A Gröbner Basis Approach to CNF-Formulae Preprocessing

This paper presents a CNF SAT-formulae transformation technique employing Gröbner bases as a means to analyze the problem structure. Gröbnerbases have been applied in the past for SAT; however, their use was primarily restricted to analyzing entire problems for proof-refutation. In contrast, this technique analyzes limited subsets of problems, and uses the derived Gröbner bases to yield new constraint-information. This information is then used to reduce problem structure, provide additional information about the problem itself, or aid other preprocessing techniques. Contrary to the precepts of contemporary techniques, the transformation often increases the problem size. However, experimental results demonstrate that our approach often improves SAT-search efficiency in a number of areas, including: solve time, conflicts, number of decisions, etc. The Boolean Satisfiability Problem (SAT) is formulated as finding solutions satisfying a set of Boolean equations, or to show that no such solutions exist (UNSAT). Such problems are often represented in Conjunctive Normal Form (CNF), whereby sets of literal-disjunctions (clauses) must be simultaneously satisfied through some variable assignment. Solving for SAT-problems involves SAT-solvers. Most are based on the DavisPutnam [1] and Davis-Logemann-Loveland [2] procedures (DPLL), which performs recursive branching and unit propagation over clauses. This technique is aided by concepts such as constraint-propagation [3], conflict analysis [4], and learning [5], which enable non-chronological backtracking [4] [6] [7], pruning the search space and reducing overall search time. The SAT solving-time is not, however, merely a function of the variables and constraints that form the core SAT problem. Problem-representation, especially in CNF, can affect how SAT-solving performs. This is especially true when SAT-instances are transformed from system designs, for validation purposes. In automated conversion, utilities can produce “unoptimized” instances for CNF—those with constraints and variables that do not provide useful information. As a result, time and resources are wasted. A recent area of research has therefore formed around CNF-formula transformation and simplification. This approach diverges from, or should be said, complements “classical” approaches to SAT-solving, those based on DPLL-solving [2] [4] [6] [7], This work is supported, in part, by a Faculty Early Career Development (CAREER) grant from the US National Science Foundation, contract No. CCF-546859. 2 Christopher Condrat and Priyank Kalla by attacking the SAT problem at its source: the constraints. Algorithms and techniques such as [8] [9] [10] simplify and transform CNF-constraints through methods such as clause subsumption, hyper-resolution, and variable elimination. The applicability of such approaches varies between problems; however, most problems can benefit from at least some level of CNF preprocessing. The goal of the preprocessor is to make the problem easier to solve, not necessarily reduce the problem size. A smaller problem implies, at the very least, that there is less the SAT-solver needs to process. However, this does not necessarily imply it is easier to solve. Indeed, some of the hardest problems, are those which have no “redundant” information present in the problem [11]. Additional information may also help clue the solver into the actual purpose of the structures in the problem represent, especially in Hybrid solvers [12]. Even the abilities of rewrite-rule-based CNF-preprocessors can be affected if constraints do not fit their simplification templates. Preprocessing approaches have traditionally concentrated on reducing the overhead— the time needed to consider constraint information—of SAT problems prior to performing the SAT search. This “overhead” comes in the form of constraints and variables that can be represented in simpler forms, or eliminated altogether. Techniques such as Hypre [8], NiVER [9], and SatELite [10] reduce this overhead through resolution-based preprocessing. 1.1 Contemporary Preprocessing Approaches HyPre [8] employs a form of binary reasoning, called “hyper-binary resolution,” in addition to the techniques found in previous preprocessors such as 2-Simplify [13]. Hyperbinary resolution performs a resolution step involving more than two input clauses. A single size-n clause and (n− 1)-binary clauses are resolved to a form that aids in SATsearch. HyPre’s ability to resolve sets of clauses to simpler forms has been relatively successful, but at the same time it can be slow. NiVER and SatELite use resolution to eliminate variables from a SAT-instance. Variable elimination, the older cousin to DPLL, finds itself on the other end of SAT solving from DPLL, where space, as opposed to time, increases exponentially. Given a variable x, and two clauses, containing the variable and its negation respectively, performing resolution on x represents the following: (x ∨ a1 ∨ . . . ∨ an) ⊗ (x′ ∨ b1 ∨ . . . ∨ bm) = (a1 ∨ . . . ∨ an ∨ b1 ∨ . . . ∨ bm) (1) where ⊗ is the resolution operator. Variable elimination is performed by resolving for a variable on all clauses that contain it as follows (for variable v): Cv = ClausesContainingLiteral(v) Cv′ = ClausesContainingLiteral(v′) Cv ⊗ v Cv′ = {c1 ⊗ c2 | c1 ∈ Cv, c2 ∈ Cv′} (2) The variable is eliminated, but at the cost of more constraints than the original set, increasing the problem size in the general case. NiVER [9] stands for “Non-Increasing Variable Elimination Resolution.” This technique attempts to overcome the size-explosion problem associated with variable elimination by only eliminating variables in a way that does not increase problem size. Some A Gröbner Basis Approach to CNF-formulae Preprocessing 3 constraint-sets resulting from variable elimination contain tautologies, which may be removed, resulting in a constraint set equal to or smaller than the original—hence “nonincreasing” variable elimination. SatELite [10] improves on NiVER by combining binary clause resolution simplification with non-increasing variable-elimination, adding new resolution rules for clause subsumption. Clause subsumption proves to be useful for simplifying clauses resulting from variable elimination, enabling an efficient clause-variable simplification procedure which can be repeated until no more reductions are possible. 2 Gröbner bases for CNF-transformation This paper presents a new CNF-formula transformation approach, exploiting the power of polynomial ring algebra, particularly Gröbner bases, to transform CNF-constraints. Gröbner bases provide a computational means to derive reduced bases of sets of polynomials. The resulting polynomials better represent the solution-set and while this process can introduce new constraints and new variables, but the problem itself is simplified— easier to solve. We show that deriving extra constraint information during preprocessing may actually improve performance. The application of Gröbner bases for SAT is nothing new. Proof-systems, such as the Nullstellensatz [14] and Gröbner/Polynomial Calculus [15] proof-systems, introduced in the mid-90s, used Gröbner bases as a means to derive proof-refutations, by generating a unit ideal from the polynomials representing the problem [16]. However, refuting an entire problem using Gröbner bases can be a time consuming, and often infeasible, task. Despite the many improvements to computation algorithms, Gröbner bases systems still have potential exponential time and space complexity, especially when analyzing large problems. Analogous to the techniques forming the foundation of Hypre, NiVER, and SatELite, a single, partial application of Gröbner bases to a SAT instance may improve solving and problem performance. By only transforming parts of the CNF-structure, the problem, as a whole, can benefit from the reduction capabilities of Gröbner bases, while leaving alone parts which are computationally infeasible. This overall strategy fits well into a CNF-SAT transformation framework, through assisting dedicated SAT-solving tools, much as contemporary CNF-preprocessors do.