Resolution Proof System Research Articles

Homomorphism Proof Systems for Unsatisfiable Formulas

A homomorphism ? of CNF formulas from H to F is a function mapping the set of literals in H to the set of literals in F and it preserves complements and clauses. If the formula H is homomorphic to the formula F, then the unsatisfiability of H implies the unsatisfiability of F. A CNF formula F is minimally unsatisfiable if F is unsatisfiable and the resulting formula deleting any one clause from F is satisfiable. MU(1) is a class of minimally unsatisfiable formulas with the deficiency of the number of clauses and variables to be one. A triple (H,?,F) is called a homomorphism proof from H of F if ? is a homomorphism from H to F. In this paper, a method from the basic matrix of MU(1) formula is used to prove that a tree resolution proof for an unsatisfiable formula F can be transformed into a homomorphism proof from a MU(1) formula for F. Whence, the homomorphism proof system from formulas in MU(1) is complete, and this proof system and the tree resolution proof system can be transformed mutually in polynomial time on the size of proof.

Towards Understanding and Harnessing the Potential of Clause Learning

Efficient implementations of DPLL with the addition of clause learning are the fastest complete Boolean satisfiability solvers and can handle many significant real-world problems, such as verification, planning and design. Despite its importance, little is known of the ultimate strengths and limitations of the technique. This paper presents the first precise characterization of clause learning as a proof system (CL), and begins the task of understanding its power by relating it to the well-studied resolution proof system. In particular, we show that with a new learning scheme, CL can provide exponentially shorter proofs than many proper refinements of general resolution (RES) satisfying a natural property. These include regular and Davis-Putnam resolution, which are already known to be much stronger than ordinary DPLL. We also show that a slight variant of CL with unlimited restarts is as powerful as RES itself. Translating these analytical results to practice, however, presents a challenge because of the nondeterministic nature of clause learning algorithms. We propose a novel way of exploiting the underlying problem structure, in the form of a high level problem description such as a graph or PDDL specification, to guide clause learning algorithms toward faster solutions. We show that this leads to exponential speed-ups on grid and randomized pebbling problems, as well as substantial improvements on certain ordering formulas.

Space complexity of random formulae in resolution

AbstractWe study the space complexity of refuting unsatisfiable random k‐CNFs in the Resolution proof system. We prove that for Δ ≥ 1 and any ϵ > 0, with high probability a random k‐CNF over n variables and Δn clauses requires resolution clause space of Ω(n/Δ1+ϵ). For constant Δ, this gives us linear, optimal, lower bounds on the clause space. One consequence of this lower bound is the first lower bound for size of treelike resolution refutations of random 3‐CNFs with clause density Δ ≫ n. This bound is nearly tight. Specifically, we show that with high probability, a random 3‐CNF with Δn clauses requires treelike refutation size of exp(Ω(n/Δ1+ϵ)), for any ϵ > 0. Our space lower bound is the consequence of three main contributions: (1) We introduce a 2‐player Matching Game on bipartite graphs G to prove that there are no perfect matchings in G. (2) We reduce lower bounds for the clause space of a formula F in Resolution to lower bounds for the complexity of the game played on the bipartite graph G(F) associated with F. (3) We prove that the complexity of the game is large whenever G is an expander graph. Finally, a simple probabilistic analysis shows that for a random formula F, with high probability G(F) is an expander. We also extend our result to the case of G‐PHP, a generalization of the Pigeonhole principle based on bipartite graphs G. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 92–109, 2003

On reducibility and symmetry of disjoint NP pairs

We consider some problems about pairs of disjoint NP sets. The theory of these sets with a natural concept of reducibility is, on the one hand, closely related to the theory of proof systems for propositional calculus, and, on the other, it resembles the theory of NP completeness. Furthermore, such pairs are important in cryptography. Among others, we prove that the Broken Mosquito Screen pair of disjoint NP sets can be polynomially reduced to Clique–Coloring pair and thus is polynomially separable and we show that the pair of disjoint NP sets canonically associated with the Resolution proof system is symmetric.

Removing irrelevant information in temporal resolution proofs

The generation of too much information prohibits efficient resolution proof search in classical logics. Subsumption is used to discard redundant information and strategies have been developed to guide the proof search avoiding irrelevant information. The extension of the resolution method to temporal logics, further magnifies this problem. Here we develop an algorithm for the removal of irrelevant information during resolution proofs for a particular resolution proof system for propositional linear-time temporal logic. The resolution system operates on formulae in a normal form. Following a phase of classical style resolution, temporal resolution is carried out by detecting sets of formulae that together represent an □-formula to be resolved with a ◊-formula. It is before the application of the temporal resolution rule that the new algorithm is applied to efficiently remove irrelevant information. The completeness of the new algorithm is discussed and its complexity considered. Its practical efficiency is demonstrated by giving timings for a set of examples.

A resolution framework for finitely-valued first-order logics

In this paper we propose a resolution proof framework on the basis of which automated proof systems for finitely-valued first-order logics ( FFO logics) can be introduced and studied. We define the notion of a first-order resolution proof system and we show that for every disjunctive FFO logic a refutationally complete resolution proof system can be constructed. Moreover, we discuss two theorem proving strategies, the polarity and set of support strategies, and we prove their completeness.

Resolution approximation of first-order logics

In this paper we investigate automated theorem proving systems represented as finite classes of resolution proof systems. We call such classes resolution approximations. For many first-order logics, theorem proving based on such representation offers a substantial saving of computer time. We show that minimal resolution approximation of finitely-valued first-order logics can be effectively constructed.

Extending resolution to resolution logics

Abstract In previous papers (O'Hearn & Stachniak 1989b, Stachniak & O'Hearn 1990, Stachniak 1989,1990a) we proposed an algebraic methodological framework for the introduction and analysis of resolution proof systems for finitely-valued logical calculi. In the present paper we extend this approach to a wider class of the so-called resolution logics.

Minimization of Resolution Proof Systems

In this paper we show that minimal resolution counterparts of strongly finite logics can be effectively constructed. Moreover, we show that the class of resolution counterparts of structural propositional logics coincides with the class of resolution counterparts of strongly finite logics.

Resolution in the Domain of Strongly Finite Logics

In this paper the notion of a resolution counterpart of a propositional logic is introduced and studied. This notion is based on a generalization of the resolution rule of J.A. Robinson. It is shown that for every strongly finite logic a refutationally complete nonclausal resolution proof system can be constructed and that the completeness of such systems is preserved with respect to the polarity and set of support strategies.

Resolution proofs of generalized pigeonhole principles

We extend results of Haken to give an exponential lower bound on the size of resolution proofs for propositional formulas encoding a generalized pigeonhole principle. These propositional formulas express the fact that there is no one-one mapping from c·n objects to n objects when c >1. As a corollary, resolution proof systems do not p-simulate constant-formula-depth Frege proof systems.