An Ultimate Trade-Off in Propositional Proof Complexity

We exhibit an unusually strong tradeoff result in propositional proof complexity that significantly deviates from the established pattern of almost all results of this kind. Namely, restrictions on one resource (width in our case) imply an increase in another resource (tree-like size) that is exponential not only with respect to the complexity of the original problem, but also to the whole class of all problems of the same bit size. More specifically, we show that for any parameter k = k(n) there are unsatisfiable k-CNFs that possess refutations of width O(k), but such that any tree-like refutation of width n1−ε/k must necessarily have double exponential size exp(nΩ(k)). This means that there exist contradictions that allow narrow refutations, but in order to keep the size of such a refutation even within a single exponent, it must necessarily use a high degree of parallelism. Our construction and proof methods combine, in a non-trivial way, two previously known techniques: the hardness escalation method based on substitution formulas and expansion. This combination results in a hardness compression approach that strives to preserve hardness of a contradiction while significantly decreasing the number of its variables. ∗University of Chicago, [email protected]; part of this work was done at Steklov Mathematical Institute, Moscow, and at Toyota Technological Institute, Chicago. Supported by the Russian Science Foundation, grant # 14-50-00005.