John V. Franco

Probabilistic analysis of the Davis Putnam procedure for solving the satisfiability problem

Abstract We look at the instance distributions used by Goldberg [3] for showing that the Davis Putnam Procedure has polynomial average complexity and show that, in a sense, all these distributions are unreasonable. We then present a ‘reasonable’ family of instance distributions F and show that for each distribution in F a variant of the Davis Putnam Procedure without the pure literal rule requires exponential time with probability 1. In addition, we show that adding subsumption still results in exponential complexity with probability 1.

Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the k satisfiability problem

Abstract Two algorithms for the k-satisfiability problem are presented and a probabilistic analysis is performed. The analysis is based on an instance distribution which is parametrized to simulate a variety of sample characteristics. The algorithms assign values to literals appearing in a given instance of k-satisfiability, one at a time, until a solution is found or it is discovered that further assignments cannot lead to finding a solution. One algorithm chooses the next literal from a unit clause if one exists and randomly from the set of remaining literals otherwise. The other algorithm uses a generalization of the unit-clause rule as a heuristic for selecting the next literal: at each step a literal is chosen randomly from a clause containing the least number of literals. The algorithms run in polynomial time and it is shown that they find a solution to a random instance of k-satisfiability with probability bounded below by a constant greater than zero for two different ranges of parameter values. It is also shown that the second algorithm mentioned finds a solution with probability approaching one for a wide range of parameter values.

Probabilistic analysis of two heuristics for the 3-satisfiability problem

An algorithm for the 3-Satisfiability problem is presented and a probabilistic analysis is performed. The analysis is based on an instance distribution which is parameterized to simulate a variety of sample characteristics. The algorithm assigns values to variables appearing in a given instance of 3-Satisfiability, one at a time, using the unit clause heuristic and a maximum occurring literal selection heuristic; at each step a variable is chosen randomly from a subset of variables which is usually large. The algorithm runs in polynomial time and it is shown that the algorithm finds a solution to a random instance of 3-Satisfiability with probability bounded from below by a constant greater than zero for a range of parameter values. The heuristics studied here can be used to select variables in a Backtrack algorithm for 3-Satisfiability. Experiments have shown that for about the same range of parameters as above the Backtrack algorithm using the heuristics finds a solution in polynomial average time.

A perspective on certain polynomial-time solvable classes of satisfiability

The scope of certain well-studied polynomial-time solvable classes of Satisfiability is investigated relative to a polynomial-time solvable class consisting of what we call matched formulas. The class of matched formulas has not been studied in the literature, probably because it seems not to contain many challenging formulas. Yet, we find that, in some sense, the matched formulas are more numerous than Horn, extended Horn, renamable Horn, q-Horn, CC-balanced, or single lookahead unit resolution (SLUR) formulas.The behavior of random k-CNF formulas generated by the constant clause-width model is investigated as n and m, the numbers of variables and clauses, go to infinity. For m/n 0.64, random formulas are matched formulas with probability tending to 1. For m/nk-1 ≥ 2k/k!, random formulas are solved by a certain polynomial-time resolution procedure with probability tending to 1.The propositional connection graph is introduced to represent clause structure for formulas with general-width clauses. Cyclic substructures are exhibited that occur with high probability and prevent formulas from being in the previously studied polynomial-time solvable classes, but do not prevent them from being in the matched class. We believe that part of the significance of this work lies in guiding the future development of polynomial-time solvable classes of Satisfiability.

On finding solutions for extended Horn formulas

Abstract We present a simple quadratic-time algorithm for solving the satisfiability problem for a special class of boolean formulas. This class properly contains the class of extended Horn formulas and balanced formulas. Previous algorithms for these classes require testing membership in the classes. However, the problem of recognizing balanced formulas is complex, and the problem of recognizing extended Horn formulas is not known to be solvable in polynomial time. Our algorithm requires no such test for membership.

Results related to threshold phenomena research in satisfiability: lower bounds

We present a history of results related to the threhold phenomena which arise in the study of random conjunctive normal form (CNF) formulas. In a companion paper (D. Achlioptas, Theoret. Comput. Sci., this volume) the major ideas used to achieve many of the lower bounds results on the location of the threshold are described in an informal, intuitive manner.

Costs of quadtree representation of nondense matrices

Abstract The quadtree representation of matrices is a uniform representation for both sparse and dense matrices which can facilitate shared manipulation on multiprocessors. This paper presents worst-case and average-case resource requirements for storing and retrieving familiar families of patterned matrices: packed, symmetric, triangular, Toeplitz, and banded. Using this representation it compares resource requirements of three kinds of permutation matrices, as examples of nondense, unpatterned matrices. Exact values for the shuffle and bit-reversal permutations (as in the fast Fourier transform) and tight bounds on the expected values from purely random permutations are derived. Two different measures, density and sparsity , are proposed from these values. Analysis of quadtree matrix addition relates density of addends to space bounds on their sum and relates their sparsity to time bounds for computing that sum.

Probabilistic analysis of the pure literal heuristic for the satisfiability problem

An algorithm for the SATISFIABILITY problem is presented and a probabilistic analysis is performed. The analysis is based on an instance distribution which is parametrized to simulate a variety of sample characteristics. The algorithm either correctly determines whether a given instance of SATISFIABILITY has a solution or gives up. It is shown that the algorithm runs in polynomial time and gives up with probability approaching zero as input size approaches infinity for a range of parameter values. This result is an improvement over the results in [3] and [4].

Computing Well-founded Semantics Faster

We address methods of speeding up the calculation of the well-founded semantics for normal propositional logic programs. We first consider two algorithms already reported in the literature and show that these, plus a variation upon them, have much improved worst-case behavior for special cases of input. Then we propose a general algorithm to speed up the calculation for logic programs with at most two positive subgoals per clause, intended to improve the worst case performance of the computation. For a logic program P in atoms A1, the speed up over the straight Van Gelder alternating fixed point algorithm (assuming worst-case behavior for both algorithms) is approximately (¦P¦/¦A¦)(1/3). For ¦P¦≥¦A¦4, the algorithm runs in time linear in ¦P¦.

SBSAT: a State-Based, BDD-Based Satisfiability Solver

We present a new approach to SAT solvers, supporting efficient implementation of highly sophisticated search heuristics over a range of propositional inputs, including CNF formulas, but particularly sets of arbitrary boolean constraints, represented as BDDs. The approach preprocesses the BDDs into state machines to allow for fast inferences based upon any single input constraint. It also simplifies the set of constraints, using a tool set similar to standard BDD engines. And it memoizes search information during an extensive preprocessing phase, allowing for a new form of lookahead, called local-function-complete lookahead. This approach has been incorporated, along with existing tools such as lemmas, to build a SAT tool we call SBSAT.