[Paper Review] PAC Quasi-automatizability of Resolution over Restricted Distributions

Brendan Juba|arXiv (Cornell University)|Apr 16, 2013

Machine Learning and Algorithms28 references3 citations

TL;DR

This paper demonstrates that resolution proof search can be quasi-automatized in quasipolynomial time under PAC-semantics when reasoning over restricted distributions—specifically, distributions with a correlation gap, such as affine distributions generalizing learning parities. The key contribution is a novel algorithm that leverages partial examples and width-restricted clause learning to efficiently detect resolution proofs, overcoming known lower bounds in the classical setting.

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)).

Motivation & Objective

• To address the challenge of reasoning over incomplete data by combining learning and inference in a unified framework under PAC-semantics.
• To investigate whether resolution proof search can be efficiently automatized when examples are partially observed and drawn from restricted distributions.
• To overcome the known difficulty of resolution automatizability, which is conjectured to be impossible in the classical setting.
• To extend the tractability of reasoning beyond restricted proof fragments (e.g., Horn clauses) to general, dag-like resolution.
• To explore whether learning from partial information can enable reasoning tasks that are infeasible in the classical two-stage learning-and-inference model.

Proposed method

• The algorithm operates under a restricted-distribution model where examples are partially revealed via a masking process $ M_{\mu} $, which reveals each literal independently with probability $ \mu $.
• It learns a low-width formula $ \psi $ from partial examples, using a width-bounded clause learning strategy to approximate the true underlying formula.
• The method applies Hoeffding's inequality to bound the probability of detecting a falsified clause in the learned formula, ensuring correctness with high probability.
• It leverages the correlation gap property of the distribution to ensure that any resolution proof has a low-width approximate version that can be detected via partial examples.
• The algorithm checks for the existence of a resolution proof by verifying whether a falsifying clause in the learned formula is witnessed in the partial examples with sufficient frequency.
• It combines techniques from PAC learning and proof complexity, using the structure of affine distributions to enable efficient proof detection.

Experimental results

Research questions

• RQ1 Can resolution proof search be quasi-automatized in quasipolynomial time when examples are partially observed and drawn from a restricted distribution?
• RQ2 Does the presence of a correlation gap in the underlying distribution enable efficient detection of resolution proofs despite incomplete information?
• RQ3 Can the combined learning-and-reasoning task outperform the classical two-stage approach of first learning rules and then reasoning on them?
• RQ4 Is it possible to achieve quasi-automatizability of general resolution under PAC-semantics, even when classical automatizability is believed to be impossible?
• RQ5 Can the results be extended to stronger proof systems such as cutting planes or polynomial calculus with resolution?

Key findings

• The algorithm achieves quasi-automatizability of general, dag-like resolution in quasipolynomial time under PAC-semantics over distributions with a correlation gap.
• The method successfully detects resolution proofs by verifying the presence of falsified clauses in the learned formula using partial examples, with high probability.
• The probability of correctly rejecting an incorrect query is at least $ 1 - \delta $, under the assumption that the clauses in the target formula have bounded width.
• The algorithm leverages the fact that each clause of width at most $ w $ is completely revealed with probability at least $ \mu^w $, enabling detection of falsifying clauses with constant probability when $ w = \Omega(\log n) $.
• The results generalize to masking processes that reveal any width-$ w $ set of literals with non-negligible probability, including $ w $-wise independent distributions.
• The work suggests that reasoning under PAC-semantics may be easier than in the classical setting, as it bypasses known lower bounds on resolution automatizability.