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On the Semantics of Place/Transition Petri Nets
, 1992
"... Abstract. In the last few years, the semantics of Petri nets has been investigated in several different ways. Apart from the classical “token game”, one can model the behaviour of Petri nets via nonsequential processes, via unfolding constructions, which provide formal relationships between nets an ..."
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Cited by 22 (10 self)
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Abstract. In the last few years, the semantics of Petri nets has been investigated in several different ways. Apart from the classical “token game”, one can model the behaviour of Petri nets via nonsequential processes, via unfolding constructions, which provide formal relationships between nets and domains, and via algebraic models, which view Petri nets as essentially algebraic theories whose models are monoidal categories. In this paper we show that these three points of view can be reconciled. More precisely, we introduce the new notion of decorated processes of Petri nets and we show that they induce on nets the same semantics as that of unfolding. In addition, we prove that the decorated processes of a net N can be axiomatized as the arrows of a symmetric monoidal category which, therefore, provides the aforesaid unification.
Towards a theory of parallel algorithms on concrete data structures
 In Semantics for Concurrency, Leicester
, 1990
"... The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of DARPA or the U.S. government. Building on Kahn and Plotkin’s theory of concrete data structures and sequential functions, ..."
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Cited by 11 (6 self)
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The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of DARPA or the U.S. government. Building on Kahn and Plotkin’s theory of concrete data structures and sequential functions, Berry and Curien defined an intensional model of sequential algorithms between concrete data structures. In this paper we report on an attempt to develop a similar intensional model of concurrent computation. We present a notion of parallel algorithm between concrete data structures, together with suitable application and currying operations. We define an intensional strictness ordering on parallel algorithms, with respect to which application is well behaved (at first order types). We define the inputoutput function computed by a parallel algorithm, and we show that every parallel algorithm computes a continuous function. Thus, a parallel algorithm may be viewed as a continuous function together with a parallel computation strategy. In contrast, a Berry
Geometric and higher order logic in terms of abstract Stone duality
 THEORY AND APPLICATIONS OF CATEGORIES
, 2000
"... The contravariant powerset, and its generalisations ΣX to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers, satisfy the Euclidean principle that φ ∧ F (φ) =φ ∧ F (⊤). Conversely, when the adjunction Σ (−) ⊣ Σ (−) is monadic, this ..."
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Cited by 7 (0 self)
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The contravariant powerset, and its generalisations ΣX to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers, satisfy the Euclidean principle that φ ∧ F (φ) =φ ∧ F (⊤). Conversely, when the adjunction Σ (−) ⊣ Σ (−) is monadic, this equation implies that Σ classifies some class of monos, and the Frobenius law ∃x.(φ(x) ∧ ψ) =(∃x.φ(x)) ∧ ψ) for the existential quantifier. In topology, the lattice duals of these equations also hold, and are related to the Phoa principle in synthetic domain theory. The natural definitions of discrete and Hausdorff spaces correspond to equality and inequality, whilst the quantifiers considered as adjoints characterise open (or, as we call them, overt) and compact spaces. Our treatment of overt discrete spaces and open maps is precisely dual to that of compact Hausdorff spaces and proper maps. The category of overt discrete spaces forms a pretopos and the paper concludes with a converse of Paré’s theorem (that the contravariant powerset functor is monadic) that characterises elementary toposes by means of the monadic and Euclidean properties together with all quantifiers, making no reference to subsets.
Bidomains and full abstraction for countable nondeterminism
 In Proceedings of FoSSaCS’06, number 3921 in LNCS
, 2006
"... Abstract. We describe a denotational semantics for a sequential functional language with random number generation over a countably infinite set (the natural numbers), and prove that it is fully abstract with respect to mayandmust testing. Our model is based on biordered sets similar to Berry’s bid ..."
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Cited by 7 (2 self)
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Abstract. We describe a denotational semantics for a sequential functional language with random number generation over a countably infinite set (the natural numbers), and prove that it is fully abstract with respect to mayandmust testing. Our model is based on biordered sets similar to Berry’s bidomains, and stable, monotone functions. However, (as in prior models of unbounded nondeterminism) these functions may not be continuous. Working in a biordered setting allows us to exploit the different properties of both extensional and stable orders to construct a Cartesian closed category of sequential, discontinuous functions, with least and greatest fixpoints having strong enough properties to prove computational adequacy. We establish full abstraction of the semantics by showing that it contains a simple, firstorder “universal typeobject ” within which all types may be embedded using functions defined by (countable) ordinal induction. 1
Bistable biorders: a sequential domain theory
 Oura) Physics of Snow and Ice
, 2005
"... Abstract. We give a simple ordertheoretic construction of a Cartesian closed category of sequential functions. It is based on bistable biorders, which are sets with a partial order — the extensional order — and a bistable coherence, which captures equivalence of program behaviour, up to permutation ..."
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Cited by 3 (0 self)
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Abstract. We give a simple ordertheoretic construction of a Cartesian closed category of sequential functions. It is based on bistable biorders, which are sets with a partial order — the extensional order — and a bistable coherence, which captures equivalence of program behaviour, up to permutation of top (error) and bottom (divergence). We show that monotone and bistable functions (which are required to preserve bistably bounded meets and joins) are strongly sequential, and use this fact to prove universality results for the bistable biorder semantics of the simplytyped lambdacalculus (with atomic constants), and an extension with arithmetic and recursion. We also construct a bistable model of SPCF, a higherorder functional programming language with nonlocal control. We use our universality result for the lambdacalculus to show that the semantics of SPCF is fully abstract. We then establish a direct correspondence between bistable functions and sequential algorithms by showing that sequential data structures give rise to bistable biorders, and that each bistable function between such biorders is computed by a sequential algorithm. 1.
Linear Programming Languages
"... Abstract. We formalize SℓPCF, namely a programming language which is able to represent linear function between coherence spaces. We give an interpretation of SℓPCF into the model of linear coherence spaces and we show that such semantics is fully abstract with respect to it. SℓPCF is not syntactical ..."
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Cited by 2 (1 self)
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Abstract. We formalize SℓPCF, namely a programming language which is able to represent linear function between coherence spaces. We give an interpretation of SℓPCF into the model of linear coherence spaces and we show that such semantics is fully abstract with respect to it. SℓPCF is not syntactically linear, namely its programs can contain the same variable more than once. Last, we address the universality problem. 1
Extensional semantics of program behaviour
, 2008
"... Game semantics is a way of characterizing programming languages and logical calculi intensionally by interpreting proofs or programs as strategies for interacting with the environment; this process of interaction can be thought of as playing a twoperson game. Games can capture precisely the behavio ..."
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Game semantics is a way of characterizing programming languages and logical calculi intensionally by interpreting proofs or programs as strategies for interacting with the environment; this process of interaction can be thought of as playing a twoperson game. Games can capture precisely the behaviour of higherorder programs in sequential languages, in a fashion which is very difficult or impossible for traditional forms of denotational semantics. This precision can be expressed formally via the properties of full abstraction, full completeness or universality. A remarkable feature of game semantics is its flexibility: games have been used to give fully abstract or fully complete models not only of logics such as linear logic [3] and functional languages such as PCF [1, 12, 38], but also languages with powerful imperative features such as references (state) [4, 2], continuations [32] and exceptions [13]. By contrast, in domain theory programs are modelled extensionally using sets (with certain ordertheoretic structure) and functions (preserving that structure). Domains have formed the basis for much research in denotational semantics by virtue of their conceptual simplicity, and now form a wideranging mathematical theory. The search for ordertheoretic characterizations of higherorder sequentiality has led to the development of several candidate notions, such as stability [5] and strong stability [6], and was arguably the driving force behind most work in denotational semantics for some years. However, a significant obstacle to the use of domains to reason about programming languages has been a lack of accurate models of imperative and concurrent features of the sort provided by game semantics. A possible basis for connecting the intensional world of games with the extensional world of domains was the remarkable observation of Cartwright and Felleisen [8]
A Logical Approach to Stable Domains
, 2006
"... Building on earlier work by GuoQiang Zhang on disjunctive information systems, and by Thomas Ehrhard, Pasquale Malacaria, and the first author on stable Stone duality, we develop a framework of disjunctive propositional logic in which theories correspond to algebraic Ldomains. Disjunctions in the ..."
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Building on earlier work by GuoQiang Zhang on disjunctive information systems, and by Thomas Ehrhard, Pasquale Malacaria, and the first author on stable Stone duality, we develop a framework of disjunctive propositional logic in which theories correspond to algebraic Ldomains. Disjunctions in the logic can be indexed by arbitrary sets (as in geometric logic) but must be provably disjoint. This raises several technical issues which have to be addressed before clean notions of axiom system and theory can be defined. We show soundness and completeness of the proof system with respect to distributive disjunctive semilattices, and prove that every such semilattice arises as the Lindenbaum algebra of a disjunctive theory. Via stable Stone duality, we show how to use disjunctive propositional logic for a logical description of algebraic Ldomains.
Maximality and Totality of Stable Functions in the Category of Stable Bifinite Domains ∗
"... This paper studies maximality and totality of stable functions in the category of stable bifinite domains. We present three main results: (1) every maximumpreserving function is a maximal element in the stable function spaces; (2) a maximal stable function f: D → E is maximumpreserving if D is maxi ..."
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This paper studies maximality and totality of stable functions in the category of stable bifinite domains. We present three main results: (1) every maximumpreserving function is a maximal element in the stable function spaces; (2) a maximal stable function f: D → E is maximumpreserving if D is maximumseparable and E is completely separable; and (3) a stable bifinite domain D is maximumseparable if and only if for any locally distributive stable bifinite domain E, each maximal stable function f: D → E is maximumpreserving.