Eugene Goldberg’s research while affiliated with University of California, Berkeley and other places

Most successful systematic SAT-solvers are descendants of the DPLL procedure and so operate on partial assignments. Using partial assignments is explained by the “enumerative semantics” of the DPLL procedure. Current clause learning SAT-solvers, in a sense, have outgrown this semantics. Instead of enumerating the search space as the DPLL procedure does, they explicitly build a resolution proof. In this paper, we suggest a semantics that, in our opinion, is more suitable for clause learning SAT-solvers. The idea is to consider a set of complete assignments not just as a part of the search space but as an “encryption” of a resolution proof or a part thereof. Importantly, a set of points encrypting a resolution proof can be dramatically smaller than the entire search space. We introduce a resolution-based SAT-solver with clause learning called FI (short for Find point Image of a proof) that is inspired by the new semantics. FI operates on complete assignments. We compare our naive implementation of FI with Minisat and BerkMin. Experiments show that FI is competitive with Minisat and BerkMin in terms of backtracks. In terms of performance, FI is slower than Minisat and BerkMin for small CNF formulas. On the other hand, even the current primitive implementation of FI is competitive with Minisat and BerkMin on large bounded model checking formulas due to its superior decision making.

We describe a new decision-making procedure for resolution-based SAT-solvers called Decision Making with a Reference Point (DMRP). In DMRP, a complete assignment called a reference point is maintained. DMRP is aimed at finding a change of the reference point under which the number of clauses falsified by the modified point is smaller than for the original one. DMRP makes it possible for a DPLL-like algorithm to perform a ”local search strategy”. We describe a SAT-algorithm with conflict clause learning that uses DMRP. Experimental results show that even a straightforward and unoptimized implementation of this algorithm is competitive with SAT-solvers like BerkMin and Minisat on practical formulas. Interestingly, DMRP is beneficial not only for satisfiable but also for unsatisfiable formulas.

Simulation and formal verification are two complementary techniques for checking the correctness of hardware and software designs. Formal verification proves that a design property holds for all points of the search space while simulation checks this property by probing the search space at a subset of points. A known fact is that simulation works surprisingly well taking into account the negligible part of the search space covered by test points. We explore this phenomenon by the example of the satisfiability problem (SAT). We believe that the success of simulation can be understood if one interprets a set of test points not as a sample of the search space, but as an "encryption" of a formal proof. We introduce the notion of a sufficient test set of a CNF formula as a test set encrypting a formal proof that this formula is unsatisfiable. We show how sufficient test sets can be built. We discuss applications of tight sufficient test sets for testing technological faults (manufacturing testing) and design changes (functional verification) and give some experimental results.

This chapter consists of two parts. In the first part we show that resolution based SAT-solvers cannot be scalable on real-life formulas unless some extra information about formula structure is known. In the second part we introduce a new way of satisfiability testing that may be used for designing more efficient and “intelligent” SAT-algorithms that will be able to take into account formula structure.

We describe a SAT-solver, BerkMin, that inherits such features of GRASP, SATO, and Chaff as clause recording, fast BCP, restarts, and conflict clause “aging”. At the same time BerkMin introduces a new decision-making procedure and a new method of clause database management. We experimentally compare BerkMin with Chaff, the leader among resolution-based SAT-solvers. Experiments show that our program is more robust than Chaff being able to solve more instances than Chaff in a reasonable amount of time.

Determinization" of resolution is usually done by employing a DPLL-like procedure that operates on partial assignments. We introduce a resolution-based SAT-solver that operates on complete assignments and give a theoretical justification for determinizing resolution in such a way. This justification is based on the notion of a point image of a resolution proof. We give experimental results confirming the viability of our approach to resolution determinization.

We consider the problem of equivalence checking of circuits N 1,N 2 with a common specification (CS). We show that circuits N 1 and N 2 have a CS iff they can be partitioned into toggle equivalent subcircuits that are connected “in the same way”. Based on this result, we formulate a procedure for checking equivalence of circuits N 1 and N 2 with specifications S 1 and S 2. This procedure not only checks equivalence of N 1 and N 2 but also verifies that S 1 and S 2 are identical. The complexity of this procedure is linear in specification size and exponential in the value of a specification parameter. Previously we considered specifications parameterized by the size of the largest subcircuit (specification granularity). In this paper we give a more general parameterization based on specification “width”.

We introduce a parameterized class M(p) of unsatisfiable formulas that specify equivalence checking of Boolean circuits. If the parameter p is fixed, a formula of M(p) can be solved in general resolution in a linear number of resolutions. On the other hand, even though there is a polynomial time deterministic algorithm that solves formulas from M(p), the order of the polynomial is a monotone increasing function of parameter p. We give reasons why resolution based SAT-algorithms should have poor performance on this very “easy” class of formulas and provide experimental evidence that this is indeed the case.

We introduce the notion of a Common Specification of circuits that generalizes the current notion of structural similarity. A CS S of circuits N1 and N2 is a circuit of multi-valued blocks from which N1 and N2 can be produced by binary encoding. We show that the equivalence checking of N1,N2 in general resolution (which a non-deterministic proof system) is linear in the number of blocks in S. However, there are reasons to believe that equivalence checking of circuits N1,N2 is hard for a deterministic algorithm if their CS is not known. We give some experimental data that substantiates this conjecture.

We show that a conjunctive normal form (CNF) formula F is unsatisfiable iff there is a set of points of the Boolean space that is stable with respect to F. So testing the satisfiability of a CNF formula reduces to looking for a stable set of points ( SSP). We give a simple algorithm for constructing a set of points that is stable with respect to a given set of clauses. Constructing an SSP can be viewed as a "natural""way of search space traversal. This naturalness of search space examination allows one to make use of the regularity of CNF formulas to be checked for satisfiability. We illustrate this point by showing that if a CNF formula is symmetric with respect to a group of permutations, it is very easy to make use of this symmetry when constructing an SSP. As an example, we show that the unsatisfiability of pigeon-hole CNF formulas can be proven by examining only a linear size set of points that can be constructed in quadratic time.