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Free-cut elimination in linear logic and an application to a feasible arithmetic

Patrick Baillot 1, 2 Anupam Das 1, 2 
2 PLUME - Preuves et Langages
LIP - Laboratoire de l'Informatique du Parallélisme
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.
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Patrick Baillot, Anupam Das. Free-cut elimination in linear logic and an application to a feasible arithmetic . Computer Science Logic 2016, Aug 2016, Marseille, France. pp. 40:1-40:18, ⟨10.4230/LIPIcs.CSL.2016.40⟩. ⟨hal-01316754⟩



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