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108
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
A Tutorial on (Co)Algebras and (Co)Induction
 EATCS Bulletin
, 1997
"... . Algebraic structures which are generated by a collection of constructors like natural numbers (generated by a zero and a successor) or finite lists and trees are of wellestablished importance in computer science. Formally, they are initial algebras. Induction is used both as a definition pr ..."
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Cited by 228 (34 self)
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. Algebraic structures which are generated by a collection of constructors like natural numbers (generated by a zero and a successor) or finite lists and trees are of wellestablished importance in computer science. Formally, they are initial algebras. Induction is used both as a definition principle, and as a proof principle for such structures. But there are also important dual "coalgebraic" structures, which do not come equipped with constructor operations but with what are sometimes called "destructor" operations (also called observers, accessors, transition maps, or mutators). Spaces of infinite data (including, for example, infinite lists, and nonwellfounded sets) are generally of this kind. In general, dynamical systems with a hidden, blackbox state space, to which a user only has limited access via specified (observer or mutator) operations, are coalgebras of various kinds. Such coalgebraic systems are common in computer science. And "coinduction" is the appropriate te...
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.
A Hidden Agenda
 Theoretical Computer Science
, 2000
"... This paper publicly reveals, motivates, and surveys the results of an ambitious hidden agenda for applying algebra to software engineering. The paper reviews selected literature, introduces a new perspective on nondeterminism, and features powerful hidden coinduction techniques for proving behaviora ..."
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Cited by 110 (23 self)
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This paper publicly reveals, motivates, and surveys the results of an ambitious hidden agenda for applying algebra to software engineering. The paper reviews selected literature, introduces a new perspective on nondeterminism, and features powerful hidden coinduction techniques for proving behavioral properties of concurrent systems, especially renements; some proofs are given using OBJ3. We also discuss where modularization, bisimulation, transition systems and combinations of the object, logic, constraint and functional paradigms t into our hidden agenda. 1 Introduction Algebra can be useful in many dierent ways in software engineering, including specication, validation, language design, and underlying theory. Specication and validation can help in the practical production of reliable programs, advances in language design can help improve the state of the art, and theory can help with building new tools to increase automation, as well as with showing correctness of the whole e...
Coalgebraic Logic
 Annals of Pure and Applied Logic
, 1999
"... We present a generalization of modal logic to logical systems which are interpreted on coalgebras of functors on sets. The leading idea is that infinitary modal logic contains characterizing formulas. That is, every modelworld pair is characterized up to bisimulation by an infinitary formula. The ..."
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Cited by 89 (0 self)
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We present a generalization of modal logic to logical systems which are interpreted on coalgebras of functors on sets. The leading idea is that infinitary modal logic contains characterizing formulas. That is, every modelworld pair is characterized up to bisimulation by an infinitary formula. The point of our generalization is to understand this on a deeper level. We do this by studying a frangment of infinitary modal logic which contains the characterizing formulas and is closed under infinitary conjunction and an operation called 4. This fragment generalizes to a wide range of coalgebraic logics. We then apply the characterization result to get representation theorems for final coalgebras in terms of maximal elements of ordered algebras. The end result is that the formulas of coalgebraic logics can be viewed as approximations to the elements of the final coalgebra. Keywords: infinitary modal logic, characterization theorem, functor on sets, coalgebra, greatest fixed point. 1 Intr...
Bisimulation for Probabilistic Transition Systems: A Coalgebraic Approach
, 1998
"... . The notion of bisimulation as proposed by Larsen and Skou for discrete probabilistic transition systems is shown to coincide with a coalgebraic definition in the sense of Aczel and Mendler in terms of a set functor. This coalgebraic formulation makes it possible to generalize the concepts to a ..."
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Cited by 75 (15 self)
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. The notion of bisimulation as proposed by Larsen and Skou for discrete probabilistic transition systems is shown to coincide with a coalgebraic definition in the sense of Aczel and Mendler in terms of a set functor. This coalgebraic formulation makes it possible to generalize the concepts to a continuous setting involving Borel probability measures. Under reasonable conditions, generalized probabilistic bisimilarity can be characterized categorically. Application of the final coalgebra paradigm then yields an internally fully abstract semantical domain with respect to probabilistic bisimulation. Keywords. Bisimulation, probabilistic transition system, coalgebra, ultrametric space, Borel measure, final coalgebra. 1 Introduction For discrete probabilistic transition systems the notion of probabilistic bisimilarity of Larsen and Skou [LS91] is regarded as the basic process equivalence. The definition was given for reactive systems. However, Van Glabbeek, Smolka and Steffen s...
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.
Coalgebraic modal logic: Soundness, completeness and decidability of local consequence
 Theoret. Comput. Sci
, 2002
"... This paper studies finitary modal logics, interpreted over coalgebras for an endofunctor, and establishes soundness, completeness and decidability results. The logics are studied within the abstract framework of coalgebraic modal logic, which can be instantiated with arbitrary endofunctors on the ca ..."
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Cited by 51 (24 self)
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This paper studies finitary modal logics, interpreted over coalgebras for an endofunctor, and establishes soundness, completeness and decidability results. The logics are studied within the abstract framework of coalgebraic modal logic, which can be instantiated with arbitrary endofunctors on the category of sets. This is achieved through the use of predicate liftings, which generalise atomic propositions and modal operators from Kripke models to arbitrary coalgebras. Predicate liftings also allow us to use induction along the terminal sequence of the underlying endofunctor as a proof principle. This induction principle is systematically exploited to establish soundness, completeness and decidability of the logics. We believe that this induction principle also opens new ways for reasoning about modal logics: Our proof of completeness does not rely on a canonical model construction, and the proof of the finite model property does not use filtrations. 1
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.
Geometry of Interaction and Linear Combinatory Algebras
, 2000
"... this paper was quite di#erent, stemming from the axiomatics of categories of tangles (although the authors were aware of possible connections to iteration theories. In fact, similar axiomatics in the symmetric case, motivated by flowcharts and "flownomials" had been developed some years earlier by S ..."
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Cited by 44 (10 self)
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this paper was quite di#erent, stemming from the axiomatics of categories of tangles (although the authors were aware of possible connections to iteration theories. In fact, similar axiomatics in the symmetric case, motivated by flowcharts and "flownomials" had been developed some years earlier by Stefanescu (Stefanescu 2000).) However, the first author realized, following a stimulating discussion with Gordon Plotkin, that traced monoidal categories provided a common denominator for the axiomatics of both the Girardstyle and AbramskyJagadeesanstyle versions of the Geometry of Interaction, at the basic level of the multiplicatives. This insight was presented in (Abramsky 1996), in which Girardstyle GoI was dubbed "particlestyle", since it concerns information particles or tokens flowing around a network, while the AbramskyJagadeesan style GoI was dubbed "wavestyle", since it concerns the evolution of a global information state or "wave". Formally, this distinction is based on whether the tensor product (i.e. the symmetric monoidal structure) in the underlying category is interpreted as a coproduct (particle style) or as a product (wave style). This computational distinction between coproduct and product interpretations of the same underlying network geometry turned out to have been partially anticipated, in a rather di#erent context, in a pioneering paper by E. S. Bainbridge (Bainbridge 1976), as observed by Dusko Pavlovic. These two forms of interpretation, and ways of combining them, have also been studied recently in (Stefanescu 2000). He uses the terminology "additive" for coproductbased (i.e. our "particlestyle") and "multiplicative" for productbased (i.e. our "wavestyle"); this is not suitable for our purposes, because of the clash with Linear Logic term...