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Free-Cut Elimination in Linear Logic and an Application to a Feasible Arithmetic

Authors Patrick Baillot, Anupam Das

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Patrick Baillot
Anupam Das

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Patrick Baillot and Anupam Das. Free-Cut Elimination in Linear Logic and an Application to a Feasible Arithmetic. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 40:1-40:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.CSL.2016.40

Abstract

We prove a general form of 'free-cut elimination' for first-order theories in linear logic, yielding normal forms of proofs where cuts are anchored to nonlogical steps. To demonstrate the usefulness of this result, we consider a version of arithmetic in linear logic, based on a previous axiomatisation by Bellantoni and Hofmann. We prove a witnessing theorem for a fragment of this arithmetic via the `witness function method', showing that the provably convergent functions are precisely the polynomial-time functions. The programs extracted are implemented in the framework of 'safe' recursive functions, due to Bellantoni and Cook, where the ! modality of linear logic corresponds to normal inputs of a safe recursive program.
Keywords
  • proof theory
  • linear logic
  • bounded arithmetic
  • polynomial time computation
  • implicit computational complexity

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