Proof Complexity of Monotone Branching Programs

Abstract

We investigate the proof complexity of systems based on positive branching programs, i.e. non-deterministic branching programs (NBPs) where, for any 0-transition between two nodes, there is also a 1-transition. Positive NBPs compute monotone Boolean functions, like negation-free circuits or formulas, but constitute a positive version of (non-uniform) $$\mathbf {N}\mathbf {L}$$ , rather than $$\mathbf {P}$$ or $$\mathbf {NC}^{1}$$ , respectively. The proof complexity of NBPs was investigated in previous work by Buss, Das and Knop, using extension variables to represent the dag-structure, over a language of (non-deterministic) decision trees, yielding the system $$\mathsf {e}\mathsf {LNDT}$$ . Our system $$\mathsf {e}\mathsf {LNDT}^{+}$$ is obtained by restricting their systems to a positive syntax, similarly to how the ‘monotone sequent calculus’ $$\mathsf {MLK}$$ is obtained from the usual sequent calculus $$\mathsf {LK}$$ by restricting to negation-free formulas. Our main result is that $$\mathsf {e}\mathsf {LNDT}^{+}$$ polynomially simulates $$\mathsf {e}\mathsf {LNDT}$$ over positive sequents. Our proof method is inspired by a similar result for $$\mathsf {MLK}$$ by Atserias, Galesi and Pudlák, that was recently improved to a bona fide polynomial simulation via works of Jeřábek and Buss, Kabanets, Kolokolova and Koucký. Along the way we formalise several properties of counting functions within $$\mathsf {e}\mathsf {LNDT}^{+}$$ by polynomial-size proofs and, as a case study, give explicit polynomial-size poofs of the propositional pigeonhole principle.