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Algebraic methods in semantics
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Needed reduction and spine strategies for the lambda calculus
Information and Computation
Optimal derivations in weak lambda-calculi and in orthogonal term rewriting systems
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An equivalence between lambda-terms
Theoretical Computer Science
Head linear reduction and pure proof net extraction
MFPS '92 Selected papers of the conference on Meeting on the mathematical foundations of programming semantics, part I : linear logic: linear logic
Parallel reductions in &lgr;-calculus
Information and Computation
Normalisation in Weakly Orthogonal Rewriting
RtA '99 Proceedings of the 10th International Conference on Rewriting Techniques and Applications
Axiomatic Rewriting Theory VI Residual Theory Revisited
RTA '02 Proceedings of the 13th International Conference on Rewriting Techniques and Applications
Discrete Normalization and Standardization in Deterministic Residual Structures
ALP '96 Proceedings of the 5th International Conference on Algebraic and Logic Programming
Game semantics and abstract machines
LICS '96 Proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science
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Information and Computation
Resource operators for λ-calculus
Information and Computation
Local Bigraphs and Confluence: Two Conjectures
Electronic Notes in Theoretical Computer Science (ENTCS)
CSL'10/EACSL'10 Proceedings of the 24th international conference/19th annual conference on Computer science logic
A semantic measure of the execution time in linear logic
Theoretical Computer Science
Böhm trees, krivine's machine and the taylor expansion of lambda-terms
CiE'06 Proceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers
Axiomatic rewriting theory i: a diagrammatic standardization theorem
Processes, Terms and Cycles
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Standardization is a fundamental notion for connecting programming languages and rewriting calculi. Since both programming languages and calculi rely on substitution for defining their dynamics, explicit substitutions (ES) help further close the gap between theory and practice. This paper focuses on standardization for the linear substitution calculus, a calculus with ES capable of mimicking reduction in lambda-calculus and linear logic proof-nets. For the latter, proof-nets can be formalized by means of a simple equational theory over the linear substitution calculus. Contrary to other extant calculi with ES, our system can be equipped with a residual theory in the sense of Lévy, which is used to prove a left-to-right standardization theorem for the calculus with ES but without the equational theory. Such a theorem, however, does not lift from the calculus with ES to proof-nets, because the notion of left-to-right derivation is not preserved by the equational theory. We then relax the notion of left-to-right standard derivation, based on a total order on redexes, to a more liberal notion of standard derivation based on partial orders. Our proofs rely on Gonthier, Lévy, and Melliès' axiomatic theory for standardization. However, we go beyond merely applying their framework, revisiting some of its key concepts: we obtain uniqueness (modulo) of standard derivations in an abstract way and we provide a coinductive characterization of their key abstract notion of external redex. This last point is then used to give a simple proof that linear head reduction --a nondeterministic strategy having a central role in the theory of linear logic-- is standard.