Some conditions implying if P=NP then P=PSPACE

TL;DR

The paper explores whether the widely conjectured separation $P\neq PSPACE$ could be reinforced by the conditional implication $P=NP \Rightarrow P=PSPACE$ under several auxiliary hypotheses. It introduces and analyzes multiple sufficient conditions (notably $A$, $A'$, and $B$) that would force $P=PSPACE$ if $P=NP$ holds, providing rigorous constructions and feasibility discussions while refraining from formal proofs. A key technical device is the $f_{M,x}$ construction, together with the novel chained$_P$ function, to formalize how a polynomial-space TM’s behavior might be captured by compact representations. The paper also presents a distinct angle via the PSPACE-complete problem $TQBF$ and a potential size-bounding property $B$ for propositional simplifications, arguing that if these properties hold (in conjunction with $P=NP$), they would yield $P=PSPACE$. While largely speculative, the work outlines concrete avenues for proving or testing these conditions and highlights the broader implications for the hardness landscape of $P$, $NP$, and $PSPACE$.

Abstract

We identify a few conditions $X$ such that $(P=NP \wedge X) \;\Rightarrow\; P=PSPACE$.