Jan Krajicek

Jan Krajicek

Motivated by the theory of proof complexity generators we consider the following $Σ^p_2$ search problem $\mbox{DD}_P$ determined by a propositional proof system $P$: given a $P$-proof $π$ of a disjunction $\bigvee_i α_i$, no two $α_i$ having an atom in common, find $i$ such that $α_i \in \mbox{TAUT}$. We formulate a hypothesis (ST) that for some strong proof system $P$ the problem $\mbox{DD}_P$ is not solvable in the student-teacher model with a p-time student and a constant number of rounds. The hypothesis follows from the existence of hard one-way permutations. We prove, using a model-theoretic assumption, that (ST) implies $NP \neq coNP$. The assumption concerns the existence of extensions of models of a bounded arithmetic theory and it is open at present if it holds.

Jan Krajicek

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).