We introduce a deep inference logical system SBVr which extends SBV [6] with Rename, a self-dual atom-renaming operator. We prove that the cut free subsystem BVr of SBVr exists. We embed the terms of linear λ-calculus with explicit substitutions into formulas of SBVr. Our embedding recalls the one of full λ-calculus into π-calculus. The proof-search inside SBVr and BVr is complete with respect to the evaluation of linear λ-calculus with explicit substitutions. Instead, only soundness of proof-search in SBVr holds. Rename is crucial to let proof-search simulate the substitution of a linear λ-terms for a variable in the course of linear β-reduction. Despite SBVr is a minimal extension of SBV its proof-search can compute all boolean functions, exactly like linear λ-calculus with explicit substitutions can do. © 2011 Springer-Verlag Berlin Heidelberg.
CITATION STYLE
Roversi, L. (2011). Linear lambda calculus and deep inference. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 6690 LNCS, pp. 184–197). https://doi.org/10.1007/978-3-642-21691-6_16
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