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24
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...
NonDeterministic Kleene Coalgebras
"... In this paper, we present a systematic way of deriving (1) languages of (generalised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of systems. This generalizes both the results of Kleene (on regular languages and deterministic finite automata) and Miln ..."
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Cited by 15 (4 self)
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In this paper, we present a systematic way of deriving (1) languages of (generalised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of systems. This generalizes both the results of Kleene (on regular languages and deterministic finite automata) and Milner (on regular behaviours and finite labelled transition systems), and includes many other systems such as Mealy and Moore machines.
The logic of recursive equations
 the Journal of Symbolic Logic
, 1998
"... JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JS ..."
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Cited by 12 (5 self)
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JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.
Reaction and Control I. Mixing Additive and Multiplicative Network Algebras
 Logic Journal of the IGPL
, 1996
"... . This paper is included in a series aiming to contribute to the algebraic theory of distributed computation. The key problem in understanding MultiAgent Systems is to find a theory which integrates the reactive part and the control part of such systems. To this end we use the calculus of flownomi ..."
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Cited by 9 (2 self)
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. This paper is included in a series aiming to contribute to the algebraic theory of distributed computation. The key problem in understanding MultiAgent Systems is to find a theory which integrates the reactive part and the control part of such systems. To this end we use the calculus of flownomials. It is a polynomiallike calculus for representing flowgraphs and their behaviours. An `additive' interpretation of the calculus was intensively developed to study control flowcharts and finite automata. For instance, regular algebra and iteration theories are included in a unified presentation. On the other hand, a `multiplicative' interpretation of the calculus of flownomials was developed to study dataflow networks. The claim of this series of papers is that the mixture of the additive and multiplicative network algebras will contribute to the understanding of distributed computation. The role of this first paper is to present a few motivating examples. To appear in Journal of IGPL....
The Algebra of Stream Processing Functions
 THEORETICAL COMPUTER SCIENCE
, 1996
"... Dataflow networks are a model of concurrent computation. They consist of a collection of concurrent asynchronous processes which communicate by sending data over FIFO channels. In this paper we study the algebraic structure of the dataflow networks and base their semantics on stream processing funct ..."
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Cited by 8 (1 self)
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Dataflow networks are a model of concurrent computation. They consist of a collection of concurrent asynchronous processes which communicate by sending data over FIFO channels. In this paper we study the algebraic structure of the dataflow networks and base their semantics on stream processing functions. The algebraic theory is provided by the calculus of flownomials which gives a unified presentation of regular algebra and iteration theories. The kernel of the calculus is an equational axiomatization called Basic Network Algebra (BNA) for flowgraphs modulo graph isomorphism. We show that the algebra of stream processing functions called SPF (used for deterministic networks) and the algebra of sets of stream processing functions called PSPF (used for nondeterministic networks) are BNA algebras. As a byproduct this shows that both semantic models are compositional. We also identify the additional axioms satisfied by the branching components that correspond to constants in these two a...
The category theoretic solution of recursive program schemes
 Proc. First Internat. Conf. on Algebra and Coalgebra in Computer Science (CALCO 2005), Lecture Notes in Computer Science
, 2006
"... Abstract. This paper provides a general account of the notion of recursive program schemes, studying both uninterpreted and interpreted solutions. It can be regarded as the categorytheoretic version of the classical area of algebraic semantics. The overall assumptions needed are small indeed: worki ..."
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Cited by 7 (2 self)
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Abstract. This paper provides a general account of the notion of recursive program schemes, studying both uninterpreted and interpreted solutions. It can be regarded as the categorytheoretic version of the classical area of algebraic semantics. The overall assumptions needed are small indeed: working only in categories with “enough final coalgebras ” we show how to formulate, solve, and study recursive program schemes. Our general theory is algebraic and so avoids using ordered, or metric structures. Our work generalizes the previous approaches which do use this extra structure by isolating the key concepts needed to study substitution in infinite trees, including secondorder substitution. As special cases of our interpreted solutions we obtain the usual denotational semantics using complete partial orders, and the one using complete metric spaces. Our theory also encompasses implicitly defined objects which are not usually taken to be related to recursive program schemes. For example, the classical Cantor twothirds set falls out as an interpreted
Algebra of Networks  Modeling simple networks, as well as complex interactive systems
 In: Proof and SystemReliability, Proc. Marktoberdorf Summer School 2001, Kluwer (2002), 4978. A., Sofronia A., Stefanescu G.: Highlevel Structured Interactive
"... The rst part of the paper contains an overview of Network Algebra (NA) book [35]. The second part introduces nite interactive systems as an abstract mathematical model of agents' behaviour and their interaction. ..."
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Cited by 2 (2 self)
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The rst part of the paper contains an overview of Network Algebra (NA) book [35]. The second part introduces nite interactive systems as an abstract mathematical model of agents' behaviour and their interaction.
Language Constructs for NonWellFounded Computation
"... Recursive functions defined on a coalgebraic datatype C may not converge if there are cycles in the input, that is, if the input object is not wellfounded. Even so, there is often a useful solution; for example, the free variables of an infinitary λterm, or the expected running time of a finitest ..."
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Cited by 2 (2 self)
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Recursive functions defined on a coalgebraic datatype C may not converge if there are cycles in the input, that is, if the input object is not wellfounded. Even so, there is often a useful solution; for example, the free variables of an infinitary λterm, or the expected running time of a finitestate probabilistic protocol. Theoretical models of recursion schemes have been well studied under the names wellfounded coalgebras, recursive coalgebras [2], corecursive algebras [4], and Elgot algebras [1]. Much of this work focuses on conditions ensuring unique or canonical solutions, e.g. when C is wellfounded. If C is not wellfounded, then there can be multiple solutions. The standard semantics of recursive programs gives a particular solution, namely the least solution in a flat Scott domain, which may not be the one we want. Unfortunately, current programming
Processes with Multiple Entries and Exits Modulo Isomorphism and Modulo Bisimulation
, 1994
"... . This paper proposes a framework for the integration of the algebra of communicating processes (ACP) and the algebra of flownomials (AF). Basically, this means to combine axiomatisations of parallel and looping operators. To this end a model of process graphs with multiple entries and exits is intr ..."
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Cited by 1 (0 self)
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. This paper proposes a framework for the integration of the algebra of communicating processes (ACP) and the algebra of flownomials (AF). Basically, this means to combine axiomatisations of parallel and looping operators. To this end a model of process graphs with multiple entries and exits is introduced. In this model the usual operations of both algebras are defined, e.g. alternative composition, sequential composition, feedback, parallel composition, left merge, communication merge, encapsulation, etc. The main results consist of correct and complete axiomatisations for process graphs modulo isomorphism and modulo bisimulation. 1
Is Observational Congruence on µExpressions Axiomatisable in Equational Horn Logic?
, 2007
"... It is well known that bisimulation on µexpressions cannot be finitely axiomatised in equational logic. Complete axiomatisations such as those of Milner and Bloom/Ésik necessarily involve implicational rules. However, both systems rely on features which go beyond pure equational Horn logic: either t ..."
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It is well known that bisimulation on µexpressions cannot be finitely axiomatised in equational logic. Complete axiomatisations such as those of Milner and Bloom/Ésik necessarily involve implicational rules. However, both systems rely on features which go beyond pure equational Horn logic: either the rules are impure by involving nonequational sideconditions, or they are schematically infinitary like the congruence rule which is not Horn. It is an open question whether these complications cannot be avoided in the prooftheoretically and computationally clean and powerful setting of secondorder equational Horn logic. This paper presents a positive and a negative result regarding axiomatisability of observational congruence in equational Horn logic. Firstly, we show how Milner’s impure rule system can be reworked into a pure Horn axiomatisation that is complete for guarded processes. Secondly, we prove that for unguarded processes, both Milner’s and Bloom/Ésik’s axiomatisations are incomplete without the congruence rule, and neither system has a complete extension in rank 1 equational axioms. It remains open whether there are higherrank equational axioms or Horn rules which would render Milner’s or Bloom / Ésik’s axiomatisations complete for unguarded processes.