various coinduction and mixed inductioncoinduction proof principles for establishing these equivalences.
On the Foundations of Corecursion
 Logic Journal of the IGPL
, 1997
"... We consider foundational questions related to the definition of functions by corecursion. This method is especially suited to functions into the greatest fixed point of some monotone operator, and it is most applicable in the context of nonwellfounded sets. We review the work on the Special Final C ..."
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Cited by 13 (1 self)
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We consider foundational questions related to the definition of functions by corecursion. This method is especially suited to functions into the greatest fixed point of some monotone operator, and it is most applicable in the context of nonwellfounded sets. We review the work on the Special Final Coalgebra Theorem of Aczel [1] and the Corecursion Theorem of Barwise and Moss [4]. We offer a condition weaker than Aczel's condition of uniformity on maps, and then we prove a result relating the operators satisfying the new condition to the smooth operators of [4]. Keywords: corecursion, coalgebra, operator on sets 1 Introduction By a stream of natural numbers we mean a pair hn; si where n 2 N and s is again a stream of natural numbers. Let f : N ! N . Consider the following function which purports to define a function from N into the streams: iter f (n) = hn; iter f f(n)i (1.1) For each n, iter f (n) is a stream, so iter f itself is a function from numbers to streams. This is an examp...
Topological Models for Higher Order Control Flow
 PROCEEDINGS OF THE 9TH INTERNATIONAL CONFERENCE ON MATHEMATICAL FOUNDATIONS OF PROGRAMMING SEMANTICS, VOLUME 802 OF LECTURE NOTES IN COMPUTER SCIENCE
, 1993
"... Semantic models are presented for two simple imperative languages with higher order constructs. In the first language the interesting notion is that of second order assignment x := s, for x a procedure variable and s a statement. The second language extends this idea by a form of higher order commun ..."
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Cited by 7 (3 self)
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Semantic models are presented for two simple imperative languages with higher order constructs. In the first language the interesting notion is that of second order assignment x := s, for x a procedure variable and s a statement. The second language extends this idea by a form of higher order communication, with statements c ! s and c ? x, for c a channel. We develop operational and denotational models for both languages, and study their relationships. Both in the definitions and the comparisons of the semantic models, convenient use is made of some tools from (metric) topology. The operational models are based on (SOSstyle) transition systems; the denotational definitions use domains specified as solutions of domain equations in a category of 1bounded complete ultrametric spaces. In establishing the connection between the two kinds of models, fruitful use is made of Rutten's processes as terms technique. Another new tool consists in the use of metric transition systems, with a metric defined on the configurations of the system. In addition to higher order programming notions, we use higher order definitional techniques, e.g., in defining the semantic mappings as fixed points of (contractive) higher order operators. By Banach's theorem, such fixed points are unique, yielding another important proof principle for our paper.
Processes and Hyperuniverses
 Proceedings of the 19th Symposium on Mathematical Foundations of Computer Science 1994, volume 841 of LNCS
, 1994
"... . We show how to define domains of processes, which arise in the denotational semantics of concurrent languages, using hypersets, i.e. nonwellfounded sets. In particular we discuss how to solve recursive equations involving settheoretic operators within hyperuniverses with atoms. Hyperuniverses ar ..."
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Cited by 7 (0 self)
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. We show how to define domains of processes, which arise in the denotational semantics of concurrent languages, using hypersets, i.e. nonwellfounded sets. In particular we discuss how to solve recursive equations involving settheoretic operators within hyperuniverses with atoms. Hyperuniverses are transitive sets which carry a uniform topological structure and include as a clopen subset their exponential space (i.e. the set of their closed subsets) with the exponential uniformity. This approach allows to solve many recursive domain equations of processes which cannot be even expressed in standard ZermeloFraenkel Set Theory, e.g. when the functors involved have negative occurrences of the argument. Such equations arise in the semantics of concurrrent programs in connection with function spaces and higher order assignment. Finally, we briefly compare our results to those which make use of complete metric spaces, due to de Bakker, America and Rutten. Introduction In the Semantics of ...
Three Metric Domains of Processes for Bisimulation
 in Proceedings of the 9th International Conference on Mathematical Foundations of Programming Semantics, LNCS
, 1993
"... A new metric domain of processes is presented. This domain is located in between two metric process domains introduced by De Bakker and Zucker. The new process domain characterizes the collection of image finite processes. This domain has as advantages over the other process domains that no complica ..."
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Cited by 4 (2 self)
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A new metric domain of processes is presented. This domain is located in between two metric process domains introduced by De Bakker and Zucker. The new process domain characterizes the collection of image finite processes. This domain has as advantages over the other process domains that no complications arise in the definitions of operators like sequential composition and parallel composition, and that image finite language constructions like random assignment can be modelled in an elementary way. As in the other domains, bisimilarity and equality coincide in this domain. The three domains are obtained as unique (up to isometry) solutions of equations in a category of 1bounded complete metric spaces. In the case the action set is finite, the three domains are shown to be equal (up to isometry). For infinite action sets, e.g., equipollent to the set of natural or real numbers, the process domains are proved not to be isometric. AMS Subject Classification (1991): 68Q55 CR Subject Classification (1991): D.3.1, F.3.2 Keywords & Phrases: process, complete metric space, bisimulation, finitely branching, image finite, sequential composition Note: This work was partially supported by the Netherlands Nationale Faciliteit Informatica programme, project Research and Education in Concurrent Systems (REX). This paper will appear in Proceedings of the Ninth Conference on the Mathematical Foundations of Programming Semantics, New Orleans, LA, USA, April 710, 1993.
A Nonwellfounded Sets Semantics for Observation Congruence over Full CCS
, 1994
"... . In the present paper we study the semantics of a minor variant of Milner's CCS process calculus. We use a compact semantic domain for processes which has been described by Aczel in [P. Aczel. Nonwellfounded Sets. Stanford University, 1988.] On the basis of the operational semantics of CCS we ..."
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Cited by 2 (1 self)
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. In the present paper we study the semantics of a minor variant of Milner's CCS process calculus. We use a compact semantic domain for processes which has been described by Aczel in [P. Aczel. Nonwellfounded Sets. Stanford University, 1988.] On the basis of the operational semantics of CCS we construct a semantic mapping from CCS into the domain. We prove that this mapping is fully abstract wrt. observation congruence, initial in a category of algebras, and final in a category of coalgebras. Furthermore, we give a logical characterization of the objects reached by it. 1. Introduction Observation congruence ([Mil80], [Mil89]) has become one of the most successful concepts in process algebra. It is the coarsest congruence contained in weak bisimulation equivalence, an equivalence which abstracts from invisible behaviour and is thus very intuitive. What is more, observation congruence usually differs from weak bisimulation equivalence only in the treatment of initial invisible tran...