[Paper Review] Every super-polynomial proof in purely implicational minimal logic has a polynomially sized proof in classical implicational propositional logic

Edward Hermann Hæusler|arXiv (Cornell University)|May 25, 2015

Logic, Reasoning, and Knowledge4 references3 citations

TL;DR

This paper demonstrates that any super-polynomially sized proof in purely implicational minimal logic can be transformed into a polynomial-sized proof in classical implicational propositional logic. The key contribution is that this transformation implies the existence of short proofs for all classical tautologies, supporting the conjecture that NP = CoNP.

ABSTRACT

In this article we show how any formula A with a proof in minimal implicational logic that is super-polynomially sized has a polynomially-sized proof in classical implicational propositional logic . This fact provides an argument in favor that any classical propositional tautology has short proofs, i.e., NP=CoNP.

Motivation & Objective

• To investigate the proof complexity gap between minimal and classical implicational logic.
• To determine whether super-polynomial proofs in minimal logic can be compressed into polynomial-sized proofs in classical logic.
• To provide evidence that all classical tautologies may have short proofs, implying NP = CoNP.

Proposed method

• Analyzing the structure of proofs in purely implicational minimal logic.
• Identifying syntactic transformations that convert minimal logic proofs into equivalent classical logic proofs.
• Applying proof-theoretic techniques to bound the size increase from super-polynomial to polynomial.
• Leveraging the completeness of classical logic to ensure proof equivalence after transformation.
• Using the Curry-Howard isomorphism implicitly to interpret proofs as terms in a typed lambda calculus.
• Establishing that the transformation preserves logical validity while drastically reducing proof size.

Experimental results

Research questions

• RQ1 Can every super-polynomial proof in purely implicational minimal logic be converted into a polynomial-sized proof in classical implicational propositional logic?
• RQ2 What structural properties of minimal logic proofs allow such a size reduction in classical logic?
• RQ3 Does this transformation provide evidence for the existence of short proofs for all classical tautologies?
• RQ4 To what extent does this result support the conjecture that NP = CoNP?
• RQ5 Are there inherent limitations in minimal logic that prevent such compression, or is the compression always possible?

Key findings

• Any formula with a super-polynomial proof in purely implicational minimal logic admits a polynomial-sized proof in classical implicational propositional logic.
• The transformation from minimal to classical logic proofs does not compromise logical validity.
• The size reduction is sufficient to suggest that all classical tautologies may have short proofs.
• The result provides indirect support for the conjecture that NP = CoNP.
• The proof method relies on syntactic rewriting and classical reasoning to compress proof length.
• The result implies that the proof complexity gap between minimal and classical logic is bounded by polynomial factors for implicational formulas.