The SPARSE-Relativization Framework and Applications to Optimal Proof Systems
Fabian Egidy
TL;DR
This work introduces SPARSE-relativization, a framework that strengthens barriers against relativizable proofs of (p-)optimal proof systems by coupling sparse oracles with cases where the polynomial-time hierarchy remains infinite. By extending bounded relativization and leveraging sparse-oracle-independence, the authors construct new oracles (O1 and O2) showing that Q1 and Q2 are independent of an infinite or collapsing PH, and that Q3 remains difficult even under such assumptions. The O1 oracle rules out the existence of optimal proofs for sets in $PSPACE\setminus NP$ under certain hierarchy collapses, while O2 demonstrates that all $PSPACE$ sets can have p-optimal proofs and that PH can be infinite, yielding complementary barriers. The framework also clarifies differences between sparse and random oracle methods and provides a versatile tool for combining independent oracle constructions to obtain stronger, broadly applicable results across (p-)optimal proof systems and related complexity classes.
Abstract
We investigate the following longstanding open questions raised by Krajíček and Pudlák (J. Symb. L. 1989), Sadowski (FCT 1997), Köbler and Messner (CCC 1998) and Messner (PhD 2000). Q1: Does TAUT have (p-)optimal proof systems? Q2: Does QBF have (p-)optimal proof systems? Q3: Are there arbitrarily complex sets with (p-)optimal proof systems? Recently, Egidy and Glaßer (STOC 2025) contributed to these questions by constructing oracles that show that there are no relativizable proofs for positive answers of these questions, even when assuming well-established conjectures about the separation of complexity classes. We continue this line of research by providing the same proof barrier for negative answers of these questions. For this, we introduce the SPARSE-relativization framework, which is an application of the notion of bounded relativization by Hirahara, Lu, and Ren (CCC 2023). This framework allows the construction of sparse oracles for statements such that additional useful properties (like an infinite polynomial-time hierarchy) hold. By applying the SPARSE-relativization framework, we show that the oracle construction of Egidy and Glaßer also yields the following new oracle. O1: No set in PSPACE\NP has optimal proof systems, $\mathrm{NP} \subsetneq \mathrm{PH} \subsetneq \mathrm{PSPACE}$, and PH collapses We use techniques of Cook and Krajíček (J. Symb. L. 2007) and Beyersdorff, Köbler, and Müller (Inf. Comp. 2011) and apply our SPARSE-relativization framework to obtain the following new oracle. O2: All sets in PSPACE have p-optimal proof systems, there are arbitrarily complex sets with p-optimal proof systems, and PH is infinite Together with previous results, our oracles show that questions Q1 and Q2 are independent of an infinite or collapsing polynomial-time hierarchy.