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Infinitary proof theory: the multiplicative additive case. (English) Zbl 1370.03077
Regnier, Laurent (ed.) et al., 25th EACSL annual conference and 30th workshop on computer science logic, CSL’16, Marseille, France, August 29 – September 1, 2016. Proceedings. Wadern: Schloss Dagstuhl – Leibniz Zentrum für Informatik (ISBN 978-3-95977-022-4). LIPIcs – Leibniz International Proceedings in Informatics 62, Article 42, 17 p. (2016).
Summary: Infinitary and regular proofs are commonly used in fixed point logics. Being natural intermediate devices between semantics and traditional finitary proof systems, they are commonly found in completeness arguments, automated deduction, verification, etc. However, their proof theory is surprisingly underdeveloped. In particular, very little is known about the computational behavior of such proofs through cut elimination. Taking such aspects into account has unlocked rich developments at the intersection of proof theory and programming language theory. One would hope that extending this to infinitary calculi would lead, e.g., to a better understanding of recursion and corecursion in programming languages. Structural proof theory is notably based on two fundamental properties of a proof system: cut elimination and focalization. The first one is only known to hold for restricted (purely additive) infinitary calculi, thanks to the work of Santocanale and Fortier; the second one has never been studied in infinitary systems. In this paper, we consider the infinitary proof system $$\mu\mathsf{MALL}^\infty$$ for multiplicative and additive linear logic extended with least and greatest fixed points, and prove these two key results. We thus establish $$\mu\mathsf{MALL}^\infty$$ as a satisfying computational proof system in itself, rather than just an intermediate device in the study of finitary proof systems.
For the entire collection see [Zbl 1351.68028].

##### MSC:
 03F52 Proof-theoretic aspects of linear logic and other substructural logics 03F05 Cut-elimination and normal-form theorems
##### Keywords:
infinitary proofs; linear logic
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