Laboratoire de l'Informatique et du Parallélisme » Linear Arithmetic

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\title{Free-cut elimination in linear logic and an application to a feasible arithmetic}

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\author{Patrick Baillot}

\author{Anupam Das}

\affil{LIP, Universit\'e de Lyon, CNRS, ENS Lyon, INRIA, Universit\'e Claude-Bernard Lyon 1, Milyon

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\EventEditors{John Q. Open and Joan R. Acces}

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\EventLongTitle{42nd Conference on Very Important Topics (CVIT 2016)}

\EventShortTitle{CVIT 2016}

\EventAcronym{CVIT}

\EventYear{2016}

\EventDate{December 24--27, 2016}

\EventLocation{Little Whinging, United Kingdom}

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\SeriesVolume{42}

\ArticleNo{23}

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\maketitle

\begin{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. 

%We conclude with some comments for further applications of the free-cut elimination result. 

 \end{abstract}

\input{introduction}

\input{preliminaries}

\input{free-cut-elim}

\input{arithmetic}

\input{bc}

\input{bc-convergence}

\input{wfm}

\input{further-work}

\input{conclusions}

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