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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Veröffentlicht in:	arXiv.org
1. Verfasser:	Steffen van Bakel
Format:	Paper
Sprache:	Englisch
Veröffentlicht:	Ithaca Cornell University Library, arXiv.org 24.05.2023
Schlagworte:	Deduction Reduction
ISSN:	2331-8422
Online-Zugang:	Volltext
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Beschreibung
Zusammenfassung:	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.
Bibliographie:	SourceType-Working Papers-1 ObjectType-Working Paper/Pre-Print-1 content type line 50
ISSN:	2331-8422
DOI:	10.48550/arxiv.2109.10447