A Linear-logical Reconstruction of Intuitionistic Modal Logic S4 [article]

Yosuke Fukuda, Akira Yoshimizu
<span title="2019-04-24">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We propose a "modal linear logic" to reformulate intuitionistic modal logic S4 (IS4) in terms of linear logic, establishing an S4-version of Girard translation from IS4 to it. While the Girard translation from intuitionistic logic to linear logic is well-known, its extension to modal logic is non-trivial since a naive combination of the S4 modality and the exponential modality causes an undesirable interaction between the two modalities. To solve the problem, we introduce an extension of
more &raquo; ... onistic multiplicative exponential linear logic with a modality combining the S4 modality and the exponential modality, and show that it admits a sound translation from IS4. Through the Curry-Howard correspondence we further obtain a Geometry of Interaction Machine semantics of the modal lambda-calculus by Pfenning and Davies for staged computation.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1904.10605v1">arXiv:1904.10605v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/oalwlgh6wjaj3pbshdvjwcuram">fatcat:oalwlgh6wjaj3pbshdvjwcuram</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200911141743/https://arxiv.org/pdf/1904.10605v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/a6/27/a627a0775c68c309eb179f0e7bf26010f4ee6020.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1904.10605v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>