of m clauses, each containing the disjunction of boolean variables from a nite set V = fv 1 ; : : : ; vng of size n [8]. Each variable occurrence with

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1 A Hybridised 3-SAT Algorithm Andrew Slater Automated Reasoning Project, Computer Sciences Laboratory, RSISE, Australian National University, 0200, Canberra April 9, Introduction In recent years much research has been done on the propositional satisability problem. Although great success has been found by using incomplete methods there are still classes of problems that remain resilient to such approaches. At the same time many algorithms and heuristics that retain completeness have been proposed and implemented. The success of these faster algorithms accompanied with better heuristics has allowed some of the more elusive problem classes to be exposed, and in general the size of the satisability problems that can be solved has grown. In addition these more advanced systems have allowed a wider empirical study of the nature of satisability problems. Much of the empirical AI that has been done has focused on the special case where each clause in the formula contains three variables, known as 3-SAT. In this report we discuss some of the development and design of software to experiment with hybridised 3-SAT algorithms. The aim of this project is to provide a simple system where dierent techniques can be isolated and combined to determine the most suitable algorithms for both particular classes of instances of the satisability problem and the general case. In particular we are interested in determining whether combining known linear time algorithms for special instances of the satisability problem with the traditional Davis Putnam style approaches which are designed for the general case will be successful in general or constrained scenarios. We present a possible approach that combines the linear time 2-SAT algorithm with a Davis-Putnam style algorithm aided by common heuristics. 2 Preliminaries The propositional satisability problem, SAT, is dened as the decision problem of determining whether a given formula in conjunctive normal form (CNF) is satisable. Formally we dene the `instance' of the problem as a conjunction 1

2 of m clauses, each containing the disjunction of boolean variables from a nite set V = fv 1 ; : : : ; vng of size n [8]. Each variable occurrence within a clause can appear positively or negatively (the negation) and these occurrences are called literals. Thus if v is a variable then v and v are literals over V. An instance of the satisability problem can now be dened as the formula F = C 1 ^ C 2 ^ : : : ^ Cn where each Ci is a disjunction of literals Ci = l 1 _ l 2 _ : : : _ lk. The problem is a prototypical NP-Complete problem. The 3-SAT problem restricts the number of literals in each clause to three. Note that an instance of the 3-SAT problem is an instance of SAT, and there exists a polynomial time transformation from any instance of SAT to 3-SAT. The 2-SAT problem restricts the number of literals in each clause to two, and was shown to be solvable in linear time by Aspvall, Plass and Tarjan [3] using a directed graph representation of the formula. The simplest, and most successful, algorithm for determining general satis- ability is the Davis Putnam Logemann Loveland algorithm [6, 5]. Essentially it chooses a literal from some clause and assigns its value to be true, then propagates this assumption through the remaining formula. Either it will terminate by nding an assignment to the variables which satises all clauses, or it will exhaust all possible decisions for that formula and nd it unsatisable. Although this is a simplied interpretation of its behaviour, the description captures the method of `proof' that it implements. More recently several modications have been suggested and implemented to improve the algorithms performance. These include heuristically driven programs from which empirical analysis has shown greater eciency, and more formal case-based algorithm's designed to show worst case upper bound complexities. For more detailed information the reader is directed to recent survey articles concerned with satisability [2, 1, 9]. 3 A Brief Overview of Previous Related Work There has been much research into SAT algorithms within the last decade. We focus here on those heuristics and algorithms that are directly related to the proposed hybrid. Ecient implementations of the Davis Putnam algorithm will use heuristics to guide or order the search. Occurrence weighting and similar statistical methods are common successful guiding techniques. One of the simplest and most eective is the MoMS heuristic [7]. This prefers the choice of assigning a propositional variable having Maximum Occurrences in clauses of Minimum Size. Several other techniques to `prune' the search space and identify redundancy have been successfully implemented, and although they are competitive, they can require a signicant overhead. Hybridisation of general satisability algorithms with linear time algorithms has been performed in the past. It is reported that some algorithms have used the linear time 2-SAT algorithm to generate an assignment that satises the existing clauses of size two, and that these algorithms slightly under perform some of the more advanced search methods [1]. Larrabee's algorithm was suc- 2

3 cessfully used to solve instances with a large initial number of binary clauses by cycling through the possible assignments for the 2-SAT component of the problem, propagating each assignment through the remaining formula [10, 11]. The algorithm was shown to remain competitive when given instances of 3-SAT. 4 A Proposed Hybrid 3-SAT Algorithm The proposed algorithm is based on the 2-SAT approach. Each binary clause represents two edges in a directed graph with vertices for each literal. For a binary clause (a _ b) we generate edges a! b and b! a. We say a reaches b if there is a directed path from a to b. Hence for two edges a! b and b! c we say a reaches c and this implicitly captures an `inference' of the binary clause a _ c this knowledge can possibly subsume a ternary clause. With a depth rst traversal of the graph we can determine the strongly connected components subgraphs where each vertex can reach every other vertex in that subgraph. If a strongly connected component contains a literal l and the negation l then the system cannot be satised. If l reaches l and l does not reach l, then l must have the truth value true is `inferred'. Additionally, the set of literals in any non-contradictory strongly connected component have equivalent truth values is also an `inference'. We use the 2-SAT method and representation as a basis for a linear time binary clause reasoning system. The controlling algorithm is a backtracking Davis Putnam style process. The central dierence is that it collects binary clauses, reasons with them, and propagates the `inferences' as well as propagating the `inferences' of unit clauses. The Davis Putnam step chooses a literal. This generates some binary clauses and eliminates some ternary clauses. The propagation step reduces the remaining 3- SAT clauses using the binary clause inferences. The propagation may subsume clauses completely, or generate more binary clauses by a partial subsumption. For example when a reaches b then the clause (a _ b _ x) is eliminated and the clause (a _ b _ x) is reduced to (b _ x). The propagation of the inferences from the basic 2-SAT algorithm rule are fairly obvious. Generating a solution for a non-contradictory 2-SAT graph is fairly straightforward. Our algorithm does not generate an intermediate solution, it only uses the rules above (and others) to generate inferences and propagate them through the rest of the ternary clauses. Propagation is continued from the binary system to the ternary system and back again until no more inferences can be made. Obviously it is more likely that an initially larger 2-SAT system will infer more information about the system as a whole. Since we begin with a 3-SAT instance which we attack with a Davis Putnam style procedure, we prefer to choose literals that will result in the most binary clauses being generated. This is similar to the MoMS approach mentioned above, but we only evaluate the number of occurrences of a literal in ternary clauses. By doing this we rapidly attain large 2-SAT systems. The backtracking component of the Davis Putnam algorithm is performed when a contradiction is identied during propagation. A formula is found satisable when no more literals can be chosen (since the 2-SAT system 3

4 remains satisable from the propagation phase). The overhead of propagation processing can be signicantly reduced by using an adjacency matrix to represent the graph. Lookup tables and statistics for variable occurrences can also be kept. Additionally we have designed a fast `clean up' system to `undo' assumptions that lead to contradictions eciently. By linking changes made to a particular depth or level in the search we can backtrack to a previous state by following a stack based list of undo operations, typically pointer swaps. 5 Discussion and Further Work We have discussed a Davis Putnam variant that reasons with and propagates unit and binary clauses as opposed to the standard process of propagating unit clauses only. The automated reasoning process for binary clauses is bases upon the fundamentals of the linear time 2-SAT algorithm. We intend to construct a full system for experimentation. It is expected that the algorithm will perform competitively with other methods but will not outperform them in general, however we hypothesise that it will perform better on problems that are unsatisable. It is likely that the algorithm is better at identifying a refutation more eciently than algorithms designed to nd a satisfying assignment. Indeed Larrabee notes that while the hard problem distribution phase transition value agrees with previous experiments, their algorithm performed better on the right hand side of the phase transition where problems are more likely to be unsatisable. We believe this is the case for some 2-SAT algorithms since they are more ecient at encapsulating contradiction in the search tree. We conjecture that our approach will not suer as badly from the nature of the heavy tailed distribution of SAT and related backtracking search methods. We intend to further investigate hybrid algorithm performance, phase transition phenomena, and proof methods for propositional and modal logics via polynomial translations of given formula to CNF. We are also interested in investigating proof search algorithm behaviour for classes of generated theorems parameterised by length and syntactic complexity. References [1] American Mathematical Society. Algorithms for the Satisability (SAT) Problem: A Survey, volume 35 of Satisability Problem: Theory and Applications, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, [2] American Mathematical Society. Finding Hard Instances of the Satisability Problem: A Survey, volume 35 of Satisability Problem: Theory and Applications, DIMACS Series in Discrete Mathematics and Theoretical Computer Science,

5 [3] Bengt Aspvall, Michael F. Plass, and Robert Endre Tarjan. A linear-time algorithm for testing the truth of certain quantied Boolean formulas. Information Processing Letters, 8(3):121{123, March See also erratum [4]. [4] Bengt Aspvall, Michael F. Plass, and Robert Endre Tarjan. Erratum: \A linear-time algorithm for testing the truth of certain quantied Boolean formulas" [Inform. Process. Lett. 8 (1979), no. 3, 121{123; MR 80b:68050]. Information Processing Letters, 14(4):195{195, June See [3]. [5] Martin Davis, George Logemann, and Donald Loveland. A machine program for theorem-proving. Communications of the ACM, 5(7):394{397, July [6] Martin Davis and Hilary Putnam. A computing procedure for quantication theory. Journal of the ACM, 7(3):201{215, July [7] J.W. Freeman. Improvements to Propositional Satisability Search Algorithms. PhD thesis, University of Pennsylvania, [8] Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York, [9] Ian P. Gent and Toby Walsh. The search for satisfaction. URL [10] Tracy Larrabee. Test pattern generation using boolean satisability. IEEE Transactions on Computer-Aided Design, pages 4{15, January [11] Tracy Larrabee and Yumi Tsuji. Evidence for a satisability threshold for random 3CNF formulas. Technical Report UCSC-CRL-92-42, University of California, Santa Cruz, Jack Baskin School of Engineering, November