Davis-Putnam Research Articles

Preface

Magic squares, chess-like problems, cryptarithmetic puzzles, and similar classes of problems have been extensively used to challenge human reasoning capabilities. Lo Shu magic square can be traced back to 650 B.C., the eight-queens problem was proposed in 1848 by the chess player Max Bazzel TWO × TWO = THREE puzzle appeared in Strand Magazine in 1924. These puzzles are nowadays widely used in constraint programming courses. The first programming language provided with constraint modelling primitives (Sketchpad) has been proposed by the Turing award winner Ivan Sutherland in his PhD thesis (1963). Logemann and Loveland, when implementing the Davis–Putnam procedure (Davis and Putnam 1960) for testing the satisfiability of a propositional formula (SAT), devised an algorithm (Davis–Putnam–Logemann–Loveland (DPLL)) that has become the core of all SAT/and Answer Set Programming solvers (50 years later). It consists in choosing an un-assigned variable, assigning it a value 0 or 1, propagating the chosen value (unit propagation), and proceeding with the alternative value, if the original assignment leads to a contradiction (backtracking). Some years later Waltz (1975) introduced the notion of domain filtering (arc-consistency-based constraint propagation). With this idea the same DPLL scheme can be used for verifying the satisfiability of a constraint satisfaction problem, where the assignment is no longer 0/1 and the unit propagation is replaced by constraint propagation. For a detailed history of these early years achievements, we refer the reader to the works by Loveland et al. (2017), Jaffar and Maher (1994), and Freuder and Mackworth (2006).

Distributed SAT

We present DPLL ABT, a distributed Satisfiability solver (SAT) (Ansotegui and Manya in IberoAm J Artif Intell 7(20):43---56, 2003) designed to solve distributed SAT problem instances. Since SAT is a particular case of constraint satisfaction, we propose a solving method based on the Asynchronous Backtracking algorithm (ABT) (Yokoo et al. in IEEE Trans Knowl Data Eng 10(5):673---685, 1998) developed for distributed constraint reasoning. In addition, we have applied the Davis-Putnam procedure (DPLL) in every agent, plus the minimum conflict heuristic in case DPLL does not detect any inconsistency. The resulting algorithm improves the performance in terms of communication cost and computational effort versus the basic ABT. The SAT instance is distributed into agents, which cooperate to solve SAT instances just sharing the minimum information. We also present the experimental results that demonstrate the performance of the method in terms of communication and execution time comparing the performance with the basic ABT algorithm.

Generalized Davis-Putnam and satisfiability problems in mathematics

The well-known Davis-Putnam Procedure used to decide the Sat Problem in Logic is shown to be combinatorial in nature and thus directly applicable to solving various Mathematical “Satisfiability” Problems.

Size Dependence of the Minimum Excitation Gap in the Quantum Adiabatic Algorithm

We study the typical (median) value of the minimum gap in the quantum version of the exact cover problem using quantum Monte Carlo simulations, in order to understand the complexity of the quantum adiabatic algorithm for much larger sizes than before. For a range of sizes N< or =128, where the classical Davis-Putnam algorithm shows exponential median complexity, the quantum adiabatic algorithm shows polynomial median complexity. The bottleneck of the algorithm is an isolated avoided-crossing point of a Landau-Zener type (collision between the two lowest energy levels only).

Conditioning probabilistic databases

Past research on probabilistic databases has studied the problem of answering queries on a static database. Application scenarios of probabilistic databases however often involve the conditioning of a database using additional information in the form of new evidence. The conditioning problem is thus to transform a probabilistic database of priors into a posterior probabilistic database which is materialized for subsequent query processing or further refinement. It turns out that the conditioning problem is closely related to the problem of computing exact tuple confidence values. It is known that exact confidence computation is an NP-hard problem. This has led researchers to consider approximation techniques for confidence computation. However, neither conditioning nor exact confidence computation can be solved using such techniques. In this paper we present efficient techniques for both problems. We study several problem decomposition methods and heuristics that are based on the most successful search techniques from constraint satisfaction, such as the Davis-Putnam algorithm. We complement this with a thorough experimental evaluation of the algorithms proposed. Our experiments show that our exact algorithms scale well to realistic database sizes and can in some scenarios compete with the most efficient previous approximation algorithms.

An Introduction to Satisfiability Algorithms

In this paper we present an introduction to satisfiability algorithms. Since most complete SAT solvers (e.g. Satz, SATO, GRASP, and Chaff) are based on the Davis-Putnam procedure, we first describe in detail that procedure. Then, we present the improvements that can be incorporated into the Davis-Putnam Procedure in order to develop a competitive SAT solver: optimized data structures, variable selection heuristics, non-chronological backtracking, conflict-driven learning, and restarts. Finally, we describe GSAT and WalkSAT, which are the most widely used local search algorithms for solving SAT.

Nagging: A scalable fault-tolerant paradigm for distributed search

This paper describes nagging, a technique for parallelizing search in a heterogeneous distributed computing environment. Nagging exploits the speedup anomaly often observed when parallelizing problems by playing multiple reformulations of the problem or portions of the problem against each other. Nagging is both fault tolerant and robust to long message latencies. In this paper, we show how nagging can be used to parallelize several different algorithms drawn from the artificial intelligence literature, and describe how nagging can be combined with partitioning, the more traditional search parallelization strategy. We present a theoretical analysis of the advantage of nagging with respect to partitioning, and give empirical results obtained on a cluster of 64 processors that demonstrate nagging's effectiveness and scalability as applied to A ∗ search, αβ minimax game tree search, and the Davis–Putnam algorithm.

MULTIRESOLUTION FOR SAT CHECKING

This paper presents a system based on new operators for handling sets of propositional clauses compactly represented by means of ZBDDs. The high compression power of such data structures allows efficient encodings of structured instances. A specialized operator for the distribution of sets of clauses is introduced and used for performing multiresolution on clause sets. Cut eliminations between sets of clauses of exponential size may then be performed using polynomial size data structures. The ZRES system, a new implementation of the Davis-Putnam procedure of 1960, solves two hard problems for resolution, that are currently out of the scope of the best SAT provers.

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 the relations between SAT and CSP enumerative algorithms

We show the equivalence between the so-called Davis–Putnam procedure (Davis et al., Comm. ACM 5 (1962) 394–397; Davis and Putnam (J. ACM 7 (1960) 201–215)) and the Forward Checking of Haralick and Elliot (Artificial Intelligence 14 (1980) 263–313). Both apply the paradigm choose and propagate in two different formalisms, namely the propositional calculus and the constraint satisfaction problems formalism. They happen to be strictly equivalent as soon as a compatible instantiation order is chosen. This equivalence is shown considering the resolution of the clausal expression of a CSP by the Davis–Putnam procedure.

New Worst-Case Upper Bounds for SAT

In 1980 Monien and Speckenmeyer proved that satisfiability of a propositional formula consisting of i>K clauses (of arbitrary length) can be checked in time of the order 2i>K / 3. Recently Kullmann and Luckhardt proved the worst-case upper bound 2i>L / 9, where i>L is the length of the input formula. The algorithms leading to these bounds are based on the i>splitting method, which goes back to the Davis–Putnam procedure. i>Transformation rules (pure literal elimination, unit propagation, etc.) constitute a substantial part of this method. In this paper we present a new transformation rule and two algorithms using this rule. We prove that these algorithms have the worst-case upper bounds 20. 30897 i>K and 20. 10299 i>L, respectively.

The Good Old Davis-Putnam Procedure Helps Counting Models

As was shown recently, many important AI problems require counting the number of models of propositional formulas. The problem of counting models of such formulas is, according to present knowledge, computationally intractable in a worst case. Based on the Davis-Putnam procedure, we present an algorithm, CDP, that computes the exact number of models of a propositional CNF or DNF formula F. Let m and n be the number of clauses and variables of F, respectively, and let p denote the probability that a literal l of F occurs in a clause C of F, then the average running time of CDP is shown to be O(nm^d), where d=-1/log(1-p). The practical performance of CDP has been estimated in a series of experiments on a wide variety of CNF formulas.

Extremal problems in logic programming and stable model computation

We study the following problem: given a class of logic programs ¢, determine the maximum number of stable models of a program from ©. We establish the maximum for the class of all logic programs with at most n clauses, and for the class of all logic programs of size at most n. We also characterize the programs for which the maxima are attained. We obtained similar results for the class of all disjunctive logic programs with at most n clauses, each of length at most m, and for the class of all disjunctive logic programs of size at most n. Our results on logic programs have direct implication for the design of algorithms to compute stable models. Several such algorithms, similar in spirit to the Davis-Putnam procedure, are described in the paper. Our results imply that there is an algorithm that finds all stable models of a program with n clauses after considering the search space of size O(3 n/3 ) in the worst case. Our results also provide some insights into the question of representability of families of sets as families of stable models of logic programs.

Meta-resolution: An algorithmic formalisation

We study the Tseitin formulae (Tseitin, 1970) on which resolution is known to be inefficient. We first give a new proof of the satisfiability condition and precise the number of solutions if any and otherwise the minimal contradictory subformula. Then, defining a new data structure, we introduce the meta-argument of that proof in the Davis-Putnam and Loveland procedure (Loveland, 1978) and obtain a new algorithm for SAT especially efficient on these formulae.

A fast parallel SAT-solver — efficient workload balancing

We present a fast parallel SAT-solver on a message based MIMD machine. The input formula is dynamically divided into disjoint subformulas. Small subformulas are solved by a fast sequential SAT-solver running on every processor, which is based on the Davis-Putnam procedure with a special heuristic for variable selection. The algorithm uses optimized data structures to modify Boolean formulas. Additionally efficient workload balancing algorithms are used, to achieve a uniform distribution of workload among the processors. We consider the communication network topologiesd-dimensional processor grid and linear processor array. Tests with up to 256 processors have shown very good efficiency-values (>0.95).

Hard random 3-SAT problems and the Davis-Putnam procedure

It is by now well known that randomly generated 3-SAT problems are very difficult to solve on the average when the ratio of clauses to variables is a constant which is approximately equal to 4.24. This difficulty appears to be algorithm-independent, but it is certainly a consequence of using the popular Davis-Putnam procedure in Loveland's form (DPL). The purpose of this paper is to try to provide an explanation of why these problems are hard for DPL by experimentally determining how their complexity decreases as the depth of their associated search trees increases. We use a highly optimized version of DPL to plot the average number of remaining variables versus search tree depth for several nontrivial sizes of critically constrained, underconstrained, and overconstrained problems. These plots have a distinct piecewise-linear shape, consisting of an initial, gradual descent from the original number of variables, and a second, steeper descent to substantially smaller subproblems. They help explain exactly why DPL performs poorly on the critically constrained problems, and why its performance is much better on under- and overconstrained problems. Also, we can use the key parameters of these plots, e.g., the depth at which rapid constraint propagation occurs, as another basis of comparison for implementations of DPL.

Critical behavior in the computational cost of satisfiability testing

In previous work, we employed finite-size scaling, a method from statistical mechanics, to explore the crossover from the SAT regime of K-SAT, where almost all randomly generated expressions are satisfiable, to the UNSAT regime, where almost all are not. In this work, we extend the experiments to cover critical behavior in the computational cost. We find that the median computational cost takes on a universal form across the transition regime. Finite-size scaling accounts for its dependence on N (the number of variables) and on M (the number of clauses in the k-CNF expression). We also inquire into the sources of the complexity by studying distributions of computational cost. In the SAT phase we observe an unusually wide range of costs. The median cost increases linearly with N, while the mean is significantly increased over the median by a small fraction of cases in which exponentially large costs are incurred. We show that the large spread in cost of finding assignments is mainly due to the variability of running time of the Davis-Putnam (DP) procedure, used to determine the satisfiability of our expressions. In particular, if we consider a single satisfiable expression and run DP many times, each time randomly relabelling the variables in the expression, the resulting distribution of costs nearly reproduces the distribution of costs encountered by running DP search once on each of many such randomly generated satisfiable expressions. There are intriguing similarities and differences between these effects and kinetic phenomena studied in statistical physics, in glasses and in spin glasses.

Exploiting data dependencies in many-valued logics

ABSTRACT The purpose of this paper is to make some pratically relevant results in automated theorem proving available to many-valued logics with suitable modifications. We are working with a notion of many-valued first-order clauses which any finitely- valued logic formula can be translated into and that has been used several times in the literature, but in an ad hoc way. We give a many-valued version of polarity which in turn leads to natural many-valued counterparts of Horn formulas, hyperresolution, and a Davis—Putnam procedure. We show that the many-valued generalizations share many of the desirable properties of the classical versions. Our results justify and generalize several earlier results on theorem proving in many-valued logics.

A Variational Method for Analysing Unit Clause Search

We expose a variational method for analysing algorithms, as applied to analyse the algorithm $\textbf{UC}$, which is the Davis-Putnam procedure for a set of clauses of three literals. The variable from a unit clause or, if there is none, the first remaining variable from a fixed list is chosen. The algorithm $\textbf{UC}$ finds all the solutions as a set of cylinders. Following the nomenclature of Purdom [J. Inform. Process., 13 (1990), pp. 449–455], we call this algorithm “unit clause backtracking with cylinders of solutions.” First we give an expression for the number of nodes of the calculation trees of all the inputs. Then we use a variational method to calculate the base $\beta $ of the principal exponential part of the average time of calculation T. This “exponential base” is the maximum of three elementary functions $f_{i}$ of four real variables, These functions are defined on the product of the half positive real line by the 3-dimensional unit real cube. We finally obtain the following short statement. Let v be the number of the variables. Let c be the number of the clauses. Let $\gamma = \frac{c}{v} > 1$. Let $\gamma $ be constant when v grows to infinity. The principal exponential part of the average time of $\textbf{UC}$ is $\beta^{v}$ where \[ \beta = \max\limits_{0 \leq \lambda \leq 1} 2^{\lambda} \left( 1 - \frac{3\lambda ^{2}}{8} + \frac{\lambda ^{3}}{4} \right)^{\gamma}. \] We mean that $\lim_{v \rightarrow \infty} T^{1/v} =\beta $. As a first consequence of our method we match $\textbf{UC}$, with the algorithm $\textbf{B}$ without rearrangement (i.e., with a fixed order for introducing the variables). This gives a proof to a conjecture of P W. Purdom [Artif. Intell., 21(1983), pp. 117–133].

Eliminating duplication with the hyper-linking strategy

The efficiency of almost all theorem proving methods suffers from a phenomenon called duplication of instances of clauses. In this paper, we present a novel technique, called the hyper-linking strategy, to eliminate such duplication. This strategy is complete for the full first-order predicate calculus. We show the effectiveness of this strategy by comparing it with other proving methods. We give empirical evidence that both the Davis-Putnam procedure and the hyper-linking strategy are comparable to each other and better than other common theorem proving strategies on propositional calculus problems. The fact that the Davis-Putnam procedure is faster than resolution and other common methods on propositional problems seems not to be appreciated by a large segment of the theorem proving community. Also, we give empirical evidence that the hyper-linking strategy is better than other common theorem proving methods on near-propositional problems like logic puzzles. We attempt to explain the superior behavior of the hyper-linking strategy and the Davis-Putnam procedure by examining the kinds of duplication that can occur during the search with the different methods. In addition, we show the completeness of the hyper-linking strategy combined with several support strategies.