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.