Mitchell, D.; Selman, B.; and Levesque, H. 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....must perform an exhaustive search for the partial solution in order to revise the bad decision. When the problem becomes very large, doing such an exhaustive search is virtually impossible. On the other hand, iterative improvement algorithms (Minton et al. 1992; Morris 1993; Selman, Levesque, Mitchell 1992) do not construct a consistent partial solution. In these algorithms, a flawed solution containing some constraint violations is revised by local changes until all constraints are satisfied. The min conflict heuristic is used as the basis for the local changes. In these algorithms, the value of ....

Mitchell, D.; Selman, B.; and Levesque, H. 1992.

....go fairly deep before reaching a dead end. Critically constrained randomly generated satisfiability problems provide a ready supply of hard test cases of arbitrary size. This discovery has lead to work on understanding (Crawford Auton 1993; Williams Hogg 1992) and solving (Selman, Levesque, Mitchell 1992) these problems. Gsat in particular appears to be well suited to solving large randomly generated critically constrained problems. However, concerns have been raised that randomly generated problems are bad test cases because they have no structure and thus may bear little resemblance to real ....

.... C : randomly chosen unsatisfied clause; With probability P, Flip a random variable in C; Otherwise (that is, with probability 1 P) Flip a variable in C resulting in the greatest decrease in the number of unsat clauses; return failure The experimental performance of gsat (Selman, Levesque, Mitchell 1992) with walk on certain problem classes is impressive. Gsat is often able to find models for randomly generated 2000 variable criticallyconstrained 3SAT problems. 5 Systematic methods (methods that are guaranteed to always to find a solution or determine that none exists) are currently not able to ....

Mitchell, D.; Selman, B.; and Levesque, H. 1992.

.... Levesque 1992) the best current estimate for the location of this crossover point is at m = 4:24n 6:21 (Crawford Auton 1993) Several recent authors have used these crossover point 3 SAT problems to measure the performance of their algorithms (Crawford Auton 1993; Selman, Levesque, Mitchell 1992). In the dynamic backtracking algorithm, step 2 leaves open the choice of which variable to select next; backtracking and backjumping have similar indeterminacies. We used the following variable selection heuristics: 1. If there is an unassigned variable with one of its two values currently ....

Mitchell, D.; Selman, B.; and Levesque, H. 1992.

....true. We conjecture that this will occur for many problems. An analogous situation has been identified in the csp literature for some wellknown problems, almost all instances are easy to solve because they are either over constrained or underconstrained (Cheeseman, Kanefsky, Taylor 1991; Mitchell, Selman, Levesque 1992). If we compare the learned axioms to those produced by our human subjects, the results are as expected. Often the humans neglected to state relevant axioms or stated them incorrectly. The machine generated axiom sets were more complete, and in some cases, the axioms were stated more concisely. ....

Mitchell, D.; Selman, B.; and Levesque, H. 1992.

....number of variables, n, and the average run time scale sub exponentially (basically as n log(n) over the range n = 25; 400. INTRODUCTION In recent years, a variety of local search routines have been proposed for (Boolean) satisfiability testing. It has been shown (Selman, Levesque, Mitchell 1992; Gu 1992; Selman, Kautz, Cohen 1994) that local search can solve a variety of realistic and randomly generated satisfiability problems much larger than conventional procedures such as Davis Putnam. The characteristic feature of local search is that it starts on a total variable assignment and works by ....

....able to give results up to 400 variables. We find that the optimal Maxflips setting scales as a simple monomial, and the mean runtime scales subexponentially, but faster than a simple power law. LOCAL SEARCH IN SAT Figure 1 gives the outline of a typical local search routine (Selman, Levesque, Mitchell 1992) to find a satisfying assignment for a set of clauses ff 1 . proc Local Search SAT Input clauses ff; Maxflips; and Maxtries for i : 1 to Maxtries do A : new total truth assignment for j : 1 to Maxflips do if A satisfies ff then return A P : select variable(ff; A) A : A with P flipped ....

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Mitchell, D.; Selman, B.; and Levesque, H. 1992.

....other in complex ways. Context limited consistency checking also becomes less useful as L becomes more than about V=2; in these cases unit propagation makes full consistency checking so easy that context limitations become superfluous. Experimental Setup We generate random 3 SAT theories using Mitchell et al. s (1992) method each clause is generated by picking three different variables at random and negating each with probability 0.5. There is no check for repeated clauses. Inconsistent theories are discarded. We then randomly select a series of L literals consistent with the theory built so far. Consistency ....

Mitchell, D.; Selman, B.; and Levesque, H. 1992.

....in cost, coupled with the general adoption of C as the programming language of choice, have made it possible for the developer of a new algorithm or heuristic to test it on large numbers of random instances. Another important advance was the recognition of the 50 satisfiable phenomenon (Mitchell, Selman, Levesque 1992), which has enabled researchers to focus on the hardest problems. It is often not clear which measures to report from large scale experiments. The usual parameter of interest is the cost of solving a problem, measured by CPU time, number of consistency checks, or size of search space. The mean and ....

....N (N Gamma 1) 2 possible; C ranges from 0 (no constraints) to 1 (a complete graph) The specific constraints are chosen randomly from a uniform distribution. We specify the parameters between angle brackets: hN; D;T; Ci. This model is the binary CSP analog of the Random KSAT model described in (Mitchell, Selman, Levesque 1992), and has been widely used by many researchers (Prosser 1996; Smith Dyer 1996) We also report some experiments with 3SAT problems, which can be viewed as a type of CSP with ternary constraints and D = 2. All experiments reported in this paper were run with parameters that produce problems in ....

Mitchell, D.; Selman, B.; and Levesque, H. 1992.

....containing universals (which are trivially unsatisfiable) are discarded. Flawed and Flawless Problems In Sat, empty and unit clauses are normally omitted in random generation methods where the number of literals in each clause varies. For example, in the constant probability model proposed in (Mitchell, Selman, Levesque 1992), each variable is included in a clause with some constant probability, but if only zero or one variable is included, the clause is discarded. An empty clause immediately makes a problem insoluble, but the reason for omitting unit clauses is more subtle. Suppose the model did not exclude unit ....

Mitchell, D.; Selman, B.; and Levesque, H. 1992.

....made a comparison of the algorithms with respect to the number of search tree nodes visited, the number of consistency checks performed, and the CPU time used. Then, we tried to investigate the phase transition in the problem we study. Cheeseman et al. 1991; Prosser, 1996; Smith and Dyer, 1996; Selman et al. 1996; Gent and Walsh, 1996; Crawford and Auton, 1996) showed that for many NP complete problems, hard problems occur arround a critical value of a control parameter. The control parameter in our problem is the ratio r of disjunctions to variables. We have chosen this parameter because of the ....

....parameter used is the ratio of clauses to variables which transfered to our problem corresponds to the ratio of disjunctions to variables. The Random Generation Model The random problem generation model used is in some ways similar to the fixed clause length model for SAT, as described in (Selman et al. 1996). For each set of problems there are four parameters: the number of constrained variables n, the number of disjuncts per disjunction k, the number of disjunctions m, and the maximum integer value L. Therefore, each problem is described by the 4 tuple k; n; m;L . As in the fixed clause length ....

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Selman, B., Mitchell, D., and Levesque, H. (1996).

.... for new problem domains All these questions have been the subject of intensive study in recent years in a large number of problem domains including for example satisfiability, graph colouring, constraint satisfaction problems, and hamiltonian circuits (Cheeseman, Kanefsky, Taylor 1991; Mitchell, Selman, Levesque 1992; Williams Hogg 1994; Smith Dyer 1996) Here, we introduce some general methods which help to answer these questions in a wide range of problems. These methods are based on a definition of the constrainedness of an ensemble of combinatorial problems. Problems which are very over constrained ....

....1=2 a , and there are l such constraints. Equation (2) gives = Gamma log 2 (1 Gamma 1 2 a ) l n That is, a constant times l=n for fixed a. The ratio l=n has been used as an order parameter for satisfiability. A phase transition in satisfiability occurs around a critical value of l=n (Mitchell, Selman, Levesque 1992). For large a, this phase transition occurs at a value close to Gamma1= log 2 (1 Gamma 1 2 a ) Kirkpatrick Selman 1994) that is around 1, as expected. In graph colouring, we are given a graph with n nodes and e edges, and wish to colour it with m colours. As a CSP, each node represents ....

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Mitchell, D.; Selman, B.; and Levesque, H. 1992.

....Figure 3: Probability of residual theory having a supermodel, as a function of number of UPIs. Instances are from the phase transition for satisfiability at n=50. optimal solutions. In this section, we briefly study the issue of supermodels and phase transitions for the case of Random 3SAT (Mitchell, Selman, Levesque 1992). Instances are characterized by n variables, m clauses, and a clause variable ratio ff = m=n. There is strong evidence that for large n, this system exhibits a phase transition at ff 4:2 (Crawford Auton 1996) Below this value, theories are almost always satisfiable; above it, they are almost ....

Mitchell, D.; Selman, B.; and Levesque, H. J. 1992.

....microscopic studies that look closely inside search may be useful in other domains. Constrainedness within satisfiability There has been considerable interest recently in encoding problems into satisfiability and solving them either with local search procedures like Gsat (Selman, Levesque, Mitchell 1992) or with the Davis Putnam decision procedure (Bayardo Schrag 1997) We therefore began our experiments by looking at how the constrainedness of satisfiability problems varies during search. The constrainedness of a satisfiability problem depends on several factors including the clause length ....

Mitchell, D.; Selman, B.; and Levesque, H. 1992.

....search procedures, such as the Davis Putnam procedure, on various classes of formulas. For example, we studied GSAT s performance on hard randomly generated formulas. Note that generating hard random formulas for testing purposes is a challenging problem by itself, see Cheeseman et al. 1991) Mitchell et al. 1992); Larrabee and Tsuji (1993) and Crawford and Auton (1993) Table 1 summarizes the results. The table shows how GSAT is indeed much faster than the Davis Putnam (DP) procedure on such hard random formulas. 3 3 During the last year, we have collected data on other backtrack procedures. Using ....

Mitchell, D., Selman, B., and Levesque, H.J. (1992).

....We felt that such techniques would fail in practice for SAT. But this appears not to be the case. The issue is clouded by the fact that some care is required to randomly generate SAT problems that are hard for even ordinary backtracking methods. 1 But once we discovered how to do this (and see Mitchell et al. 1992) for details) we found that GSAT s local search was very good at finding models for the hardest formulas we could generate. Because model finding is NP hard, we cannot expect GSAT to solve it completely and exactly within tolerable resource bounds. What we will claim, however, is that the ....

....average time on a class of random formulas. However, Franco and Paull (1983) demonstrated that the instances considered by Goldberg were so satisfiable that an algorithm that simply guessed truth assignments would find a satisfying one just as quickly as DP This issue is discussed in detail in (Mitchell et al. 1992). Formulas are generated using the uniform distribution or fixed clause length model. For each class of formulas, we choose the number of variables N , the number of literals per clause K, and the number of clauses L. Each instance is obtained by generating L random clauses each containing K ....

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Mitchell, D., Selman, B., and Levesque, H.J. (1992).

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Science 264:1297--1301. Mitchell, D.; Selman, B.; and Levesque, H. 1992.

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Mitchell, D., Selman, B., and Levesque, H. (1992).

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Forthcoming. Mitchell, D.; Selman, B.; and Levesque, H. 1992.

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Science 264:1297--1301. Mitchell, D.; Selman, B.; and Levesque, H. 1992.

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