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Invariants, Bisimulations and the Correctness of Coalgebraic Refinements
 Techn. Rep. CSIR9704, Comput. Sci. Inst., Univ. of Nijmegen
, 1997
"... . Coalgebraic specifications are used to formally describe the behaviour of classes in objectoriented languages. In this paper, a general notion of refinement between two such coalgebraic specifications is defined, capturing the idea that one "concrete" class specification realises the behaviour of ..."
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Cited by 12 (4 self)
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. Coalgebraic specifications are used to formally describe the behaviour of classes in objectoriented languages. In this paper, a general notion of refinement between two such coalgebraic specifications is defined, capturing the idea that one "concrete" class specification realises the behaviour of the other, "abstract" class specification. Two (complete) prooftechniques are given to establish such refinements: one involving an invariant (a predicate that is closed under transitions) on the concrete class, and one involving a bisimulation (a relation that is closed under transitions) between the concrete and the abstract class. The latter can only be used if the abstract class is what we call totally specified. Parts of the underlying theory of invariants and bisimulations in a coalgebraic setting are included, involving least and greatest invariants and connections between invariants and bisimulations. Also, the proofprinciples are illustrated in examples (which are fully formalise...
Generic trace theory
 International Workshop on Coalgebraic Methods in Computer Science (CMCS 2006), volume 164 of Elect. Notes in Theor. Comp. Sci
, 2006
"... Trace semantics has been defined for various nondeterministic systems with different input/output types, or with different types of “nondeterminism ” such as classical nondeterminism (with a set of possible choices) vs. probabilistic nondeterminism. In this paper we claim that these various forms ..."
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Cited by 8 (4 self)
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Trace semantics has been defined for various nondeterministic systems with different input/output types, or with different types of “nondeterminism ” such as classical nondeterminism (with a set of possible choices) vs. probabilistic nondeterminism. In this paper we claim that these various forms of “trace semantics” are instances of a single categorical construction, namely coinduction in a Kleisli category. This claim is based on our main technical result that an initial algebra in
NonDeterministic Extensions of Untyped λcalculus
 INFO. AND COMP
, 1995
"... The main concern of this paper is the study of the interplay between functionality and non determinism. Indeed the first question we ask is whether the analysis of parallelism in terms of sequentiality and non determinism, which is usual in the algebraic treatment of concurrency, remains correct in ..."
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Cited by 6 (0 self)
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The main concern of this paper is the study of the interplay between functionality and non determinism. Indeed the first question we ask is whether the analysis of parallelism in terms of sequentiality and non determinism, which is usual in the algebraic treatment of concurrency, remains correct in presence of functional application and abstraction. We identify non determinism in the setting of λcalculus with the absence of the ChurchRosser property plus the inconsistency of the equational theory obtained by the symmetric closure of the reduction relation. We argue in favour of a distinction between non determinism and parallelism, due to the conjunctive nature of the former in contrast to the disjunctive character of the latter. This is the basis of our analysis of the operational and denotational semantics of non deterministiccalculus, which is the classical calculus plus a choice operator, and of our election of bounded indeterminacy as the semantical counterpart of conjunctive non determinism. This leads to operational semantics based on...
CSP, Partial Automata, and Coalgebras
, 1999
"... Based on the theory of coalgebras the paper builds a bridge between CSP and automata theory. We show that the concepts of processes in [4] coincide with the concepts of states for special, namely, final partial automata. Moreover, we show how the deterministic and nondeterministic operations in [4] ..."
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Cited by 2 (0 self)
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Based on the theory of coalgebras the paper builds a bridge between CSP and automata theory. We show that the concepts of processes in [4] coincide with the concepts of states for special, namely, final partial automata. Moreover, we show how the deterministic and nondeterministic operations in [4] can be interpreted in a compatible way by constructions on the semantical level of automata. Especially, we are able to interpret each finite process expression as representing a finite partial automaton with a designated initial state. In such a way we provide a new method for solving recursive process equations which is based on the concept of final automata. That is, there is no need to impose a cpo structure on the set of processes to describe the solutions of recursive process equations. 1 Introduction For people usually working on model theory or semantics of formal specifications it becomes often very hard to approach the area of process calculi and process algebras. There are proces...
On Coalgebras and Final Semantics: Progress Report
"... This report summarises research undertaken by the author as part of his DPhil in computation. A draft table of contents of the thesis, a proposed timetable for submission, and a list of publications are also included. ..."
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This report summarises research undertaken by the author as part of his DPhil in computation. A draft table of contents of the thesis, a proposed timetable for submission, and a list of publications are also included.
Testing for Simulation and Bisimulation in Labelled Markov Processes
, 2003
"... This paper presents a fundamental study of similarity and bisimilarity for labelled Markov processes: a particular class of probabilistic labelled transition systems. The main results characterize similarity as a testing preorder and bisimilarity as a testing equivalence. ..."
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This paper presents a fundamental study of similarity and bisimilarity for labelled Markov processes: a particular class of probabilistic labelled transition systems. The main results characterize similarity as a testing preorder and bisimilarity as a testing equivalence.
Coalgebraic Representation Theory of Fractals (Extended Abstract)
"... We develop a representation theory in which a point of a fractal specified by metric means (by a variant of an iterated function system, IFS) is represented by a suitable equivalence class of infinite streams of symbols. The framework is categorical: symbolic representatives carry a final coalgebra; ..."
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We develop a representation theory in which a point of a fractal specified by metric means (by a variant of an iterated function system, IFS) is represented by a suitable equivalence class of infinite streams of symbols. The framework is categorical: symbolic representatives carry a final coalgebra; an IFSlike metric specification of a fractal is an algebra for the same functor. Relating the two there canonically arises a representation map, much like in America and Rutten’s use of metric enrichment in denotational semantics. A distinctive feature of our framework is that the canonical representation map is bijective. In the technical development, gluing of shapes in a fractal specification is a major challenge. On the metric side we introduce the notion of injective IFS to be used in place of conventional IFSs. On the symbolic side we employ Leinster’s presheaf framework that uniformly addresses necessary identification of streams—such as.0111... =.1000... in the binary expansion of real numbers. Our leading example is the unit interval I = [0, 1].
NonDeterministic untyped λcalculus  A study about explicit non determinism in higherorder functional calculi
, 1991
"... ..."
An introduction to (co)algebra and (co)induction
"... Algebra is a wellestablished part of mathematics, dealing with sets with operations satisfying certain properties, like groups, rings, vector spaces, etcetera. Its results are essential throughout mathematics and other sciences. Universal algebra is a part of algebra in which algebraic structures a ..."
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Algebra is a wellestablished part of mathematics, dealing with sets with operations satisfying certain properties, like groups, rings, vector spaces, etcetera. Its results are essential throughout mathematics and other sciences. Universal algebra is a part of algebra in which algebraic structures are studied at a high