[Paper Review] Bouncing threads for infinitary and circular proofs

David Baelde, Amina Doumane+2 more|arXiv (Cornell University)|May 17, 2020

Logic, programming, and type systems6 references4 citations

TL;DR

This paper introduces a generalized validity criterion for infinitary and circular proofs in multiplicative additive linear logic with fixed points (μMALL∞), extending prior work to handle axioms and cuts. By introducing 'bouncing threads'—threads that can revisit formulas through cyclic paths—it ensures soundness and cut elimination, proving the criterion undecidable in general but decidable under specific restrictions.

ABSTRACT

We generalize the validity criterion for the infinitary proof system of the multiplicative additive linear logic with fixed points. Our criterion is designed to take into account axioms and cuts. We show that it is sound and enjoys the cut elimination property. We finally study its decidability properties, and prove that it is undecidable in general but becomes decidable under some restrictions.

Motivation & Objective

• To extend the validity criterion for infinitary proofs in μMALL∞ to handle cuts and axioms, which previous criteria excluded.
• To provide a computational interpretation of cut elimination in circular proofs by ensuring productivity via the same validity condition.
• To formalize a new notion of 'bouncing threads' that allows threads to revisit formulas through cyclic paths, enhancing expressiveness.
• To analyze decidability of the validity criterion, showing it is undecidable in general but decidable under certain constraints.
• To establish a Curry-Howard correspondence for μMALL∞ by linking proof validity to program productivity and termination.

Proposed method

• Introduce a generalized validity criterion based on 'bouncing threads' that trace formulas through cyclic paths in non-wellfounded proof trees.
• Define a sliced proof system to handle additive connectives, enabling modular treatment of cut-elimination and validity.
• Introduce the concept of 'persistent slices' to maintain thread consistency across cut-elimination steps.
• Develop a trace-based reduction strategy for cut elimination, showing that the trace of a reduction sequence forms a valid truncated proof.
• Use a reduction to the halting problem of 2-counter machines to prove undecidability of the validity criterion.
• Identify a hierarchy of decidable validity conditions by restricting thread behavior and proof structure.

Experimental results

Research questions

• RQ1 Can a validity criterion for circular proofs in μMALL∞ be extended to handle both cuts and axioms?
• RQ2 Does the same validity condition that ensures soundness also guarantee productivity in cut-elimination?
• RQ3 Is the validity of circular pre-proofs decidable, and if not, what restrictions make it decidable?
• RQ4 Can the notion of threads be generalized to allow revisiting formulas through cycles, improving proof compositionality?
• RQ5 What is the computational complexity of deciding whether a circular pre-proof is valid?

Key findings

• The proposed bouncing thread criterion is sound and ensures cut elimination in μMALL∞, even for proofs with cuts and axioms.
• The criterion is undecidable in general, as shown by a reduction to the halting problem of 2-counter machines.
• The validity problem is Σ⁰₁-complete, meaning it is recursively enumerable but not co-r.e.
• Under certain structural restrictions, such as bounded thread depth or limited cyclic reuse, the validity criterion becomes decidable.
• The trace of any cut-elimination sequence is a valid truncated proof, establishing a connection between reduction and proof structure.
• The framework supports a Curry-Howard interpretation where valid circular proofs correspond to productive, terminating programs in coinductive type systems.