Eugene Goldberg

Boundary Points and Resolution

We use the notion of boundary points to study resolution proofs. Given a CNF formula F , a lit (x )-boundary point is a complete assignment falsifying only clauses of F having the same literal lit (x ) of variablex . A lit (x )-boundary point mandates a resolution on variable x . Adding the resolvent of this resolution to F eliminates this boundary point. Any resolution proof has to eventually eliminate all boundary points of F . Hence one can study resolution proofs from the viewpoint of boundary point elimination. We use equivalence checking formulas to compare proofs of their unsatisfiability built by a conflict driven SAT-solver and very short proofs tailored to these formulas. We show experimentally that in contrast to proofs generated by this SAT-solver, almost every resolution of a specialized proof eliminates a boundary point. This implies that one may use the share of resolutions eliminating boundary points as a metric for proof quality.

Sat-solving based on boundary point elimination

We study the problem of building structure-aware SAT-solvers based on resolution. In this study, we use the idea of treating a resolution proof as a process of Boundary Point Elimination (BPE). We identify two problems of using SAT-algorithms with Conflict Driven Clause Learning (CDCL) for structure-aware SAT-solving. We introduce a template of resolution based SAT-solvers called BPE-SAT that is based on a few generic implications of the BPE concept. BPE-SAT can be viewed as a generalization of CDCL SAT-solvers and is meant for building new structure-aware SAT-algorithms. We give experimental results substantiating the ideas of the BPE approach. In particular, to show the importance of structural information we compare an implementation of BPE-SAT and state-of-the-art SAT-solvers on narrow CNF formulas.

Quantifier elimination via clause redundancy

We consider the problem of existential quantifier elimination for Boolean formulas in conjunctive normal form. Recently we presented a new method for solving this problem based on the machinery of Dependency sequents (D-sequents). The essence of this method is to add to the quantified formula implied clauses until all the clauses with quantified variables become redundant. A D-sequent is a record of the fact that a set of quantified variables is redundant in some subspace. In this paper, we introduce a quantifier elimination algorithm based on a new type of D-sequents called clause D-sequents that express redundancy of clauses rather than variables. Clause D-sequents significantly extend our ability to introduce and algorithmically exploit redundancy, as our experimental results show.

Equivalence checking by logic relaxation

We introduce a new framework for Equivalence Checking (EC) of Boolean circuits based on a general technique called Logic Relaxation (LoR). LoR is meant for checking if a propositional formula G has only “good” satisfying assignments specified by a design property. The essence of LoR is to relax G into a formula Grlx and compute a set S that contains all assignments that satisfy Grlx but do not satisfy G. If all bad satisfying assignments are in S, formula G can have only good ones and the design property in question holds. Set S is built by a procedure called partial quantifier elimination. The appeal of EC by LoR is twofold. First, it facilitates generation of powerful inductive proofs. Second, proving inequiv-alence comes down to checking the existence of some assignments satisfying Grlx i.e. a simpler version of the original formula. We give experimental evidence that supports our approach.

Partial Quantifier Elimination

We consider the problem of Partial Quantifier Elimination (PQE). Given formula ∃ X[F(X,Y) ∧ G(X,Y)], where F, G are in conjunctive normal form, the PQE problem is to find a formula F *(Y) such that F * ∧ ∃ X[G] ≡ ∃ X[F ∧ G]. We solve the PQE problem by generating and adding to F clauses over the free variables that make the clauses of F with quantified variables redundant in ∃ X[F ∧ G]. The traditional Quantifier Elimination problem (QE) can be viewed as a degenerate case of PQE where G is empty so all clauses of the input formula with quantified variables need to be made redundant. The importance of PQE is threefold. First, in non-degenerate cases, PQE can be solved more efficiently than QE. Second, many problems are more naturally formulated in terms of PQE rather than QE. Third, an efficient PQE-algorithm will enable new methods of model checking and SAT-solving. We describe a PQE algorithm based on the machinery of dependency sequents and give experimental results showing the promise of PQE.

Software for Quantifier Elimination in Propositional Logic

We consider the following problem of Quantifier Elimination (QE). Given a Boolean CNF formula F where some variables are existentially quantified, find a logically equivalent CNF formula that is free of quantifiers. Solving this problem comes down to finding a set of clauses depending only on free variables that has the following property: adding the clauses of this set to F makes all the clauses of F with quantified variables redundant. To solve the QE problem we developed a tool meant for handling a more general problem called partial QE. This tool builds a set of clauses adding which to F renders a specified subset of clauses with quantified variables redundant. In particular, if the specified subset contains all the clauses with quantified variables, our tool performs QE.