Proof complexity of positive branching programs

Anupam Das, Avgerinos Delkos

Research output: Working paper/PreprintPreprint

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, just like negation-free circuits or formulas, but constitute a positive version of (non-uniform) NL, rather than P or NC1, 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 eLNDT. Our system eLNDT+ is obtained by restricting their systems to a positive syntax, similarly to how the 'monotone sequent calculus' MLK is obtained from the usual sequent calculus LK by restricting to negation-free formulas.

Our main result is that eLNDT+ polynomially simulates eLNDT over positive sequents. Our proof method is inspired by a similar result for MLK by Atserias, Galesi and Pudl\'ak, that was recently improved to a bona fide polynomial simulation via works of Je\v{r}\'abek and Buss, Kabanets, Kolokolova and Kouck\'y. Along the way we formalise several properties of counting functions within eLNDT+ by polynomial-size proofs and, as a case study, give explicit polynomial-size poofs of the propositional pigeonhole principle.
Original languageEnglish
PublisherarXiv
Pages1-31
Number of pages31
DOIs
Publication statusPublished - 12 Feb 2021

Bibliographical note

31 pages, 5 figures

Keywords

  • cs.CC
  • cs.LO
  • math.LO

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