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Proofnets: The parallel syntax for prooftheory
 Logic and Algebra
, 1996
"... The paper is mainly concerned with the extension of proofnets to additives, for which the best known solution is presented. It proposes two cutelimination procedures, the lazy one being in linear time. The solution is shown to be compatible with quantifiers, and the structural rules of exponential ..."
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Cited by 88 (1 self)
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The paper is mainly concerned with the extension of proofnets to additives, for which the best known solution is presented. It proposes two cutelimination procedures, the lazy one being in linear time. The solution is shown to be compatible with quantifiers, and the structural rules of exponentials are also accommodated. Traditional prooftheory deals with cutelimination; these results are usually obtained by means of sequent calculi, with the consequence that 75 % of a cutelimination proof is devoted to endless commutations of rules. It is hard to be happy with this, mainly because: ◮ the structure of the proof is blurred by all these cases; ◮ whole forests have been destroyed in order to print the same routine lemmas; ◮ this is not extremely elegant. However oldfashioned prooftheory, which is concerned with the ritual question: “isthattheoryconsistent? ” never really cared. The situation changed when subtle algorithmic aspects of cutelimination became prominent: typically
Geometry of Interaction III: Accommodating the Additives
 In: Advances in Linear Logic, LNS 222,CUP, 329–389
, 1995
"... The paper expounds geometry of interaction, for the first time in the full case, i.e. for all connectives of linear logic, including additives and constants. The interpretation is done within a C ∗algebra which is induced by the rule of resolution of logic programming, and therefore the execution f ..."
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Cited by 29 (5 self)
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The paper expounds geometry of interaction, for the first time in the full case, i.e. for all connectives of linear logic, including additives and constants. The interpretation is done within a C ∗algebra which is induced by the rule of resolution of logic programming, and therefore the execution formula can be presented as a simple logic programming loop. Part of the data is public (shared channels) but part of it can be viewed as private dialect (defined up to isomorphism) that cannot be shared during interaction, thus illustrating the theme of communication without understanding. One can prove a nilpotency (i.e. termination) theorem for this semantics, and also its soundness w.r.t. a slight modification of familiar sequent calculus in the case of exponentialfree conclusions. 1
The Algebraic Theory of Interaction Nets
, 1995
"... : The theory of interaction nets, invented by Lafont, is reexamined from the algebraic hypergraph rewriting perspective. Supersimple nets are defined and discussed, and some related classes of nets, the polysimple and monosimple classes, are defined and investigated. Their static properties are est ..."
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Cited by 7 (3 self)
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: The theory of interaction nets, invented by Lafont, is reexamined from the algebraic hypergraph rewriting perspective. Supersimple nets are defined and discussed, and some related classes of nets, the polysimple and monosimple classes, are defined and investigated. Their static properties are established, and the invariants that need to be preserved by rewriting are investigated in detail. It is shown that in the general case, contextspecific information may be used to ensure that rules actually preserve the characteristics of the rewritten net. Subordinate agents, which like logical constants for falsity may be introduced only in nonvoid contexts, are presented, and the ramifications of the theory in their presence are investigated, relating it to the simple and semisimple classes of Lafont. Under suitable conditions, describable in purely combinatorial terms, net rewriting systems possess ChurchRosser and Strong Normalisation properties usually associated with rewriting systems...
Types and Dynamics in Partially Additive Categories
, 1995
"... this paper can be directly applied in order to give a semantics of reduction for the multiplicative fragment of linear logic: in order to capture larger fragments it should be extended dealing with infinitary objects and more generally, this approach allows to investigate an algebraic theory of comp ..."
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this paper can be directly applied in order to give a semantics of reduction for the multiplicative fragment of linear logic: in order to capture larger fragments it should be extended dealing with infinitary objects and more generally, this approach allows to investigate an algebraic theory of computation based on the cutelimination of proofs. 6 Conclusions and Related Works
Dynamics of Algorithms
"... This paper represents an extended version of [16]. In the next section we describe partially additive categories. In section 3 we present the execution of pseudoalgorithms in the framework of the partially additive categories as a kind of iteration of endomorphisms. Moreover, we consider deadlock fr ..."
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This paper represents an extended version of [16]. In the next section we describe partially additive categories. In section 3 we present the execution of pseudoalgorithms in the framework of the partially additive categories as a kind of iteration of endomorphisms. Moreover, we consider deadlock free pseudoalgorithms as to be characterized by a summability condition; concepts about structural aspects are organized via type discipline and type compatible dynamics.