[Paper Review] Removal of Quantifiers by Elimination of Boundary Points

Eugene Goldberg, Panagiotis Manolios|arXiv (Cornell University)|Apr 8, 2012

Formal Methods in Verification15 references3 citations

TL;DR

This paper introduces a novel quantifier elimination method for Boolean CNF formulas by eliminating boundary points through resolvent clauses and dependency sequents (D-sequents). The approach ensures compositional, scalable QEP solving by terminating when all removable boundary points are removed, significantly reducing clause explosion and outperforming traditional methods in model checking benchmarks.

ABSTRACT

We consider the problem of elimination of existential quantifiers from a Boolean CNF formula. Our approach is based on the following observation. One can get rid of dependency on a set of variables of a quantified CNF formula F by adding resolvent clauses of F eliminating boundary points. This approach is similar to the method of quantifier elimination described in [9]. The difference of the method described in the present paper is twofold: {\bullet} branching is performed only on quantified variables, {\bullet} an explicit search for boundary points is performed by calls to a SAT-solver Although we published the paper [9] before this one, chrono- logically the method of the present report was developed first. Preliminary presentations of this method were made in [10], [11]. We postponed a publication of this method due to preparation of a patent application [8].

Motivation & Objective

• To address the inefficiency and non-compositional nature of existing QEP solvers that rely on enumerating satisfying assignments or generating excessive resolvents.
• To develop a termination condition for quantifier elimination that avoids exponential blowup in formula size while ensuring correctness.
• To enable compositional QEP solving by identifying and eliminating redundant variables via removable boundary points.
• To introduce D-sequents as a formal mechanism to record and propagate variable redundancy across subformulas.
• To improve scalability and performance in hardware and software verification tasks, particularly in model checking.

Proposed method

• The method identifies X′-boundary points in a CNF formula F(X,Y), where a complete assignment falsifies F and every falsified clause contains a variable from X′, with minimality enforced over subsets of X′.
• A boundary point is removable if no satisfying assignment of F can be obtained by flipping variables in X; such points are eliminated by adding resolvent clauses that do not involve variables from X′.
• The algorithm performs branching only on quantified variables X, using a SAT solver to explicitly search for boundary points.
• It maintains and derives D-sequents of the form (F, X′, q) → X′′, where q is a partial assignment to X, to record that variables in X′′ are redundant in F after q is applied.
• The algorithm terminates when no removable X′-boundary points remain, and the final formula F*(Y) is obtained by removing all clauses containing variables from X.
• The D-sequent framework enables compositional reasoning by allowing independent processing of subformulas when no boundary points remain in a subspace.

Experimental results

Research questions

• RQ1 Can a termination condition for QEP be designed that avoids the exponential blowup of enumerating satisfying assignments while ensuring correctness?
• RQ2 Can the notion of boundary points be generalized beyond single variables to arbitrary subsets X′ ⊆ X to support compositional reasoning?
• RQ3 Can dependency sequents (D-sequents) be used to formally capture and propagate variable redundancy in quantifier elimination?
• RQ4 Does a QEP solver based on boundary point elimination and D-sequents achieve better scalability and performance than existing methods in practical verification tasks?
• RQ5 To what extent does the new method preserve or improve compositional behavior compared to BDD-based or resolution-based solvers?

Key findings

• The proposed DDS algorithm solved more benchmarks than DP and EnumSA in forward model checking and dramatically more in backward model checking on the HWMCC’10 suite.
• DDS_impl solved 758 benchmarks in backward model checking, outperforming EnumSA and DP, though it required more time per benchmark due to solving the most complex instances.
• The method is compositional because it can independently process subformulas when no removable boundary points exist in a subspace, enabling efficient handling of decomposable formulas.
• The use of boundary point elimination reduces the number of resolvents generated compared to the DP procedure, leading to improved efficiency.
• The D-sequent framework allows for a formal, sound, and terminating derivation of variable redundancy, avoiding the need for exhaustive assignment enumeration.
• Preliminary experiments with a simple implementation show exponential performance gains in cases where the formula is composed of independent subformulas.