Relating size and width in variants of Q-resolution

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Highlights

  • We investigate the classical size-width technique for QBF resolution.

  • The technique holds for level-ordered tree-like Q-resolution.

  • The size-width relation fails for universal and long-distance Q-resolution.

Abstract

In their influential paper ‘Short proofs are narrow – resolution made simple’ [3], Ben-Sasson and Wigderson introduced a crucial tool for proving lower bounds on the lengths of proofs in the resolution calculus. Over a decade later their technique for showing lower bounds on the size of proofs, by examining the width of all possible proofs, remains one of the most effective lower bound techniques in propositional proof complexity.

We continue the investigation begun in [6] into the application of this technique to proof systems for quantified Boolean formulas. We demonstrate a relationship between the size of proofs in level-ordered Q-Resolution and the width of proofs in Q-Resolution. In general, however, the picture is not positive, and for most stronger systems based on Q-Resolution, the size-width relation of [3] fails, answering an open question from [6].

Introduction

Proof complexity aims to understand the strength and limitations of various systems of logic. In particular, we seek upper and lower bounds on the size of proofs, and to develop general methods for finding such bounds. Resolution is a refutational system for propositional logic, with close connections to modern SAT solvers [7]. An important tool for proving lower bounds on the length of Resolution refutations was introduced in [3]. Ben-Sasson and Wigderson showed that whenever a short resolution refutation exists, a narrow refutation (i.e., of small width) can be constructed from it; so conversely if every refutation of some family of formulas must contain a clause of large width, then no small refutation can exist. In this context, width refers to the maximum number of literals in any clause in the proof.

The authors of [6] began the study of possible relationships between size, width and space of refutations in the context of resolution-based proof systems for quantified Boolean formulas (QBF). Understanding which lower bound techniques are effective for QBF is of great importance (cf. [4], [5]); however, the findings of [6] show that size-width relations in the spirit of [3] fail in Q-Resolution, both tree-like and DAG-like. This was shown by presenting a specific class of formulas with short proofs, but requiring large width (even when just counting existential variables).

This investigation is continued here by considering three additional QBF proof systems: level-ordered Q-Resolution, universal Q-Resolution (QU-Res), and long-distance Q-Resolution (LDQ-Res). While QU-Res is a natural counterpart to propositional Resolution, level-ordered and long-distance Q-Resolution are motivated by their connections to QBF solving [10], [12], [14]. In particular, tree-like level-ordered Q-Resolution corresponds to the basic QDPLL algorithm, which lifts the classic DPLL algorithm to QBF and underlies a number of solving approaches.

Implicit assumptions underlying Ben-Sasson and Wigderson's argument break down in the context of Q-Resolution due to the restrictions imposed by the quantifier prefix (cf. [6]). We show that the original argument of [3] can be lifted to QBF in level-ordered Q-Resolution and relate the proof size in that system to the width of Q-Resolution refutations. This holds only for the tree-like systems and cannot be lifted to DAG-like level-ordered Q-Resolution. In contrast, we lift the negative results of [6] to the stronger systems of QU-Res and LDQ-Res, thus answering a question of [6].

Section snippets

Quantified Boolean formulas

Quantified Boolean logic is an extension of propositional logic in which variables may be universally as well as existentially quantified. We consider quantified Boolean formulas (QBFs) in closed prenex conjunctive normal form, denoted Φ=Qϕ. In the quantifier prefix, Q=Q1X1QmXm, the Xi are disjoint sets of variables, and Qi{,}. The matrix ϕ is a formula in conjunctive normal form over the variables in i=1mXi. A variable xXi is at quantification level i, written lv(x)=i. We say that x is

Negative results

We revisit the counterexample for the size-width relation in tree-like Q-Resolution from [6].

Proposition 2

[6]

There is a family of false QBF sentences Φn over O(n2) variables, such that w(Φn)=3 and in tree-like Q-Resolution S(Φn)=nO(1), and w(Φn)=Ω(n).

To prove this proposition, the following QBFs, introduced in [11], are used.Φn=x1,1x1,nxn,nza1an,b1bn,y0yn,p0pni,j=1n(xi,jzai)i,j=1n(¬xi,j¬zbj)¬y0i=1n(yi1¬ai¬yi)yn¬p0j=1n(pj1¬bj¬pj)pn

There are O(n2)-size tree-like Q-Resolution

Relating size and width between tree-like and tree-like level-ordered Q-resolution

Ben-Sasson and Wigderson [3] show that it is possible to construct a narrow (tree-like) Resolution proof from a short Resolution proof. We will show that the same argument can be applied to a level-ordered Q-Resolution refutation, but that the constructed proof may not remain level-ordered. First we examine the reason that the proof must be level-ordered.

Suppose we have a Resolution refutation of some propositional formula ϕ. The final step in the proof resolves x and ¬x. So we also have a

Conclusion

We have demonstrated that the result of [3] can be lifted to relate two variants of Q-Resolution, highlighting an interesting relationship between level-ordered and non level-ordered proofs in Q-Resolution. Level-ordered Q-Resolution is important since it corresponds to the QDPLL algorithm that underlies some modern QBF solving algorithms, so a mechanism to lower bound the size of proofs is useful in understanding the strength of search-based QBF solvers. Removing either the restriction that

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Supported by grant no. 60842 from the John Templeton Foundation.

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