A model for generating random quantified boolean formulas

2006

We investigate geometrical properties of the random K-satisfiability problem using the notion of x-satisfiability: a formula is x-satisfiable is there exist two SAT assignments differing in N x variables. We show the existence of a sharp threshold for this property as a function of the clause density. For large enough K, we prove that there exists a region of clause density, below the satisfiability threshold, where the landscape of Hamming distances between SAT assignments experiences a gap: pairs of SAT-assignments exist at small x, and around x = 1 2 , but they do not exist at intermediate values of x. This result is consistent with the clustering scenario which is at the heart of the recent heuristic analysis of satisfiability using statistical physics analysis (the cavity method), and its algorithmic counterpart (the survey propagation algorithm). Our method uses elementary probabilistic arguments (first and second moment methods), and might be useful in other problems of computational and physical interest where similar phenomena appear.

Theoretical Computer Science, 2008

We investigate geometrical properties of the random K-satisfiability problem using the notion of x-satisfiability: a formula is x-satisfiable is there exist two SAT assignments differing in N x variables. We show the existence of a sharp threshold for this property as a function of the clause density. For large enough K, we prove that there exists a region of clause density, below the satisfiability threshold, where the landscape of Hamming distances between SAT assignments experiences a gap: pairs of SAT-assignments exist at small x, and around x = 1 2 , but they donot exist at intermediate values of x. This result is consistent with the clustering scenario which is at the heart of the recent heuristic analysis of satisfiability using statistical physics analysis (the cavity method), and its algorithmic counterpart (the survey propagation algorithm). The method uses elementary probabilistic arguments (first and second moment methods), and might be useful in other problems of computational and physical interest where similar phenomena appear.

2005

We investigate geometrical properties of the random K-satisfiability problem using the notion of x-satisfiability: a formula is x-satisfiable is there exist two SAT assignments differing in N x variables. We show the existence of a sharp threshold for this property as a function of the clause density. For large enough K, we prove that there exists a region of clause density, below the satisfiability threshold, where the landscape of Hamming distances between SAT assignments experiences a gap: pairs of SAT-assignments exist at small x, and around x = 1 2 , but they donot exist at intermediate values of x. This result is consistent with the clustering scenario which is at the heart of the recent heuristic analysis of satisfiability using statistical physics analysis (the cavity method), and its algorithmic counterpart (the survey propagation algorithm). The method uses elementary probabilistic arguments (first and second moment methods), and might be useful in other problems of computational and physical interest where similar phenomena appear.

2002

Abstract We study the satisfiability of random Boolean expressions built from many clauses with K variables per clause (K-satisfiability). Expressions with a ratio α of clauses to variables less than a threshold α c are almost always satisfiable, whereas those with a ratio above this threshold are almost always unsatisfiable. We show the existence of an intermediate phase below α c, where the proliferation of metastable states is responsible for the onset of complexity in search algorithms.

2008

Many studies focus on the generation of hard SAT instances. The hardness is usually measured by the time it takes SAT solvers to solve the instances. In this preliminary study, we focus on the generation of instances that have computational properties that are more similar to real-world instances. In particular, instances with the same degree of difficulty, measured in terms of the tree-like resolution space complexity. It is known that industrial instances, even with a great number of variables, can be solved by a clever solver in a reasonable amount of time. One of the reasons may be their relatively small space complexity, compared with randomly generated instances. We provide two generation methods of k-SAT instances, called geometrical and the geo-regular, as generalizations of the uniform and regular k-CNF generators. Both are based on the use of a geometric probability distribution to select variables. We study the phase transition phenomena and the hardness of the generated instances as a function of the number of variables and the base of the geometric distribution. We prove that, with these two parameters we can adjust the difficulty of the problems in the phase transition point. We conjecture that this will allow us to generate random instances more similar to industrial instances, of interest for testing purposes.

Comput. Artif. Intell., 2001

In the last decade a lot of effort has been invested into both theoretical and experimental analysis of SAT phase transition. However, a deep theoretical understanding of this phenomenon is still lacking. Besides, many of experimental results are based on some assumptions that are not supported theoretically. In this paper we introduce the notion of SAT-equivalence and we prove that some restrictions often used in SAT experiments don't make an impact on location of a crossover point. We consider several fixed and random clause length SAT models and relations between them. We also discuss one new SAT model and report on a detected phase transition for it.

Journal of Physics: Conference Series, 2010

Random Boolean formulae, generated by a growth process of noisy logical gates are analyzed using the generating functional methodology of statistical physics. We study the type of functions generated for different input distributions, their robustness for a given level of gate error and its dependence on the formulae depth and complexity and the gates used. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding typical-case phase transitions. Results for error-rates, function-depth and sensitivity of the generated functions are obtained for various gate-type and noise models.

2003

We present a novel approach to solving Quantified Boolean Formulae (QBFs), exploiting the power of stochastic local search methods for SAT. This makes the search process different in some interesting ways from conventional QBF solvers. First, the resulting solver is incomplete, as it can terminate without a definite result. Second, we can take advantage of the high level of optimisations in a conventional stochastic SAT algorithm.

2000

We present a satisability,tester Qsat for quantied,Boolean formulae and a restriction QsatCNF of Qsat to unquantied,conjunctive normal form formulae. Qsat makes use of procedures which replace subformulae of a formula by equivalent formulae. By a sequence of such replacements, the original formula can be simplied to true or false. It may,also be necessary to transform the original formula to

1998

The high computational complexity of advanced reasoning tasks such as belief revision and planning calls for efficient and reliable algorithms for reasoning problems harder than NP. In this paper we propose Evaluate, an algorithm for evaluating Quantified Boolean Formulae, a language that extends propositional logic in a way such that many advanced forms of propositional reasoning, e.g., reasoning about knowledge, can be easily formulated as evaluation of a QBF. Algorithms for evaluation of QBFs are suitable for the experimental analysis on a wide range of complexity classes, a property not easily found in other formalisms. Evaluate is based on a generalization of the Davis-Putnam procedure for SAT, and is guaranteed to work in polynomial space. Before presenting Evaluate, we discuss all the abstract properties of QBFs that we singled out to make the algorithm more efficient. We also briefly mention the main results of the experimental analysis, which is reported elsewhere.