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172
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
Abstract behavior types: A foundation model for components and their composition
 SCIENCE OF COMPUTER PROGRAMMING
, 2003
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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...
Automata and coinduction (an exercise in coalgebra
 LNCS
, 1998
"... The classical theory of deterministic automata is presented in terms of the notions of homomorphism and bisimulation, which are the cornerstones of the theory of (universal) coalgebra. This leads to a transparent and uniform presentation of automata theory and yields some new insights, amongst which ..."
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Cited by 62 (16 self)
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The classical theory of deterministic automata is presented in terms of the notions of homomorphism and bisimulation, which are the cornerstones of the theory of (universal) coalgebra. This leads to a transparent and uniform presentation of automata theory and yields some new insights, amongst which coinduction proof methods for language equality and language inclusion. At the same time, the present treatment of automata theory may serve as an introduction to coalgebra.
A Coinductive Calculus of Component Connectors
, 2002
"... Reo is a recently introduced channelbased coordination model, wherein complex coordinators, called connectors, are compositionally built out of simpler ones. Using a more liberal notion of a channel, Reo generalises existing dataflow networks. In this paper, we present a simple and transparent sema ..."
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Cited by 58 (25 self)
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Reo is a recently introduced channelbased coordination model, wherein complex coordinators, called connectors, are compositionally built out of simpler ones. Using a more liberal notion of a channel, Reo generalises existing dataflow networks. In this paper, we present a simple and transparent semantical model for Reo, in which connectors are relations on timed data streams. Timed data streams constitute a characteristic of our model and consist of twin pairs of separate data and time streams. Furthermore, coinduction is our main reasoning principle and we use it to prove properties such as connector equivalence.
Observational logic
 In Algebraic Methodology and Software Technology (AMAST'98
, 1999
"... Abstract. We present an institution of observational logic suited for statebased systems specifications. The institution is based on the notion of an observational signature (which incorporates the declaration of a distinguished set of observers) and on observational algebras whose operations are r ..."
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Cited by 53 (10 self)
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Abstract. We present an institution of observational logic suited for statebased systems specifications. The institution is based on the notion of an observational signature (which incorporates the declaration of a distinguished set of observers) and on observational algebras whose operations are required to be compatible with the indistinguishability relation determined by the given observers. In particular, we introduce a homomorphism concept for observational algebras which adequately expresses observational relationships between algebras. Then we consider a flexible notion of observational signature morphism which guarantees the satisfaction condition of institutions w.r.t. observational satisfaction of arbitrary firstorder sentences. From the proof theoretical point of view we construct a sound and complete proof system for the observational consequence relation. Then we consider structured observational specifications and we provide a sound and complete proof system for such specifications by using a general, institutionindependent result of [6]. 1
ManySorted Coalgebraic Modal Logic: a Modeltheoretic Study
 Theoretical Informatics and Applications
, 2001
"... This paper gives a semantical underpinning for a manysorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in objectoriented languages. These systems will be described as coalgebras of socalled polynomial functors, built up from constants an ..."
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Cited by 53 (3 self)
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This paper gives a semantical underpinning for a manysorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in objectoriented languages. These systems will be described as coalgebras of socalled polynomial functors, built up from constants and identities, using products, coproducts and powersets. The semantical account involves Boolean algebras with operators indexed by polynomial functors, called MBAOs, for Manysorted Boolean Algebras with Operators, combining standard (categorical) models of modal logic and of manysorted predicate logic.
Circular Coinductive Rewriting
 In Proceedings of Automated Software Engineering 2000
, 2000
"... Circular coinductive rewriting is a new method for proving behavioral properties, that combines behavioral rewriting with circular coinduction. This method is implemented in our new BOBJ behavioral specification and computation system, which is used in examples throughout this paper. These examples ..."
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Cited by 46 (11 self)
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Circular coinductive rewriting is a new method for proving behavioral properties, that combines behavioral rewriting with circular coinduction. This method is implemented in our new BOBJ behavioral specification and computation system, which is used in examples throughout this paper. These examples demonstrate the surprising power of circular coinductive rewriting. The paper also sketches the underlying hidden algebraic theory and briefly describes BOBJ and some of its algorithms.
HigherOrder Pushdown Trees Are Easy
, 2002
"... We show that the monadic secondorder theory of an infinite tree recognized by a higherorder pushdown automaton of any level is decidable. We also show that trees recognized by pushdown automata of level n coincide with trees generated by safe higherorder grammars of level n. Our decidability resu ..."
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Cited by 43 (2 self)
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We show that the monadic secondorder theory of an infinite tree recognized by a higherorder pushdown automaton of any level is decidable. We also show that trees recognized by pushdown automata of level n coincide with trees generated by safe higherorder grammars of level n. Our decidability result extends the result of Courcelle on algebraic (pushdown of level 1) trees and our own result on trees of level 2.
Hiding More of Hidden Algebra
 FM'99  Formal Methods
, 1999
"... This paper generalizes the hidden algebra approach to allow: (P1) operations with multiple hidden arguments, and (P2) defining behavioral equivalence with a subset of operations, in addition to the already present (P3) builtin data types, (P4) nondeterminism, (P5) concurrency, and (P6) noncongruen ..."
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Cited by 42 (15 self)
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This paper generalizes the hidden algebra approach to allow: (P1) operations with multiple hidden arguments, and (P2) defining behavioral equivalence with a subset of operations, in addition to the already present (P3) builtin data types, (P4) nondeterminism, (P5) concurrency, and (P6) noncongruent operations. All important results generalize, but more elegant formulations use the new institution in Section 5. Behavioral satisfaction appeared 1981 in [20], hidden algebra 1989 in [9], multiple hidden arguments 1992 in [1], congruent and behavioral operations in [1, 18], behavioral equivalence defined by a subset of operations in [1], and noncongruent operations in [5]; all this was previously integrated in [21], but this paper gives new examples, institutions, and results relating hidden algebra to information hiding. We assume familiarity with basics of algebraic specification, e.g., [11, 13].