A reduction of proof complexity to computational complexity for $AC^0[p]$ Frege systems

We give a general reduction of lengths-of-proofs lower bounds for constant depth Frege systems in DeMorgan language augmented by a connective counting modulo a prime p (the so called AC[p] Frege systems) to computational complexity lower bounds for search tasks involving search trees branching upon values of maps on the vector space of low degree polynomials over Fp. In 1988 Ajtai [2] proved that the unsatisfiable set (¬PHPn) of propositional formulas ∨ j∈[n] pij and ¬pi1j ∨ ¬pi2j and ¬pij1 ∨ ¬pij2 for all i ∈ [n + 1] = {1, . . . , n + 1} , all i1 6= i2 ∈ [n + 1], j ∈ [n], and all i ∈ [n + 1], j1 6= j2 ∈ [n] respectively, expressing the failure of the pigeonhole principle (PHP), has for no d ≥ 1 a polynomial size refutation in a Frege proof system operating only with DeMorgan formulas of depth at most d. Subsequently Kraj́ıček [18] established an exponential lower bound for these so called AC Frege proof systems (for different formulas) and Kraj́ıček, Pudlák and Woods [23] and Pitassi, Beame and Impagliazzo [26] improved independently (and announced jointly in [7]) Ajtai’s bound for PHP to exponential. All these papers employ some adaptation of the random restriction method that has been so successfully applied earlier in circuit complexity (cf. [1, 14, 31, 15]). Razborov [28] invented already in 1987 an elegant method, simplified and generalized by Smolensky [30], to prove lower bounds even for AC[p] circuits, p a prime. Thus immediately after the lower bounds for AC Frege systems were proved researchers attempted to adapt the Razborov-Smolensky method to proof complexity and to prove lower bounds also for AC[p] Frege systems. This turned out to be rather elusive and no lower bounds for the systems were proved, although some related results were obtained. Ajtai [3, 4, 5], Beame et.al.[6] and Buss et.al.[9] proved lower bounds for AC Frege systems in DeMorgan language augmented by the so called modular counting principles as extra axioms (via degree lower bounds for the Nullstellensatz proof system in [6, 9]),