The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs

Abstract

Abstract. In recent years there has been some progress in our understanding of the proof-search problem for very low-depth proof systems, e.g. proof systems that manipulate formulas of very low complexity such as clauses (i.e. resolution), DNF-formulas (i.e. R(k) systems), or polynomial inequalities (i.e. semi-algebraic proof systems). In this talk I will overview this progress. I will start with bounded-width resolution, whose specialized proof-search algorithm is as easy as uninteresting, but whose proof-search problem is unintentionally solved by certain versions of conflict-driven clause-learning algorithms with restarts. I will continue with R(k) systems, whose proof-search problem turned out to hide the complexity of certain two-player games of interest in the area of systems synthesis and verification. And I will close with bounded-degree semialgebraic proof systems, whose proof-search problem turned out to hide the complexity of systems of linear equations over finite fields, among other problems.

Automating Resolution is NP-Hard

“…Hence, the above results left open whether weak proof systems, in particular those having feasible interpolation such as Resolution [29], were (weakly) automatizable. We refer to [3] for a survey, and focus from now on on Resolution.…”

Automating Resolution is NP-Hard

“…On the more technical side, we would like to know whether the formulas REF(F, p(n)) are hard for Resolution where F ranges over unsatisfiable CNF formulas with n variables and p is some fixed polynomial. We conjecture that this is the case but we only succeeded in establishing a width lower bound 3 . Of course, one can define analogous formulas P -REF(F, s) for any proof system P .…”

“…If i = 0, we find v 0 , v 1 as the pre-images of v ′ 0 , v ′ 1 under t. Otherwise i ∈ [n − k] and we choose v 0 , v 1 ∈ B i such that h ′ is injective. This can be done because |B i | 3w by (3) or (4), so by ( 5)…”