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).