Technology Category
Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesReasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
Wider systems for linear logic with fixed points: proof theory and complexity
🤖 AI Summary
This study investigates the provability complexity of infinitary, well-founded proof systems in linear logic extended with least and greatest fixed points. By introducing transfinite branching rules to construct infinitary proofs, and combining cut elimination with focalization strategies, the authors employ a refined rank measure on formulas to tightly control proof tree heights. This approach establishes an exact correspondence between provability in the system and levels of the hyperarithmetical hierarchy. The main contribution is the first complete characterization showing that the provability strength of this fixed-point linear logic precisely coincides with the $\omega^{\alpha^\omega}$-th level of the hyperarithmetical hierarchy, where $\alpha$ is a computable ordinal, thereby pinpointing its proof-theoretic complexity within the higher-order arithmetical hierarchy.
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Application Category
📝 Abstract
We investigate infinitary wellfounded systems for linear logic with fixed points, with transfinite branching rules indexed by some closure ordinal $\alpha$ for fixed points. Our main result is that provability in the system for some computable ordinal $\alpha$ is complete for the $\omega^{\alpha^\omega}$ level of the hyperarithmetical hierarchy. To this end we first develop proof theoretic foundations, namely cut elimination and focussing results, to control both the upper and lower bound analysis. Our arguments employ a carefully calibrated notion of formula rank, calculating a tight bound on the height of the (cut-free) proof search space.
Problem
Research questions and friction points this paper is trying to address.
linear logic
fixed points
proof theory
complexity
hyperarithmetical hierarchy
Innovation
Methods, ideas, or system contributions that make the work stand out.
linear logic
fixed points
infinitary proof systems
cut elimination
hyperarithmetical hierarchy
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