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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 43 (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...
The Weakest Completion Approach to the Probabilistic Semantics
, 2000
"... A standard program starts its execution in an initial state, and terminates (if it ever does) in one of a set of final states. Its behaviour can be modelled by a binary relation between the initial and final states. The difference between the standard and the probabilistic semantics is that the form ..."
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Cited by 1 (0 self)
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A standard program starts its execution in an initial state, and terminates (if it ever does) in one of a set of final states. Its behaviour can be modelled by a binary relation between the initial and final states. The difference between the standard and the probabilistic semantics is that the former tells us which final states are or are not possible, whereas the latter tells us the probability with which they may occur. This paper presents a link between the probabilistic and the imperative programming using the weakest completion. We demonstrate how the probabilistic semantics can be derived directly from the standard relational one using the type embedding and healthiness condition of real programs. Carroll Morgan is an adjunct professor in the department of computer science at New South Wales University in Australia. He conducts research in the area of refinement theories and formal methods applied to software engineering and applications to parallel and distributed computing, ...
Unifying Theories of Programming with Monads
"... Abstract. The combination of probabilistic and nondeterministic choice in program calculi is a notoriously tricky problem, and one with a long history. We present a simple functional programming approach to this challenge, based on algebraic theories of computational effects. We make use of the powe ..."
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Abstract. The combination of probabilistic and nondeterministic choice in program calculi is a notoriously tricky problem, and one with a long history. We present a simple functional programming approach to this challenge, based on algebraic theories of computational effects. We make use of the powerful abstraction facilities of modern functional languages, to introduce the choice operations as a little embedded domainspecific language rather than having to define a language extension; we rely on referential transparency, to justify straightforward equational reasoning about program behaviour. 1