Formulas In Conjunctive Normal Form Research Articles

A satisfiability procedure for quantified Boolean formulae

We present a satisfiability tester Q SAT for quantified Boolean formulae and a restriction Q SAT CNF of Q SAT to unquantified conjunctive normal form formulae. Q SAT makes use of procedures which replace subformulae of a formula by equivalent formulae. By a sequence of such replacements, the original formula can be simplified to true or false . It may also be necessary to transform the original formula to generate a subformula to replace. Q SAT CNF eliminates collections of variables from an unquantified clause form formula until all variables have been eliminated. Q SAT and Q SAT CNF can be applied to hardware verification and symbolic model checking. Results of an implementation of Q SAT CNF are described, as well as some complexity results for Q SAT and Q SAT CNF . Q SAT runs in linear time on a class of quantified Boolean formulae related to symbolic model checking. We present the class of “long and thin” unquantified formulae and give evidence that this class is common in applications. We also give theoretical and empirical evidence that Q SAT CNF is often faster than Davis and Putnam-type satisfiability checkers and ordered binary decision diagrams (OBDDs) on this class of formulae. We give an example where Q SAT CNF is exponentially faster than BDDs.

The Efficiency of Resolution and Davis--Putnam Procedures

We consider several problems related to the use of resolution-based methods for determining whether a given boolean formula in conjunctive normal form is satisfiable. First, building on the work of Clegg, Edmonds, and Impagliazzo in [Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, Philadelphia, PA, 1996, ACM, New York, 1996, pp. 174--183], we give an algorithm for unsatisfiability that when given an unsatisfiable formula of F finds a resolution proof of F. The runtime of our algorithm is subexponential in the size of the shortest resolution proof of F. Next, we investigate a class of backtrack search algorithms for producing resolution refutations of unsatisfiability, commonly known as Davis--Putnam procedures, and provide the first asymptotically tight average-case complexity analysis for their behavior on random formulas. In particular, for a simple algorithm in this class, called ordered DLL, we prove that the running time of the algorithm on a randomly generated k-CNF formula with n variables and m clauses is $2^{\Theta(n(n/m)^{1/(k-2)})}$ with probability $1-o(1)$. Finally, we give new lower bounds on $\mbox{res}(F)$, the size of the smallest resolution refutation of F, for a class of formulas representing the pigeonhole principle and for randomly generated formulas. For random formulas, Chvatal and Szemeredi [J. ACM, 35 (1988), pp. 759--768] had shown that random 3-CNF formulas with a linear number of clauses require exponential size resolution proofs, and Fu [On the Complexity of Proof Systems, Ph. D. thesis, University of Toronto, Toronto, ON, Canada, 1995] extended their results to k-CNF formulas. These proofs apply only when the number of clauses is $\Omega(n \log n)$. We show that a lower bound of the form $2^{n^{\gamma}}$ holds with high probability even when the number of clauses is $n^{(k+2)/4-\epsilon}$.

Solving satisfiability problems using reconfigurable computing

This paper reports on an innovative approach for solving satisfiability problems for propositional formulas in conjunctive normal form (SAT) by creating a logic circuit that is specialized to solve each problem instance on field programmable gate arrays (FPGAs). This approach has become feasible due to recent advances in reconfigurable computing and has opened up an exciting new research field in algorithm design. SAT is an important subclass of constraint satisfaction problems, which can formalize a wide range of application problems. We have developed a series of algorithms that are suitable for a logic circuit implementation, including an algorithm whose performance is equivalent to the Davis-Putnam procedure with powerful dynamic variable ordering. Simulation results show that this method can solve a hard random 3-SAT problem with 400 variables within 1.6 min at a clock rate of 10 MHz. Faster speeds can be obtained by increasing the clock rate. Furthermore, we have actually implemented a 128-variable 256-clause problem instance on FPGAs.

On subclasses of minimal unsatisfiable formulas

We consider the minimal unsatisfiablity problem MU( k) for propositional formulas in conjunctive normal form (CNF) over n variables and n+ k clauses, where k is fixed. k is called the difference. Any formula in MU( k) can be split into two minimal unsatisfiable formula. For such splittings we investigate the size of the differences of the resulting formulas in comparison to the difference of the initial formula. Based on these results we prove that MU( k) for fixed k is in NP, and for MU(2) we present a simple and unique characterization.

New Upper Bounds for Maximum Satisfiability

The (unweighted) Maximum Satisfiability problem (MaxSat) is: Given a Boolean formula in conjunctive normal form, find a truth assignment that satisfies the largest number of clauses. This paper describes exact algorithms that provide new upper bounds for MaxSat. We prove that MaxSat can be solved in time O(|F|·1.3803K), where |F| is the length of a formula F in conjunctive normal form and K is the number of clauses in F. We also prove the time bounds O(|F|·1.3995k), where k is the maximum number of satisfiable clauses, and O(1.1279|F|), for the same problem. For Max2Sat this implies a bound of O(1.2722K).

Resolution remains hard under equivalence

The basis of this paper is the observation that for several proof systems for propositional formulas in conjunctive normal form there are families of hard examples for which the shortest proof requires superpolynomially many steps. We will show that – under the assumption NP≠coNP – it is impossible to transform formulas into a logically equivalent formula by adding polynomially many clauses such that it can be decided in polynomial time whether a clause is a consequence of the enlarged formula. This result especially holds for resolution, i.e. for all polynomials p and q there exists a formula Φ∈ CNF, such that for each equivalent formula Ψ∈ CNF with | Ψ|⩽ q(| Φ|) there is a clause π with Φ⊨π for which the shortest resolution proof Ψ ⊢ RES π requires more than p(| Φ|+| π|) resolution steps.

Complexity analysis of propositional resolution with autarky pruning

An algorithm called “Modoc”, which has been introduced elsewhere, enhances propositional model elimination with autarky pruning, and other features. The model elimination method is based on linear resolution, and is designed to produce refutations of formulas in conjunctive normal form (CNF). Informally, an autarky is a “self-sufficient” model for some clauses, but which does not affect the remaining clauses of the formula. Modoc finds a model if it fails to find a refutation, essentially by combining autarkies. Although the original motivation for autarky pruning was to extract a model when the refutation attempt failed, practical experience has shown that it also greatly increases the performance, by reducing the amount of search redundancy. This paper presents a worst-case analysis of Modoc as a function of the number of propositional variables in the formula. The analysis sheds light on why autarky pruning improves the performance, compared to “standard” model elimination. A worst-case analysis of the original algorithm of Davis, Putnam, Loveland and Logemann (DPLL) is also presented. The Modoc analysis yields a worst-case upper bound that is not as strong as the best known upper bound for model-searching satisfiability methods, on general propositional CNF. However, it is the first time a nontrivial upper bound on non-Horn formulas has been shown for any resolution-based refutation procedure.

Cutting seriation for approximate #SAT resolution

We propose here a general method for approximating the number of solutions of a boolean formula in conjunctive normal form F. By applying the principle "divise to resolve", this method reduces considerably the computational complexity. It is based on cutting a seriation established on an incidence data table associated with F. Moreover, the independence probability concept is finely exploited. Theoretical justification and intensive experimentation validate the proposed method.

Learning to ask relevant questions

This paper describes an effective technique for relevant questioning in expert systems whose knowledge base is encoded in a propositional formula in conjunctive normal form. The methodology does not require initial knowledge about the relationships between questions. Instead, the system learns such relationships over time as follows. After each session, the system analyzes its questioning, deduces how it could have obtained each conclusion without asking irrelevant questions, and records the relevant questions and answers in so-called processed dialogues. When a question is to be selected in a subsequent session, the system measures the relevancy of questions using the processed dialogues, ranks the questions according to that measure, and asks the highest-ranked question next. We have used the methodology in an expert system that handles industrial chemical exposure management. In that application, the system learned rather quickly to ask relevant questions and became just as effective as a human expert.

Recent Advances in AI Planning

The past five years have seen dramatic advances in planning algorithms, with an emphasis on propositional methods such as GRAPHPLAN and compilers that convert planning problems into propositional conjunctive normal form formulas for solution using systematic or stochastic SAT methods. Related work, in the context of spacecraft control, advances our understanding of interleaved planning and execution. In this survey, I explain the latest techniques and suggest areas for future research.

An Efficient Algorithm to Generate Prime Implicants

In this paper, an efficient recursive algorithm is presented to compute the set of prime implicants of a propositional formula in conjunctive normal form (CNF). The propositional formula is represented as a (0,1)-matrix, and a set of 1’s across its columns are termed as paths. The algorithm finds the prime implicants as the prime paths in the matrix using the divide-and-conquer technique. The algorithm is based on the principle that the prime implicant of a formula is the concatenation of the prime implicants of two of its subformulae. The set of prime paths containing a specific literal and devoid of a literal are characterized. Based on this characterization, the formula is recursively divided into subformulae to employ the divide-and-conquer paradigm. The prime paths of the subformulae are then concatenated to obtain the prime paths of the formula. In this process, the number of subsumption operations is reduced. It is also shown that the earlier algorithm based on prime paths has some avoidable computations that the proposed algorithm avoids. Besides being more efficient, the proposed algorithm has the additional advantage of being suitable for the incremental method, without recomputing prime paths for the updated formula. The subsumption operation is one of the crucial operations for any such algorithms, and it is shown that the number of subsumption operation is reduced in the proposed algorithm. Experimental results are presented to substantiate that the proposed algorithm is more efficient than the existing algorithms.

Computation of prime implicants using matrix and paths

In this paper, an efficient algorithm to compute the set of prime implicants of a propositional formula in Conjunctive Normal Form (CNF) is presented. The proposed algorithm uses a concept of representing the formula as a binary matrix and computing paths through the matrix as implicants. The algorithm finds the prime implicants as the prime paths using the divide-and-conquer technique. The proposed algorithm can be used for knowledge compilation, Clause Maintenance Systems where the knowledge base is propositional formulae. Moreover, the algorithm is easily adaptable to the incremental mode of computation where an earlier formula is updated by a set of clauses.

A note on the monotone complexity of 2-REF

Sharpening an argument in (Cook and Luby, 1988), we show that the problem of recognizing refutable propositional logic formulas in conjunctive normal form with two disjuncts is monotonic p-projection equivalent to the problem st-DCON of directed connectivity between two distinguished vertices s, t of a directed graph. It follows by (Karchmer and Wigderson, 1990) that the monotonic depth of bounded fan-in boolean circuits for 2- ref is Θ(log 2( n)).

Cutting planes, connectivity, and threshold logic

Originating from work in operations research the cutting plane refutation systemCP is an extension of resolution, where unsatisfiable propositional logic formulas in conjunctive normal form are recognized by showing the non-existence of boolean solutions to associated families of linear inequalities. Polynomial sizeCP proofs are given for the undirecteds-t connectivity principle. The subsystemsCPq ofCP, forq≥2, are shown to be polynomially equivalent toCP, thus answering problem 19 from the list of open problems of [8]. We present a normal form theorem forCP2-proofs and thereby for arbitraryCP-proofs. As a corollary, we show that the coefficients and constant terms in arbitrary cutting plane proofs may be exponentially bounded by the number of steps in the proof, at the cost of an at most polynomial increase in the number of steps in the proof. The extensionCPLE+, introduced in [9] and there shown top-simulate Frege systems, is proved to be polynomially equivalent to Frege systems. Lastly, since linear inequalities are related to threshold gates, we introduce a new threshold logic and prove a completeness theorem.

Convergence properties of optimization algorithms for the SAT problem

The satisfiability (SAT) problem is a basic problem in computing theory. Presently, an active area of research on SAT problem is to design efficient algorithms to find a solution for a satisfiable conjunctive normal form (CNF) formula. A new formulation, the universal SAT problem model, which transforms the SAT problem on Boolean space into an optimization problem on real space has been developed (J. Gu, 1988; 1992; 1994). Many optimization techniques, such as the steepest descent method, Newton's method, and the coordinate descent method, can be used to solve the universal SAT problem. We prove that when the initial solution is sufficiently close to the optimal solution, the steepest descent method has a linear convergence ratio /spl beta/<1, Newton's method has a convergence ratio of order two, and the convergence ratio of the steepest descent method is approximately (1-/spl beta//m) for the universal SAT problem with m variables. An algorithm based on the coordinate descent method for the universal SAT problem is also presented. Experimental results show that this algorithm is more efficient than some previous ones in finding a solution for certain classes of the satisfiable CNF formulas.

A Relationship Between CNF and DNF Systems Derivable from Examples

In learning from examples, the main goal is to use a collection of positive examples and a collection of negative examples to derive a Boolean expression which satisfies the requirements imposed by the examples. In order to represent such a Boolean expression, the conjunctive normal form (CNF) and the disjunctive normal form (DNF) have been proposed. This paper makes two contributions. First it shows how to use any DNF algorithm to derive a CNF formula (or vice-versa). Furthermore, it demonstrates how to make efficient use of DNF algorithms which cannot handle a large number of positive (or negative) examples by using them as negative (or positive) examples and deriving CNF (or vice-versa). Therefore, the findings of this paper can be used to solve efficiently large scale learning problems. Two learning algorithms are used to illustrate the above issues. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

BREAKUP: a preprocessing algorithm for satisfiability testing of CNF formulas.

An algorithm called BREAKUP, which processes CNF formulas by separating them into components, is introduced. BREAKUP is then used to speed up the testing of some first-order formulas for satisfi- ability using Iwama's IS Algorithm. The complexity of this algorithm is shown to be on the order of O(nc-nv), where nc is the number of clauses and nv is the number of variables. 1 Introduction Connectedness is an attribute of graphs that can be useful in deciding the satisfiability of propositional formulas in conjunctive normal form (CNF). Preprocessing to separate a formula into components is generally fast and easy, and can significantly speed up testing of the formula for satisfiability. BREAKUP, the algorithm introduced here, will be used in con- junction with Iwama's IS Algorithm (4),(5) and will be shown to significantly speed up (as compared to using IS alone) the processing of some formulas from Gilmore (3) and Davis and Putnam (2). A CNF formula is a conjunction of clauses where each clause is a disjunc- tion of literals. A literal is a propositional variable or the negation of a propo- sitional variable. Given a pair of literals {p9-*p}9 each will be called the mate of the other (cf. Davis (1)). Clearly, we can assume that no clause contains both a literal and its mate. There are two ways to think of clauses in a formula as con- nected. First, two clauses c,d, are connected if there exists a chain of clauses starting with c and ending with d such that for any two adjacent clauses x,y, either y contains a literal that occurs in x or y contains the mate of a literal that occurs in x. Alternatively, c and d can be said to be connected if there is a chain of clauses starting with c and ending with d such that if x and y are adjacent then y contains the mate of a literal that occurs in x. In either case, a set of clauses is said to be connected if every pair in the set is connected and a component is a maximally connected set of the clauses of the formula.

A hierarchy of tractable satisfiability problems

Propositional satisfiability (SAT) is a well-known NP-complete problem. We define a hierarchy Ω 0, Ω 1,… of classes of formulae such that for any class Ω k , SAT is solvable in O( n k + 1 ) time. The basic class Ω 0 contains all formulae in conjunctive normal form (CNF) where each conjunct is a Horn clause, and allformulae in CNF where each conjunct is a binary clause. This hierarchy improves upon the hierarchy defined by Gallo and Scutellá. We also present a sound and complete algorithm for solving SAT that takes time O( n k + 1 ) for any formula in the class Ω k .

Exact satisfiability, a natural extension of set partition, and its average case behavior

The problem of determining whether a Boolean formula in conjunctive normal form is satisfiable in such a way that in each clause exactly one literal is set true, and all the other literals are set false is called the exact satisfiability problem. The exact satisfiability problem is well known to be NP-complete [5] and it contains the well known set partitioning problem as a special case. We study here the average time complexity of a simple backtracking strategy for solving the exact satisfiability problem under two probability models, the constant density model and the constant degree model. For both models we present results sharply separating classes of instances solvable in low degree polynomial time in the average from classes for which superpolynomial or exponential time is needed in the average.

Davis-Putnam resolution versus unrestricted resolution

A resolution proof of an unsatisfiable propositional formula in conjunctive normal form is a “Davis-Putnam resolution proof” iff there exists a sequence of the variables of the formula such thatx is eliminated (with the resolution rule) beforey on any branch of the proof tree representing the resolution proof, only ifx is beforey in this sequence. Davis-Putnam resolution is one of several resolution restrictions. It is complete. We present an infinite family of unsatisfiable propositional formulas and show: These formulas have unrestricted resolution proofs whose length is polynomial in their size. All Davis-Putnam resolution proofs of these formulas are of superpolynomial length. In the terminology of [4, definition 1.5]: Davis-Putnam resolution does notp-simulate unrestricted resolution.