Decision Problems for Linear Logic with Least and Greatest Fixed Points

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 ( µ MALL ), in particular, recent systems based on “circular” and “non-wellfounded” reasoning. In this paper, we show that µ MALL is undecidable. More explicitly, we show that the general non-wellfounded system is Π 01 -hard via a reduction to the non-halting of Minsky machines, and thus is strictly stronger than its circular counterpart (which is in Σ 01 ). Moreover, we show that the restriction of these systems to theorems with only the least fixed points is already Σ 01 -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 Σ 01 -complete.