Short Resolution Proofs Research Articles

Toward Verifying Nonlinear Integer Arithmetic

We eliminate a key roadblock to efficient verification of nonlinear integer arithmetic using CDCL SAT solvers, by showing how to construct short resolution proofs for many properties of the most widely used multiplier circuits. Such short proofs were conjectured not to exist. More precisely, we give n O (1) size regular resolution proofs for arbitrary degree 2 identities on array, diagonal, and Booth multipliers and n O (log n ) size proofs for these identities on Wallace tree multipliers.

The Resolution Complexity of Independent Sets and Vertex Covers in Random Graphs

We consider the problem of providing a resolution proof of the statement that a given graph with n vertices and average degree roughly Δ does not contain an independent set of size k. For randomly chosen graphs and k � n/3, we show that such proofs asymptotically almost surely require size roughly exponential in n/Δ6. This, in particular, implies a 2� (n) lower bound for constant degree graphs, and for Δ � n 1/6, shows that there are almost always no short resolution proofs for k as large as n/3 even though a maximum independent set is likely to be much smaller, roughly n 5/6 in size. Our result implies that for graphs that are not too dense, almost all instances of the independent set problem are hard for resolution. Further, it provides an unconditional exponential lower bound on the running time of resolution-based search algorithms for finding a maximum independent set or approximating it within a factor of Δ/(6 ln Δ). We also give relatively simple upper bounds for the problem and show them to be tight for the class of exhaustive backtracking algorithms. We deduce similar complexity results for the related vertex cover problem on random graphs, proving, in particular, that no polynomial-time resolution-based method can achieve an approximation within a factor of 3/2.

Resolution lower bounds for the weak pigeonhole principle

We prove that any Resolution proof for the weak pigeonhole principle, with n holes and any number of pigeons, is of length Ω(2 n ϵ ), (for some global constant ϵ > 0). One corollary is that a certain propositional formulation of the statement NP ⊄ P / poly does not have short Resolution proofs.

Combining satisfiability techniques from AI and OR

The recent effort to integrate techniques from the fields of artificial intelligence and operations research has been motivated in part by the fact that scientists in each group are often unacquainted with recent (and not so recent) progress in the other field. Our goal in this paper is to introduce the artificial intelligence community to pseudo-Boolean representation and cutting plane proofs, and to introduce the operations research community to restricted learning methods such as relevance-bounded learning. Complete methods for solving satisfiability problems are necessarily bounded from below by the length of the shortest proof of unsatisfiability; the fact that cutting plane proofs of unsatisfiability can be exponentially shorter than the shortest resolution proof can thus in theory lead to substantial improvements in the performance of complete satisfiability engines. Relevance-bounded learning is a method for bounding the size of a learned constraint set. It is currently the best artificial intelligence strategy for deciding which learned constraints to retain and which to discard. We believe these two elements or some analogous form of them are necessary ingredients to improving the performance of satisfiability algorithms generally. We also present a new cutting plane proof of the pigeonhole principle that is of size n2, and show how to implement some intelligent backtracking techniques using pseudo-Boolean representation.

Resolution remains hard under equivalence

The basis of this paper is the observation that for several proof systems for propositional formulas in conjunctive normal form there are families of hard examples for which the shortest proof requires superpolynomially many steps. We will show that – under the assumption NP≠coNP – it is impossible to transform formulas into a logically equivalent formula by adding polynomially many clauses such that it can be decided in polynomial time whether a clause is a consequence of the enlarged formula. This result especially holds for resolution, i.e. for all polynomials p and q there exists a formula Φ∈ CNF, such that for each equivalent formula Ψ∈ CNF with | Ψ|⩽ q(| Φ|) there is a clause π with Φ⊨π for which the shortest resolution proof Ψ ⊢ RES π requires more than p(| Φ|+| π|) resolution steps.

Short resolution proofs for a sequence of tricky formulas

A class of propositional formulas, encoding the property that every finite, transitive digraph with no two-cycles must have a source, has been investigated by Krishnamurty and conjectured as hard for resolution. In this note we prove, opposed to that conjecture, that there are proofs of polynomial lengths (or even linear in the lengths of the formulas) of those formulas.