PAC Quasi-automatizability of Resolution over Restricted Distributions

Abstract

We consider principled alternatives to unsupervised learning in data mining by situating the learning task in the context of the subsequent analysis task. Specifically, we consider a query-answering (hypothesis-testing) task: In the combined task, we decide whether an input query formula is satisfied over a background distribution by using input examples directly, rather than invoking a two-stage process in which (i) rules over the distribution are learned by an unsupervised learning algorithm and (ii) a reasoning algorithm decides whether or not the query formula follows from the learned rules. In a previous work (2013), we observed that the learning task could satisfy numerous desirable criteria in this combined context -- effectively matching what could be achieved by agnostic learning of CNFs from partial information -- that are not known to be achievable directly. In this work, we show that likewise, there are reasoning tasks that are achievable in such a combined context that are not known to be achievable directly (and indeed, have been seriously conjectured to be impossible, cf. (Alekhnovich and Razborov, 2008)). Namely, we test for a resolution proof of the query formula of a given size in quasipolynomial time (that is, "quasi-automatizing" resolution). The learning setting we consider is a partial-information, restricted-distribution setting that generalizes learning parities over the uniform distribution from partial information, another task that is known not to be achievable directly in various models (cf. (Ben-David and Dichterman, 1998) and (Michael, 2010)).