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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 91 (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
A jump from parallel to sequential proofs. multiplicatives
 of Lecture Notes in Computer Science
, 2006
"... Abstract. We introduce a new class of multiplicative proof nets, Jproof nets, which are a typed version of Faggian and Maurel’s multiplicative Lnets. In Jproof nets, we can characterize nets with different degrees of sequentiality, by gradual insertion of sequentiality constraints. As a byproduct ..."
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Cited by 3 (2 self)
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Abstract. We introduce a new class of multiplicative proof nets, Jproof nets, which are a typed version of Faggian and Maurel’s multiplicative Lnets. In Jproof nets, we can characterize nets with different degrees of sequentiality, by gradual insertion of sequentiality constraints. As a byproduct, we obtain a simple proof of the sequentialisation theorem. 1
Reversible, Irreversible and Optimal
 in Electronic Notes in Theoretical Computer Science
, 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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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 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)x)(f)x)y y ! (f(y (y x)x)(f)x)y y where the succ...