A Closer Look at Some Recent Proof Compression-Related Claims

TL;DR

This paper critically examines recent claims about proof compression in propositional logic, showing that previous proofs of polynomial size bounds are flawed, thus invalidating related complexity class equalities.

Contribution

The paper identifies gaps in prior proof compression arguments and demonstrates that their polynomial size bounds do not hold, challenging claims about NP=PSPACE.

Findings

Previous proof compression claims are incorrect.

Polynomial size bounds for minimal propositional logic proofs are not established.

Consequently, the proof that NP=PSPACE based on these bounds is invalid.

Abstract

Gordeev and Haeusler [GH19] claim that each tautology $\rho$ of minimal propositional logic can be proved with a natural deduction of size polynomial in $|\rho|$. This builds on work from Hudelmaier [Hud93] that found a similar result for intuitionistic propositional logic, but for which only the height of the proof was polynomially bounded, not the overall size. They arrive at this result by transforming a proof in Hudelmaier's sequent calculus into an equivalent tree-like proof in Prawitz's system of natural deduction, and then compressing the tree-like proof into an equivalent DAG-like proof in such a way that a polynomial bound on the height and foundation implies a polynomial bound on the overall size. Our paper, however, observes that this construction was performed only on minimal implicational logic, which we show to be weaker than the minimal propositional logic for which they claim the result (see Section 4.2). Simply extending the logic systems used to cover minimal propositional logic would not be sufficient to recover the results of the paper, as it would entirely disrupt proofs of a number of the theorems that are critical to proving the main result. Relying heavily on their aforementioned work, Gordeev and Haeusler [GH20] claim to establish NP=PSPACE. The argument centrally depends on the polynomial bound on proof size in minimal propositional logic. Since we show that that bound has not been correctly established by them, their purported proof does not correctly establish NP=PSPACE.