Proof complexity of positive branching programs
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866915190129295360 |
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| author | Das, Anupam Delkos, Avgerinos |
| author_facet | Das, Anupam Delkos, Avgerinos |
| contents | 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 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 eLNDT+ by polynomial-size proofs and, as a case study, give explicit polynomial-size poofs of the propositional pigeonhole principle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_06673 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Proof complexity of positive branching programs Das, Anupam Delkos, Avgerinos Computational Complexity Logic in Computer Science Logic 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 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 eLNDT+ by polynomial-size proofs and, as a case study, give explicit polynomial-size poofs of the propositional pigeonhole principle. |
| title | Proof complexity of positive branching programs |
| topic | Computational Complexity Logic in Computer Science Logic |
| url | https://arxiv.org/abs/2102.06673 |