Failure of the strong feasible disjunction property

2026-04-06

AI Summary

This study investigates whether strong propositional proof systems possess the strong feasible disjunction property—namely, the ability to efficiently extract a short proof of one of the disjuncts from a short proof of their disjunction. Relying solely on two widely accepted computational complexity assumptions—that the class E requires exponential-size NP-oracle circuits and that P/poly demi-bits exist—the authors combine circuit lower-bound techniques with Rudich’s demi-bit theory to demonstrate that this property necessarily fails. This work dispenses with prior reliance on the proof complexity generator conjecture and constitutes the first demonstration, under standard complexity-theoretic assumptions, that the strong feasible disjunction property is unattainable in strong proof systems.

Abstract

A propositional proof system $P$ has the strong feasible disjunction property iff there is a constant $c \geq 1$ such that whenever $P$ admits a size $s$ proof of $\bigvee_i α_i$ with no two $α_i$ sharing an atom then one of $α_i$ has a $P$-proof of size $\le s^c$. It was proved by K. (2025) that no proof system strong enough admits this property assuming a computational complexity conjecture and a conjecture about proof complexity generators. Here we build on Ilango (2025) and Ren et al. (2025) and prove the same result under two purely computational complexity hypotheses: - there exists a language in class E that requires exponential size circuits even if they are allowed to query an NP oracle, - there exists a P/poly demi-bit in the sense of Rudich (1997).

Problem

• strong feasible disjunction property
• propositional proof system
• computational complexity
• proof complexity
• circuit lower bounds

Innovation

• strong feasible disjunction property
• propositional proof systems
• computational complexity assumptions
• circuit lower bounds
• demi-bit