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.

中文翻译：

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正分支程序的证明复杂性

我们研究基于正分支程序（即非确定性分支程序（NBP））的系统的证明复杂性，其中对于两个节点之间的任何0转换，也存在1转换。正NBP计算单调布尔函数，就像无求反的电路或公式一样，但分别构成（非均匀）NL的正版本，而不是P或NC1。Buss，Das和Knop在先前的工作中对NBP的证明复杂性进行了研究，使用扩展变量来表示dag结构（通过非确定性决策树的语言），得出系统eLNDT。我们的系统eLNDT +是通过将其系统限制为正语法而获得的，类似于通过限制无负数的公式从通常的顺序演算LK中获得“单调顺序演算” MLK的方式。我们的主要结果是eLNDT +在正序列上多项式模拟eLNDT。我们的证明方法的灵感来自于Atserias，Galesi和Pudl \'ak的MLK相似结果，最近通过Je \ v {r} \'abek和Buss，Kabanets，Kolokolova和库克 在此过程中，我们通过多项式大小的证明形式化了eLNDT +中计数函数的几种性质，并作为案例研究，给出了命题信鸽原理的明确的多项式大小po。