Proof Nets and Explicit Substitutions
Roberto Di Cosmo,Delia Kesner,Emmanuel Polonovski +2 more
- pp 63-81
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The simulation technique introduced in [10] is refined to show strong normalization of λ-calculi with explicit substitutions via termination of cut elimination in proof nets and a version of typed λl with named variables is proposed which helps to better understand the complex mechanism of the explicit weakening notation.Abstract:
We refine the simulation technique introduced in [10] to show strong normalization of λ-calculi with explicit substitutions via termination of cut elimination in proof nets [13]. We first propose a notion of equivalence relation for proof nets that extends the one in [9], and we show that cut elimination modulo this equivalence relation is terminating. We then show strong normalization of the typed version of the λl-calculus with de Bruijn indices (a calculus with full composition defined in [8]) using a translation from typed λl to proof nets. Finally, we propose a version of typed λl with named variables which helps to better understand the complex mechanism of the explicit weakening notation introduced in the λl-calculus with de Bruijn indices [8].read more
Citations
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Book ChapterDOI
Confluence of pure differential nets with promotion
TL;DR: It is proved that pure differential nets are Church-Rosser modulo such equivalences, which generalizes to linear logic regular proof nets, and uses a result of finiteness of developments, given by strong normalization when blocking a suitable notion of "new" cuts.
Dissertation
Computing with sequents and diagrams in classical logic - calculi *X, dX and ©X
TL;DR: This Ph.D. thesis addresses the problem of giving computational interpretation to proofs in classical logic by presenting three calculi reflecting different approaches in the study of this area, and presents the dX calculus, the diagrammatic calculus for classical logic, whose diagrams originate from *X-terms.
Journal ArticleDOI
The conservation theorem for differential nets
Michele Pagani,Paolo Tranquilli +1 more
TL;DR: The conservation theorem for differential nets – the graph-theoretical syntax of the differential extension of Linear Logic (Ehrhard and Regnier's DiLL) is proved, which turns the quest for strong normalisation into one for non-erasing weak normalisation (WN), and indeed this result is used to prove SN of simply typed DiLL.
Journal ArticleDOI
Labelled calculi of resources
TL;DR: This paper defines two labelled calculi with explicit substitutions and resource management, and observes a tight relationship between labels and the dynamics of substitutions, which will guide the design of a third calculus that combines the advantages of the previous two.
Translating types and effects with state monads and linear logic
TL;DR: It is proved that the stratification condition on regions, already used in type and effect systems to assure termination, is equivalent to completely avoid the use of recursion in the types used in the translation, obtaining a logical characterization of stratification.
References
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Linear logic
TL;DR: This column presents an intuitive overview of linear logic, some recent theoretical results, and summarizes several applications oflinear logic to computer science.
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Fiftieth volume of theoretical computer science
TL;DR: This contribution was made possible only by the miraculous fact that the first members of the Editorial Board were sharing the same conviction about the necessity of Theoretical Computer Science.
Proceedings ArticleDOI
Explicit substitutions
TL;DR: The λ&sgr;-calculus is a refinement of the λ-Calculus where substitutions are manipulated explicitly, and provides a setting for studying the theory of substitutions, with pleasant mathematical properties.
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Geometry of Interaction 1: Interpretation of System F
TL;DR: The chapter describes the development of a semantics of computation free from the twin drawbacks of reductionism and subjectivism and that a representative class of algorithms can be modelized by means of standard mathematics.