Master s Project Proposal: Enhancements of SAT-Solvers for Some Combinatorial NAE-SAT Problems

Transcription

1 Master s Project Proposal: Enhancements of SAT-Solvers for Some Combinatorial NAE-SAT Problems Nicolette Nicolosi Department of Computer Science Rochester Institute of Technology Rochester, NY nan2563@cs.rit.edu June 2, 2013 Committee Members Chair: Signature: Date: Prof. Stanis law Radziszowski spr@cs.rit.edu Reader: Signature: Date: Prof. Hans-Peter Bischof hpb@cs.rit.edu Observer: Signature: Date: Prof. Zack Butler zjb@cs.rit.edu 1

2 Abstract Improvements on specific subsets of SAT relevant to certain NP-complete problems can yield further results in both fields. This project proposes to utilize knowledge of phase transition boundaries for variants of SAT to work toward improving one or more existing SAT-solvers to specifically target improvements for not-all-equivalent satisfiability (NAE-SAT). This is particularly applicable to the field of Ramsey theory because expressions that are created when searching for non-monochromatic complete subgraphs will tend to be in NAE-SAT form. The improvements will include modifications to the solver logic to recognize satisfying assignments for NAE-SAT, which may allow for reduction of large SAT formulas that typically represent NAE-SAT instances. Because the typical NAE-SAT representation contains approximately twice as many clauses as a similar regular SAT instance, improvements to the solver logic may lead to significant reductions in running times for these cases due to the ability to greatly decrease the input formula size. Additionally, there may be a different phase transition point for these NAE-SAT instances, which may lead to improvements for problems that lie near the regular SAT phase transition point due to the ability to express the problem in multiple ways. These modifications will be tested on both Ramsey problems and randomly generated SAT instances. 2

3 Contents 1 Introduction 4 2 Background Satisfiability SAT-Solvers Phase Transition Ramsey Theory Project Approach Theory Software Software Tools SAT-Solver Design Languages Project Results Deliverables Evaluation Proposed Outline Milestones List of Figures 1 A CNF formula Phase Transition for 3-SAT A 2-coloring of K 5 that does not contain any monochromatic K Monochromatic triangles on K List of Algorithms 1 DPLL Algorithm CDCL Algorithm

4 1 Introduction There are constant improvements being made in approximating solutions for boolean satisfiability (SAT) problems. This coupled with the wide availability of general and specialized SAT-solvers makes it easy to see the utility of expressing other NP-complete problems in the form of a SAT problem [17]. Improvements on specific subsets of SAT relevant to these problems can yield further results in both fields. This project proposes to work toward improving one or more existing SAT-solvers to specifically target producing new results in the field of Ramsey theory. In particular, targeting improvements for not-all-equivalent satisfiability (NAE-SAT) problems seems as though it would be useful for solving Ramsey problems. 2 Background 2.1 Satisfiability Satisfiability is the problem of determining a qualifying assignment of logical values for a boolean expression that results in the expression evaluating to true. A boolean satisfiability expression typically consists of a series of clauses (disjunctions of literals) joined by conjunctions. This is known as Conjunctive Normal Form (CNF). See Figure 1 for an example. (x y) (y z) Figure 1: A CNF formula The general case of SAT (k-sat, where k is the exact number of literals per clause) has been widely studied. 3-SAT is a subset of the generalized k-sat problem that requires that each clause contain exactly three literals - this and all variants of k-sat where k > 2 are NP-complete. NAE-SAT is a subproblem within SAT which requires that a satisfying assignment contain no clause which has all literals equal in value. [16] [13] As an example, consider the CNF formula from Figure 1. A qualifying assignment in regular SAT would set x = true, y = true, and z = true. However, this assignment is not valid in NAE-SAT because x and y cannot both be true, so we must assign x = true, y = false, and z = true. 2.2 SAT-Solvers The International SAT Competition is held on a yearly basis to compare performance of entered SAT-solvers on three categories of SAT problems - application (typical problems one might see in real life), maliciously crafted (designed to trip up most solvers), and random. Recent successful solvers include minisat, glucose, SATzilla, and ppfolio. 4

5 There are typically two approaches to writing improved SAT-solvers - enhancements to the DPLL or CDCL (conflict driven clause learning) algorithms, and using a portfolio of other solvers in addition to some sort of heuristic for choosing the best solver for a given problem. minisat and variants on it use the first approach, while SATzilla uses a portfoliobased approach that trains a preprocessor to categorize SAT problems and then uses that knowledge to choose the most effective solvers to try. ppfolio by itself uses a naive portfolio based approach. The basic SAT-solver algorithm is DPLL, named for the authors Davis, Putnam, Logemann, and Loveland [4]. It is a chronological backtracking algorithm that systematically examines all possible truth assignments to find a satisfying assignment. A clause is considered in one of four possible states - unsatisfied, satisfied, unit, or unresolved. An unsatisfied clause has all literals assigned to false (0), while a satisfied clause has one or more literals assigned to true (1). A clause is considered unit if all literals but one are assigned to false, and the remaining literal is unassigned. An unresolved clause is one that is in none of these states. If a clause is unit, then the unassigned literal must be assigned true for the clause to be satisfied - this is known as the unit clause rule. Iteratively applying this rule is known as unit propagation or Boolean Constraint Propagation (BCP) and is done after branching steps to identify variables that must be assigned certain values. If this results in an unsatisfied clause, there is a conflict and backtracking occurs. [8] This can be done recursively, but is typically implemented iteratively - see Algorithm 1. Conflict-Driven Clause Learning (CDCL) is an improved approach that came from the original DPLL backtracking algorithm, which relies on trying to set a literal to a value and then adjusting that assignment based on the conflicts that arise. CDCL is a non-chronological backtracking algorithm that is designed to be more intelligent than DPLL in that it uses additional techniques such as learning clauses from backtracking for future use, periodically restarting the backtracking search, and heuristic branching techniques [7]. Each variable has a number of properties that CDCL solvers can take into account, including the value (ν(x i ) {0, u, 1}), antecedent (α(x i ) ϕ {NIL}), and decision level (δ(x i ) { 1, 0, 1,..., X }). A variable s value is implied if it is set as a result of applying the unit clause rule. The antecedent of the variable is the unit clause ω used to imply this value - unassigned or decision variables (variables used for branching) have a NIL antecedent. The decision level of a variable is simply the depth within the decision tree at which the variable is assigned to true or false - unassigned variables have a decision level of 1. The decision level of an arbitrary variable can be calculated as follows: δ(x i ) = max({0} {δ(x j ) x j ω x j x i }) [8] The basic algorithm is shown in Algorithm 2. The main drawback to this approach is that it rapidly becomes difficult to preserve learned clauses due to memory constraints. However, not preserving learned clauses can lead to repeating conflict resolutions. Solvers approach this problem in different ways - a naive approach is to only remember the most recently learned clauses. Another simple approach is to prefer to remember shorter clauses to reduce memory usage. The Chaff algorithm and the minisat [5] implementation of it track the activity of a learnt clause in resolving conflicts - more used clauses are preserved while less used ones are pruned. glucose uses a new method 5

6 Algorithm 1 Iterative DPLL Backtracking Algorithm [11] function RESOLVE-CONFLICT(c) d = most recent decision not tried for all values if d == NULL then return false end if Set d to opposite value Mark d tried Undo invalidated implications return true end function function BCP(c) while NOT CONFLICT do Apply the unit clause rule until satisfiable or a conflict is found end while end function while φ contains unassigned literals do Assign a value to an unassigned literal and push a record of this to the decision stack while NOT BCP() do if NOT RESOLVE-CONFLICT() then return UNSAT end if end while end while return SAT that attempts to predict whether a clause is likely to be used in the future [3]. The Chaff algorithm [11] features a number of improvements on DPLL/CDCL algorithms. It introduces the concept of watched literals, which prevents unnecessary checking of clauses during unit propagation by having two literals in a clause be watched - until one of those two literals is set to 0, it isn t necessary to attempt unit propagation. For deciding which variable and state should be selected, Chaff uses a new decision heuristic called Variable State Independent Decaying Sum (VSIDS). This heuristic attempts to first satisfy recent conflict clauses, and has low overhead since it does not rely on the variable state and the statistics are only updated when a new conflict is discovered. Another optimization involves occasionally clearing the current variable state and restarting the search with the knowledge of the learned clauses. Minisat provides a minimal implementation of an improved variant on the Chaff algorithm that is designed to be easily extensible. Glucose is an extension of this solver that identifies glue clauses and calculates Literal Blocks Difference (LBD) for clauses to deter- 6

7 Algorithm 2 Conflict-Driven Clause Learning (CDCL) (modified from Rintanen[14]) function CDCL(C) Initialize v to satisfy all unit clauses in C Extend v by unit propagation with v and C Initialize decision level to 0 while C does not contain empty clauses do Choose a variable a with an unassigned value v(a) Increase the decision level by 1 Assign v(a) to either true or false Extend v by unit propagation with v and C if v causes a conflict in a clause in C then Infer a new clause c and add it to C (learn a new clause) Undo implied assignments caused by a so that c is not falsified end if end while end function mine whether they should be kept. A block is defined by Audemard and Simon [3] as all literals of the same level in the decision tree. If the literals of a clause are partitioned by decision level, the number of subsets that result from such a partitioning is called the Literal Blocks Distance (LBD) of that clause. When a clause is learned, glucose computes and also stores the LBD of the clause. Glue clauses are defined as clauses which have an LBD of 2 - these clauses are important because they only contain one variable of the last decision level. This variable will be glued to the block of literals that was propagated previous to this point by the solver. Another important feature of glucose is aggressive deletion of clauses - half of the learned clauses are deleted at regular conflict intervals to increase the efficiency of the unit propagation step. The increased efficiency from using the LBD measure allows for this aggressive clause deletion without negatively affecting overall performance because this measure is designed to preserve good clauses instead of every clause. The portfolio-based approach to creating SAT-solvers is typically successful because it can rely on the strengths of all the component solvers while mitigating their weaknesses by using other solvers that work better for those cases. Ppfolio takes a naive approach by simply starting the 5 component solvers either sequentially or in parallel [15]. SATzilla relies on a pattern recognition-based approach that trains the system initially on a large data set containing previously used SAT problems. This training is done to determine classes of SAT problems and which solvers work best for these types of problems. When attempting to solve novel instances, SATzilla will first attempt to categorize the problem based on what it knows about the previous instances it has seen. It then will select the two predicted best performing solvers for that class to attempt to find a satisfying assignment - the second solver is used if the first is unable to find a satisfying assignment in the given amount of time. [18] 7

8 2.3 Phase Transition Many problems exhibit behavior such that we can predict whether or not an instance of the problem will be easy to solve. In the general case, there appears a point at which instances of certain problems rapidly become very difficult and instances that are further away from this point become much easier to solve. For satisfiability, we can compare the ratio of variables to clauses (α) to the fraction of unsatisfiable problems [6] or to the computation cost [10] to approximate points at which boolean satisfiability expressions will likely be satisfiable or not, and how quickly such a determination can be made. See Figure 2 for an example graph illustrating phase transition. Figure 2: An example graph illustrating phase transition for 3-SAT for varying numbers of total literals (N)[6] A paper by Kirkpatrick [6] shows that for 3-SAT, α appears to be around A paper by Achlioptas et al. [2] studies the phenomenon of phase transition for the NAE-SAT problem and is particularly relevant to this project. This paper has found that the phase transition boundary for random NAE-SAT lies somewhere in 2 < α < 2.1. This is consistent with the finding that the boundary for general 3-SAT is about 4.2 given the statement by Achlioptas et al. [2] that NAE-3SAT instances typically contain twice as many clauses as instances of general 3-SAT. 8

9 2.4 Ramsey Theory Ramsey theory is an area within combinatorics that studies combinatorial objects which contain smaller objects - the classical 2-color definition of a Ramsey number R(r, s) is the smallest positive integer n such that for any complete 2-colored graph on R(r, s) vertices, there exists either a complete subgraph on r vertices that is entirely one color, or a complete subgraph on s vertices that is completely the other color [12]. As an example, we can show that R(3, 3) = 6 by showing that R(3, 3) > 5 and R(3, 3) 6. We can easily show that R(3, 3) > 5 because it is possible to 2-color a K 5 without a monochromatic K 3 (see Figure 3). Figure 3: A 2-coloring of K 5 that does not contain any monochromatic K 3 To show that R(3, 3) 6, we assume we have a 2-colored K 6. For any vertex v in the graph, we know that there are 5 edges connected to v, and thus we know that at least 3 must be the same color. If any of the three vertices at the end of these edges are connected to each other by an edge of the same color, then there is a monochromatic triangle. If not, then those three edges are all the opposite color and make up a monochromatic triangle of that color. Thus, any 2-colored K 6 must contain a monochromatic K 3, so R(3, 3) 6. See Figure 4 for an illustration. Because this is expressed as a 2 color graph coloring problem, it can be easily converted to a satisfiability problem with the literals representing whether or not a particular edge is a specific color. An example of expressing a Ramsey problem as a SAT problem is given in [17], which describes a graph G with two colors F and T. We consider each edge to be a boolean variable. When seeing if G (K 3, J 4 ), we define two types of clauses to check that there is no K 3 formed in color F and no J 4 formed in color T. To check the case where no K 3 (containing edges {e 1, e 2, e 3 }) is formed in color F, we include the clause (e 1 e 2 e 3 ) which will force at least one edge to have color T. For the other case, where no J 4 (containing 9

10 Figure 4: Monochromatic triangles on K 6 - if edge d or e were red instead of blue, there will be a monochromatic triangle. If both edges are blue as shown, coloring edge f either color will produce a monochromatic triangle edges {e 1, e 2, e 3, e 4, e 5 }) is formed in color F, we include the clause (e 1 e 2 e 3 e 4 e 5 ), which, similarly to the previous clause, will force at least one edge to have the color F. Enhancements to solving NAE-SAT problems may be useful since the expressions that we will create when searching for non-monochromatic complete subgraphs will tend to be in NAE-SAT form - that is, if we have three edges {e 1, e 2, e 3 }, we will check (e 1 e 2 e 3 ) (e 1 e 2 e 3 ) [2]. 3 Project Approach 3.1 Theory This project will study current state of the art SAT-solvers and their approaches with the goal of modifying one or two to produce improved results for NAE-SAT. I will modify glucose (an extension of minisat) by either building a layer on top that calls the original solver or 10

11 editing the solver source code to include optimizations of the CDCL algorithm for NAE- SAT - it may be necessary to combine these approaches. Once the SAT-solver modifications are made, I will test the modified and original solvers on randomly generated and Ramsey problems, and possibly other similar problems, and try to determine if there is a phase transition point specific to these problems expressed as NAE-SAT and where the threshold for that lies. As a second test, I will include my new solver in a portfolio-based solver such as SATzilla or ppfolio. I will then measure the effectiveness of my solver both by comparing its performance to the performance of the unmodified solver that I started with and by comparing it against the other solvers in the portfolio. 3.2 Software 3.3 Software Tools I will generate graphs using both custom tools that I will write and the nauty software package by Brendan McKay [9]. I will also write a tool to convert instances of these graphs to SAT problems in CNF form so that they can be input to the SAT-solver. The SAT-solvers that I will modify will be glucose and SATzilla or ppfolio. 3.4 SAT-Solver Design Glucose is based on minisat, which uses a Conflict-Driven Clause Learning (CDCL) algorithm. For this part I will have to write something to recognize NAE-SAT instances and additionally include some optimizations such as reducing the problem to a more easily solved form. Reducing the problem may work for certain cases where we have knowledge of the typical form the SAT problem will take. Because NAE-SAT has additional restrictions on what is considered a satisfying assignment, I will likely need to include some logic to reject assignments that would be considered satisfying for SAT but not NAE-SAT. SATzilla is a portfolio-based SAT-solver that includes 31 leading solvers. It initially trains a preprocessor to recognize different classes of SAT problems and classifies them based on which solvers produce the fastest output. Time-permitting, I may add my solver to SATzilla s portfolio and then retrain my instance to include a set of NAE-SAT problems. A simpler portfoliobased modification might be to add my solver to the naive ppfolio solver since it simply runs all component solvers at once on different cores, or in sequence, and does not require any retraining. 3.5 Languages The graphs to be tested will be expressed as SAT problems in conjunctive normal form (CNF) using the DIMACS format [1]. Glucose and ppfolio are written in C, while SATzilla uses Java and Ruby. The modifications to glucose will be written in C, while any preprocessing and helper code will either be written in C or an interpreted language such as Ruby. If it 11

12 is feasible to run tests using a portfolio solver such as SATzilla or ppfolio, I will likely not need to write much, if any, additional code for these tests. 4 Project Results 4.1 Deliverables The project deliverables will include the project code, any test files, my results, and a written report containing an in-depth discussion of the topic, my project approach, and my results Evaluation The main outcomes expected from this project are a comparison between NAE-SAT and regular SAT, exploration of the phase transition boundary for NAE-SAT and comparison of this value with the phase transition boundary for regular SAT, tests of the performance of a SAT-solver modified for NAE-SAT on random instances, and tests of this solver on instances generated from select Ramsey and Folkman problems Proposed Outline The following is a proposed outline for my final report. 1. Introduction I will provide a high-level overview of the project and its intended outcomes. I will also provide a brief discussion of the motivation for the project. 2. Background Satisfiability SAT-Solvers Phase Transition Ramsey Theory 3. Problem Description This section will discuss the application of the background information discussed in the previous section and the purpose of the project. 4. Algorithms and Implementation The algorithms that were used will be described, as will the implementation details, system design, and usage instructions. 5. Results and Discussion The results will be presented and discussed. 12

13 6. Conclusion The work and results will be summarized, and ideas for future work will be briefly presented. 7. References I expect to have approximately 20 to 30 references. 4.2 Milestones The following timeline for project milestones and completion is proposed: Research and Test SAT-solvers April 21, 2013 Proposal completion May 20, 2013 Baseline tests May 30, 2013 Initial SAT-solver modifications June 7, 2013 Tests on modified solvers complete June 14, 2013 Initial report draft June 19, 2013 Final report draft sent to committee June 21, 2013 Final Report June 25, 2013 Defense July 5, 2013 References [1] Dimacs CNF format. [2] Dimitris Achlioptas, Arthur Chtcherba, Gabriel Istrate, and Cristopher Moore. The Phase Transition in 1-in-k SAT and NAE 3-SAT. In Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms, SODA 01, pages , Philadelphia, PA, USA, Society for Industrial and Applied Mathematics. [3] Gilles Audemard and Laurent Simon. Predicting Learnt Clauses Quality in Modern SAT Solvers. In Proceedings of the 21st international joint conference on Artificial intelligence, pages Morgan Kaufmann Publishers Inc., [4] Martin Davis, George Logemann, and Donald Loveland. A Machine Program for Theorem-proving. Communications of the ACM, 5(7): , [5] Niklas Eén and Niklas Sörensson. An Extensible SAT-Solver. In Theory and Applications of Satisfiability Testing, pages Springer,

14 [6] Scott Kirkpatrick, Bart Selman, et al. Critical Behavior in the Satisfiability of Random Boolean Expressions. Science-AAAS-Weekly Paper Edition-including Guide to Scientific Information, 264(5163): , [7] João P Marques-Silva and Karem A. Sakallah. GRASP: A Search Algorithm for Propositional Satisfiability. Computers, IEEE Transactions on, 48(5): , [8] Joao Marques-Silve, Ines Lynce, and Sharad Malik. Handbook of Satisfiability. IOS Press, [9] Brendan D McKay. Nauty users guide (version 2.4). Computer Science Dept., Australian National University, [10] Rémi Monasson, Riccardo Zecchina, Scott Kirkpatrick, Bart Selman, and Lidror Troyansky. Determining Computational Complexity from Characteristic Phase Transitions. Nature, 400(6740): , [11] Matthew W Moskewicz, Conor F Madigan, Ying Zhao, Lintao Zhang, and Sharad Malik. Chaff: Engineering an Efficient SAT Solver. In Proceedings of the 38th annual Design Automation Conference, pages ACM, [12] Stanis law P Radziszowski. Small Ramsey Numbers [13] Vishwambhar Rathi, Erik Aurell, Lars Rasmussen, and Mikael Skoglund. Bounds on Threshold of Regular Random k-sat. Theory and Applications of Satisfiability Testing SAT 2010, pages , [14] Jussi Rintanen. Heuristics for Planning with SAT. In Principles and Practice of Constraint Programming CP 2010, pages Springer, [15] Olivier Roussel. Description of ppfolio. SAT Competition 2011, [16] Thomas J Schaefer. The Complexity of Satisfiability Problems. In Proceedings of the tenth annual ACM symposium on Theory of computing, pages , [17] Daniel S Shetler, Michael A Wurtz, and Stanis law P Radziszowski. On Some Multicolor Ramsey Numbers Involving k 3 + e and k 4 e. SIAM Journal on Discrete Mathematics, 26(3): , [18] Lin Xu, Frank Hutter, Holger H Hoos, and Kevin Leyton-Brown. SATzilla: Portfoliobased Algorithm Selection for SAT. Journal of Artificial Intelligence Research, 32(1): ,