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105
Nuclear and Trace Ideals in Tensored *Categories
, 1998
"... We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The comp ..."
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Cited by 28 (9 self)
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We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored categories, all morphisms are nuclear, and in the tensored category of Hilbert spaces, the nuclear morphisms are the HilbertSchmidt maps. We also introduce two new examples of tensored categories, in which integration plays the role of composition. In the first, mor...
A Relational Model of NonDeterministic Dataflow
 In CONCUR'98, volume 1466 of LNCS
, 1998
"... . We recast dataflow in a modern categorical light using profunctors as a generalisation of relations. The well known causal anomalies associated with relational semantics of indeterminate dataflow are avoided, but still we preserve much of the intuitions of a relational model. The development fits ..."
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Cited by 27 (13 self)
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. We recast dataflow in a modern categorical light using profunctors as a generalisation of relations. The well known causal anomalies associated with relational semantics of indeterminate dataflow are avoided, but still we preserve much of the intuitions of a relational model. The development fits with the view of categories of models for concurrency and the general treatment of bisimulation they provide. In particular it fits with the recent categorical formulation of feedback using traced monoidal categories. The payoffs are: (1) explicit relations to existing models and semantics, especially the usual axioms of monotone IO automata are read off from the definition of profunctors, (2) a new definition of bisimulation for dataflow, the proof of the congruence of which benefits from the preservation properties associated with open maps and (3) a treatment of higherorder dataflow as a biproduct, essentially by following the geometry of interaction programme. 1 Introduction A fundament...
Strong Normalization of Explicit Substitutions via Cut Elimination in Proof Nets
, 1997
"... In this paper, we show the correspondence existing between normalization in calculi with explicit substitution and cut elimination in sequent calculus for Linear Logic, via Proof Nets. This correspondence allows us to prove that a typed version of the #xcalculus [30, 29] is strongly normalizing, as ..."
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Cited by 24 (5 self)
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In this paper, we show the correspondence existing between normalization in calculi with explicit substitution and cut elimination in sequent calculus for Linear Logic, via Proof Nets. This correspondence allows us to prove that a typed version of the #xcalculus [30, 29] is strongly normalizing, as well as of all the calculi isomorphic to it such as # # [24], # s [19], # d [21], and # f [11]. In order to achieve this result, we introduce a new notion of reduction in Proof Nets: this extended reduction is still confluent and strongly normalizing, and is of interest of its own, as it correspond to more identifications of proofs in Linear Logic that differ by inessential details. These results show that calculi with explicit substitutions are really an intermediate formalism between lambda calculus and proof nets, and suggest a completely new way to look at the problems still open in the field of explicit substitutions.
Optimality and Inefficiency : What Isn't a Cost Model of the Lambda Calculus?
 In Proceedings of the 1996 ACM SIGPLAN International Conference on Functional Programming
, 1996
"... We investigate the computational efficiency of the sharing graphs of Lamping [Lam90], Gonthier, Abadi, and L'evy [GAL92], and Asperti [Asp94], designed to effect socalled optimal evaluation, with the goal of reconciling optimality, efficiency, and the clarification of reasonable cost models f ..."
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Cited by 19 (2 self)
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We investigate the computational efficiency of the sharing graphs of Lamping [Lam90], Gonthier, Abadi, and L'evy [GAL92], and Asperti [Asp94], designed to effect socalled optimal evaluation, with the goal of reconciling optimality, efficiency, and the clarification of reasonable cost models for the calculus. Do these graphs suggest reasonable cost models for the calculus? If they are optimal, are they efficient? We present a brief survey of these optimal evaluators, identifying their common characteristics, as well as their shared failures. We give a lower bound on the efficiency of sharing graphs by identifying a class of terms that are normalizable in \Theta(n) time, and require \Theta(n) "fan interactions, " but require\Omega\Gammaq n ) bookkeeping steps. For [GAL92], we analyze this anomaly in terms of the dynamic maintenance of deBruijn indices for intermediate terms. We give another lower bound showing that sharing graphs can do \Omega\Gammao n ) work (via fan interactio...
The Bologna Optimal Higherorder Machine
 Journal of Functional Programming
, 1996
"... gzipped PostScript format via anonymous FTP from the area ftp.cs.unibo.it:/pub/TR/UBLCS or via WWW at ..."
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Cited by 19 (0 self)
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gzipped PostScript format via anonymous FTP from the area ftp.cs.unibo.it:/pub/TR/UBLCS or via WWW at
Designs, Disputes And Strategies
, 2002
"... Important progresses in logic are leading to interactive and dynamical models. Geometry of Interaction and Games Semantics are two major examples. Ludics, initiated by Girard, is a further step in this direction. The objects ..."
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Cited by 18 (6 self)
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Important progresses in logic are leading to interactive and dynamical models. Geometry of Interaction and Games Semantics are two major examples. Ludics, initiated by Girard, is a further step in this direction. The objects
Elementary Complexity and Geometry of Interaction
, 2000
"... We introduce a geometry of interaction model given by an algebra of clauses equipped with resolution (following [Gir95]) into which proofs of Elementary Linear Logic can be interpreted. In order to extend geometry of interaction computation (the so called execution formula) to a wider class of prog ..."
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Cited by 17 (6 self)
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We introduce a geometry of interaction model given by an algebra of clauses equipped with resolution (following [Gir95]) into which proofs of Elementary Linear Logic can be interpreted. In order to extend geometry of interaction computation (the so called execution formula) to a wider class of programs in the algebra than just those coming from proofs, we define a variant of execution (called weak execution). Its application to any program of clauses is shown to terminate with a bound on the number of steps which is elementary in the size of the program. We establish that weak execution coincides with standard execution on programs coming from proofs. Keywords: Elementary Linear Logic, Geometry of interaction, Complexity, Semantics.
Reversible, Irreversible and Optimal lambdamachines (Extended Abstract)
, 1996
"... There are two quite different possibilities for implementing linear head reduction in calculus. Two ways which we are going to explain briefly here in the introduction and in details in the body of the paper. The paper itself is concerned with showing an unexpectedly simple relation between these ..."
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Cited by 16 (1 self)
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There are two quite different possibilities for implementing linear head reduction in calculus. Two ways which we are going to explain briefly here in the introduction and in details in the body of the paper. The paper itself is concerned with showing an unexpectedly simple relation between these two ways, which we term reversible and irreversible, namely that the latter may be obtained as a natural optimization of the former. Keywords: calculus, abstract machines, geometry of interaction, reversible computations. 1 Introduction Notation. We denote the application of U to V by (U)V , e.g., the Church integer ¯ 2 will be fx (f)(f)x. Linear head reduction. But what is exactly linear head reduction, to begin with. It is a variant of head reduction, where one substitutes at each step the leftmost occurrence of c fl1996 Elsevier Science B. V. Danos & Regnier variable whenever it is engaged into a redex, as in: (f (f )(f)x)y y ! (f(y y)(f)x)y y ! (f(y (f )x)(f)x)y y ! (f(y (y y)...