Types and dynamics in partially additive categories

Mascari, Gianfranco; Pedicini, Marco

doi:10.1017/cbo9780511662508.008

Cited by 3 publications

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“…In [17], we showed that GoI situations with a special class of traced monoidal categories, known as unique decomposition categories as the underlying category, completely capture Girard's GoI 1 in [11]. It should be noted that Mascari and Pedicini [28] have also used partially additive categories (special case of unique decomposition categories) in their study of GoI. The present and previous work by the author is independent of [28].…”

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confidence: 69%

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Towards a geometry of recursion

Haghverdi

2011

Theoretical Computer Science

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a b s t r a c tAny mathematical theory of algorithms striving to offer a foundation for programming needs to provide a rigorous definition for an abstract algorithm. The works reported by Girard (1988) in [10] and by Moschovakis ( , 1995 in [29][30][31] are among the best examples of such attempts. They both try to offer a mathematically precise and rigorous formulation of an abstract algorithm, intend to keep the algorithmic flavour present and take the notion of recursion as primary and central. The present work is motivated by Girard's GoI 2 paper (Girard (1988) [10], which offers a treatment of recursion in terms of fixed points of linear functions. It is situated in the context of the geometry of interaction (GoI) program and is carried out in the concrete setting of the space of bounded linear maps on a Hilbert space. In this paper, we extend the work in Girard (1988) [10] to the context of traced unique decomposition categories, once again emphasizing the role of abstract trace in the theory of computing. We show that traced unique decomposition categories enriched over partially additive monoids or their variations suffice to axiomatize and hence extend the work in Girard's GoI 2 paper. The theory developed here allows us to formulate an abstract notion of an algorithm as a pair of morphisms in a traced unique decomposition category, an abstract notion of computation as the execution formula (defined using the trace operator) applied to an algorithm, and finally a notion of deadlock-freeness for algorithms. In addition, we can treat recursive definitions, fixed points and fixed point operators in a uniform way in terms of traced unique decomposition categories.

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confidence: 69%

“…It should be noted that Mascari and Pedicini [28] have also used partially additive categories (special case of unique decomposition categories) in their study of GoI. The present and previous work by the author is independent of [28]. This paper is motivated by the work in [10] and thus we shall give a brief account on the motivations for the work in [10].…”

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confidence: 97%

Towards a geometry of recursion

Haghverdi

2011

Theoretical Computer Science

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“…One purpose has been to provide a bridge between operational and denotational semantics. In this context we recall the work in Abramsky et al [1994] on full abstraction, which also gave rise to the study of game semantics for programming languages, and the work in Mascari and Pedicini [1998] on partially additive categories, recently extended by Haghverdi and Scott [2004] to cope with the traced monoidal categories originally proposed • M. Pedicini and F. Quaglia by Joyal et al [1996]. Another purpose has been constructing operational semantics for classical logic in Laurent [2001], successively addressed by Führmann and Pym [2004] through also using traced monoidal categories.…”

Section: Algebraic Setting For the Geometry Of Interactionmentioning

confidence: 99%

Pelcr

Pedicini

Quaglia

2007

ACM Trans. Comput. Logic

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In this article we present the implementation of an environment supporting Lévy's optimal reduction for the λ-calculus on parallel (or distributed) computing systems. In a similar approach to Lamping's, we base our work on a graph reduction technique, known as directed virtual reduction, which is actually a restriction of Danos-Regnier virtual reduction.The environment, which we refer to as PELCR (parallel environment for optimal lambdacalculus reduction), relies on a strategy for directed virtual reduction, namely half combustion. While developing PELCR we adopted both a message aggregation technique, allowing reduction of the communication overhead, and a fair policy for distributing dynamically originated load among processors.We also present an experimental study demonstrating the ability of PELCR to definitely exploit the parallelism intrinsic to λ-terms while performing the reduction. We show how PELCR allows achieving up to 70-80% of the ideal speedup on last generation multiprocessor computing systems. As a last note, the software modules have been developed with the C language and using a standard interface for message passing, that is, MPI, thus making PELCR itself a highly portable software package.

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An introduction to idempotency

Gunawardena¹

1998

Idempotency

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Types and dynamics in partially additive categories

Cited by 3 publications

References 15 publications

Towards a geometry of recursion

Towards a geometry of recursion

Pelcr

An introduction to idempotency

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