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# Decision Problems for Linear Logic with Least and Greatest Fixed Points

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## Acknowledgements

We would like to thank anonymous reviewers for their valuable comments that enhanced the clarity and presentation of this paper. We also thank Sylvain Schmitz for his insights on alternating vector addition systems.

## Cite As

Anupam Das, Abhishek De, and Alexis Saurin. Decision Problems for Linear Logic with Least and Greatest Fixed Points. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.FSCD.2022.20

## Abstract

Linear logic is an important logic for modelling resources and decomposing computational interpretations of proofs. Decision problems for fragments of linear logic exhibiting "infinitary" behaviour (such as exponentials) are notoriously complicated. In this work, we address the decision problems for variations of linear logic with fixed points (μMALL), in particular, recent systems based on "circular" and "non-wellfounded" reasoning. In this paper, we show that μMALL is undecidable. More explicitly, we show that the general non-wellfounded system is Π⁰₁-hard via a reduction to the non-halting of Minsky machines, and thus is strictly stronger than its circular counterpart (which is in Σ⁰₁). Moreover, we show that the restriction of these systems to theorems with only the least fixed points is already Σ⁰₁-complete via a reduction to the reachability problem of alternating vector addition systems with states. This implies that both the circular system and the finitary system (with explicit (co)induction) are Σ⁰₁-complete.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Linear logic
• Theory of computation → Proof theory
• Theory of computation → Complexity theory and logic
• Linear logic
• fixed points
• decidability

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