🤖 AI Summary
This work investigates the proof complexity of the proof system eLNDT⁺ induced by positive nondeterministic branching programs (positive NBPs). Focusing on positive Boolean sequents—those containing only monotone Boolean functions—we introduce, for the first time, a syntactically restricted variant of eLNDT⁺ whose inference rules strictly mirror the structural constraints of positive NBPs. Our contributions are threefold: (1) We prove that eLNDT⁺ polynomially simulates eLNDT for all positive sequents; (2) we provide explicit polynomial-size proofs within eLNDT⁺ for the pigeonhole principle (PHP) and counting function properties—first such constructions in this framework; (3) by integrating techniques from monotone proof complexity, extension variables, and MLK-style sequent calculus restrictions, we establish positive NBPs as the proof-theoretic foundation for “positive NL.” This work bridges a fundamental theoretical gap between positive computation models and proof complexity.
📝 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 Pudlak, that was recently improved to a bona fide polynomial simulation via works of Jeřabek and Buss, Kabanets, Kolokolova and Koucký. 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.