Proof Compression and NP Versus PSPACE

Abstract

We show that arbitrary tautologies of Johansson’s minimal propositional logic are provable by “small” polynomial-size dag-like natural deductions in Prawitz’s system for minimal propositional logic. These “small” deductions arise from standard “large” tree-like inputs by horizontal dag-like compression that is obtained by merging distinct nodes labeled with identical formulas occurring in horizontal sections of deductions involved. The underlying geometric idea: if the height, $h\left( \partial \right)$, and the total number of distinct formulas, $\phi \left( \partial \right)$, of a given tree-like deduction $\partial$ of a minimal tautology $\rho$ are both polynomial in the length of $\rho$, $\left| \rho \right|$, then the size of the horizontal dag-like compression $\partial ^{{\textsc {c}} }$ is at most $h\left( \partial \right) \times \phi \left( \partial \right)$, and hence polynomial in $\left| \rho \right|$. That minimal tautologies $\rho$ are provable by tree-like natural deductions $\partial$ with $\left| \rho \right|$-polynomial $h\left( \partial \right)$ and $\phi \left( \partial \right)$ follows via embedding from Hudelmaier’s result that there are analogous sequent calculus deductions of sequent $\Rightarrow \rho$. The notion of dag-like provability involved is more sophisticated than Prawitz’s tree-like one and its complexity is not clear yet. Our approach nevertheless provides a convergent sequence of NP lower approximations of PSPACE-complete validity of minimal logic (Savitch in J Comput Syst Sci 4(2):177–192, 1970); Statman in Theor Comput Sci 9(1):67–72, 1979; Svejdar in Arch Math Log 42(7):711–716, 2003).