Adding Negation to Lambda Mu
We present \(\cal L\), an extension of Parigot's \(\lambda\mu\)-calculus by adding negation as a type constructor, together with syntactic constructs that represent negation introduction and elimination. We will define a notion of reduction that extends \(\lambda\mu\)'s reduction system wi...
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| Published in: | arXiv.org |
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| Main Author: | |
| Format: | Paper |
| Language: | English |
| Published: |
Ithaca
Cornell University Library, arXiv.org
24.05.2023
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| Subjects: | |
| ISSN: | 2331-8422 |
| Online Access: | Get full text |
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| Summary: | We present \(\cal L\), an extension of Parigot's \(\lambda\mu\)-calculus by adding negation as a type constructor, together with syntactic constructs that represent negation introduction and elimination. We will define a notion of reduction that extends \(\lambda\mu\)'s reduction system with two new reduction rules, and show that the system satisfies subject reduction. Using Aczel's generalisation of Tait and Martin-L\"of's notion of parallel reduction, we show that this extended reduction is confluent. Although the notion of type assignment has its limitations with respect to representation of proofs in natural deduction with implication and negation, we will show that all propositions that can be shown in there have a witness in \(\cal L\). Using Girard's approach of reducibility candidates, we show that all typeable terms are strongly normalisable, and conclude the paper by showing that type assignment for \(\cal L\) enjoys the principal typing property. |
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| Bibliography: | SourceType-Working Papers-1 ObjectType-Working Paper/Pre-Print-1 content type line 50 |
| ISSN: | 2331-8422 |
| DOI: | 10.48550/arxiv.2109.10447 |