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Interaction Categories and the Foundations of Typed Concurrent Programming
 In Deductive Program Design: Proceedings of the 1994 Marktoberdorf Summer School, NATO ASI Series F
, 1995
"... We propose Interaction Categories as a new paradigm for the semantics of functional and concurrent computation. Interaction categories have specifications as objects, processes as morphisms, and interaction as composition. We introduce two key examples of interaction categories for concurrent compu ..."
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We propose Interaction Categories as a new paradigm for the semantics of functional and concurrent computation. Interaction categories have specifications as objects, processes as morphisms, and interaction as composition. We introduce two key examples of interaction categories for concurrent computation and indicate how a general axiomatisation can be developed. The upshot of our approach is that traditional process calculus is reconstituted in functorial form, and integrated with type theory and functional programming.
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 82 (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...
A typed calculus of synchronous processes
 In Proceedings of IEEE Symposium on Logic in Computer Science
, 1995
"... ..."
A Formalization of Viewpoints
 FUNDAMENTA INFORMATICAE
, 1995
"... We present a formalisation for the notion of viewpoint , a construct meant for expressing several varieties of relativised truth. The formalisation consists in a logic which extends first order predicate calculus through an axiomatization of provability and with the addition of proper reflection rul ..."
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Cited by 36 (3 self)
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We present a formalisation for the notion of viewpoint , a construct meant for expressing several varieties of relativised truth. The formalisation consists in a logic which extends first order predicate calculus through an axiomatization of provability and with the addition of proper reflection rules. The extension is not conservative, but consistency is granted. Viewpoints are defined as set of reified metalevel sentences. A proof theory for viewponts is developed which enables to carry out proofs of sentences involving several viewpoints. A semantic account of viewpoints is provided, dealing with issues of self referential theories and paradoxes, and exploiting the notion of contextual entailment . Notions such as beliefs, knowledge, truth and situations can be uniformly modeled as provability in specialised viewpoints, obtained by imposing suitable constraints on viewpoints.
Intuitionistic Sets and Ordinals
 Journal of symbolic Logic
, 1996
"... Transitive extensional well founded relations provide an intuitionistic notion of ordinals which admits transfinite induction. However these ordinals are not directed and their successor operation is poorly behaved, leading to problems of functoriality. We show how to make the successor monotone by ..."
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Cited by 9 (2 self)
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Transitive extensional well founded relations provide an intuitionistic notion of ordinals which admits transfinite induction. However these ordinals are not directed and their successor operation is poorly behaved, leading to problems of functoriality. We show how to make the successor monotone by introducing plumpness, which strengthens transitivity. This clarifes the traditional development of successors and unions, making it intuitionistic; even the (classical) proof of trichotomy is made simpler. The definition is, however, recursive, and, as their name suggests, the plump ordinals grow very rapidly. Directedness must be defined hereditarily. It is orthogonal to the other four conditions, and the lower powerdomain construction is shown to be the universal way of imposing it. We treat ordinals as ordertypes, and develop a corresponding set theory similar to Osius’ transitive set objects. This presents Mostowski’s theorem as a reflection of categories, and settheoretic union is a corollary of the adjoint functor theorem. Mostowski’s theorem and the rank for some of the notions of ordinal are formulated and proved without the axiom of replacement, but this seems to be unavoidable for the plump rank. The comparison between sets and toposes is developed as far as the identification of replacement with completeness and there are some suggestions for further work in this area. Each notion of set or ordinal defines a free algebra for one of the theories discussed by Joyal and Moerdijk, namely joins of a family of arities together with an operation s satisfying conditions such as x ≤ sx, monotonicity or s(x ∨ y) ≤ sx ∨ sy. Finally we discuss the fixed point theorem for a monotone endofunction s of a poset with least element and directed joins. This may be proved under each of a variety of additional hypotheses. We explain why it is unlikely that any notion of ordinal obeying the induction scheme for arbitrary predicates will prove the pure result.
Hypergraphdb: A generalized graph database
 In WAIM Workshops
, 2010
"... Abstract. We present HyperGraphDB, a novel graph database based on generalized hypergraphs where hyperedges can contain other hyperedges. This generalization automatically reifies every entity expressed in the database thus removing many of the usual difficulties in dealing with higherorder relati ..."
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Cited by 5 (0 self)
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Abstract. We present HyperGraphDB, a novel graph database based on generalized hypergraphs where hyperedges can contain other hyperedges. This generalization automatically reifies every entity expressed in the database thus removing many of the usual difficulties in dealing with higherorder relationships. An open twolayered architecture of the data organization yields a highly customizable system where specific domain representations can be optimized while remaining within a uniform conceptual framework. HyperGraphDB is an embedded, transactional database designed as a universal data model for highly complex, large scale knowledge representation applications such as found in artificial intelligence, bioinformatics and natural language processing.
Modelling SIGNAL in Interaction Categories
 THEORY AND FORMAL METHODS 1993: PROCEEDINGS OF THE FIRST IMPERIAL COLLEGE DEPARTMENT OF COMPUTING WORKSHOP ON THEORY AND FORMAL METHODS. SPRINGERVERLAG WORKSHOPS IN COMPUTER SCIENCE
, 1993
"... Abramsky has recently proposed Interaction Categories as a new paradigm for the semantics of sequential and parallel computation. Working with the category SProc of synchronous processes, which is a key example of an Interaction Category, we study synchronous dataflow as part of a programme of gain ..."
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Cited by 3 (1 self)
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Abramsky has recently proposed Interaction Categories as a new paradigm for the semantics of sequential and parallel computation. Working with the category SProc of synchronous processes, which is a key example of an Interaction Category, we study synchronous dataflow as part of a programme of gaining experience in the use of Interaction Categories. After making some general points about representing dataflow in SProc, we present a detailed model of the synchronous dataflow language Signal. We demonstrate that dataflow is a model of concurrency which can easily be treated in a typed framework, and that the structure of Interaction Categories is appropriate for describing real concurrent languages.
Towards a unified treatment of induction, I: the general recursion theorem, unfinished draft manuscript
, 1996
"... The recursive construction of a function f: A → Θ consists, paradigmatically, of finding a functor T and maps α: A → TA and θ: TΘ → Θ such that f = α; Tf; θ. The role of the functor T is to marshall the recursive subarguments, and apply the function f to them in parallel. This equation is called pa ..."
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Cited by 2 (0 self)
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The recursive construction of a function f: A → Θ consists, paradigmatically, of finding a functor T and maps α: A → TA and θ: TΘ → Θ such that f = α; Tf; θ. The role of the functor T is to marshall the recursive subarguments, and apply the function f to them in parallel. This equation is called partial correctness of the recursive program, because we have also to show that it terminates, i.e. that the recursion (coded by α) is well founded. This may be done by finding another map g: A → N, called a loop variant, where N is some standard well founded srtucture such as the natural numbers or ordinals. In set theory the functor T is the covariant powerset; in the study of the free algebra for a free theory Ω (such as in proof theory) it is the polynomial Σr∈Ω(−)ar(r), and it is often something very crude. We identify the properties of the category of sets needed to prove the general recursion theorem, that these data suffice to define f uniquely. For any pullbackpreserving functor T, a structure similar to the von Neumann hierarchy is developed which analyses the free Talgebra if it exists, or deputises for it otherwise. There is considerable latitude in the choice of ambient category, the functor T and the class of predicates admissible in the induction scheme. Free algebras, set theory, the familiar ordinals and novel forms of them which have arisen in theoretical computer science are treated in a uniform fashion. The central idea in the paper is a categorical definition of well founded coalgebra α: A. TA, namely that any pullback diagram of the form
Why sets?
 PILLARS OF COMPUTER SCIENCE: ESSAYS DEDICATED TO BORIS (BOAZ) TRAKHTENBROT ON THE OCCASION OF HIS 85TH BIRTHDAY, VOLUME 4800 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2008
"... Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besi ..."
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Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.
Perspectives and the Referential Use of Definite Descriptions in Dialogue
"... This is an investigation into the pragmatics of the referential use of de nite descriptions. We examine situations where a speaker uses a description to induce a hearer to pick out a certain object from a set of mutually given objects, in order to state some proposition about this object. ..."
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Cited by 2 (1 self)
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This is an investigation into the pragmatics of the referential use of de nite descriptions. We examine situations where a speaker uses a description to induce a hearer to pick out a certain object from a set of mutually given objects, in order to state some proposition about this object.