Improving Diversification in Local Search for Propositional Satisfiability

Transcription

1 Improving Diversification in Local Search for Propositional Satisfiability by Thach-Thao Nguyen Duong Bachelor of Information Technology, University of Science, Vietnam (2006) Master of Computer Science, University of Science, Vietnam (2009) Institute for Integrated and Intelligent Systems Science, Environment, Engineering and Technology Griffith University A thesis submitted in fulfillment of the requirements of the degree of Doctor of Philosophy Febuary, 2014

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3 Abstract In recent years, the Propositional Satisfiability (SAT) has become standard for encoding real world complex constrained problems. SAT has significant impacts on various research fields in Artificial Intelligence (AI) and Constraint Programming (CP). SAT algorithms have also been successfully used in solving many practical and industrial applications that include electronic design automation, default reasoning, diagnosis, planning, scheduling, image interpretation, circuit design, and hardware and software verification. The most common representation of a SAT formula is the Conjunctive Normal Form (CNF). A CNF formula is a conjunction of clauses where each clause is a disjunction of Boolean literals. A SAT formula is satisfiable if there is a truth assignment for each variable such that all clauses in the formula are satisfied. Solving a SAT problem is to determine a truth assignment that satisfies a CNF formula. SAT is the first problem proved to be NP-complete [20]. There are many algorithmic methodologies to solve SAT. The most obvious one is systematic search; however another popular and successful approach is stochastic local search (SLS). Systematic search is usually referred to as complete search or backtrack-style search. In contrast, SLS is a method to explore the search space by randomisation and perturbation operations. Although SLS is an incomplete search method, it is able to find the solutions effectively by using limited time and resources. Moreover, some SLS solvers can solve hard SAT problems in a few minutes while these problems could be beyond the capacity of systematic search solvers. Due to the widespread demands for efficient SLS for SAT (SLS-SAT) and also due to the fact that many large-scale problems encoded successfully into SAT, methods to boost the performance of the SLS-SAT solvers are highly desired. Although SLS-SAT have been rigorously investigated in the recent decades, there are still various open areas that should be addressed to further boost the performance of SLS-SAT. Such open areas include improving diversification phases to widely explore the search space and improving the intensification to find solutions faster, escaping local i

4 ii Abstract minima intelligently, and controlling the diversification and intensification during the search. All of these issues require handling the trade-off between intensification and diversification. In this thesis, we proposed and developed novel strategies to enhance SLS performance in both intensification and diversification phases. We aimed to verify empirically the efficiency and robustness of our proposed strategies on structured and random instances. Since local search and systematic search have different effects and impacts on solving structured and random instances, optimal settings of enhancement strategies are learnt separately on different instance sets. The outcomes showed that optimal parameter settings depend on the types of problem instance. Our four strategies to improve performance of SLS for SAT are listed below: Learning strategies on variables that are involved in stagnation situation with a view to efficiently preventing local search from falling into traps. [Part of this work was published as regular papers at AAAI-2012, ECAI-2012, and AI-2012 conferences] Exploiting clause weights in clause selection at stagnation to help SLS escape local minima. [Part of this work was published as a regular paper at IJCAI-2013 conference] Decreasing greediness in the scoring function in the diversification phases to boost SLS to reach solutions faster especially in structured instances [Part of this work was published as a regular paper and was awarded the Best Paper Award at AI-2013 conference] Backtracking from local minima to identify and revert the conflicts as an alternative way to escape local minima. [Part of this work was published as a regular paper at SCAI-2013 conference] All of these enhancement strategies are tested on verification benchmarks, and SAT 2011 and 2012 competition datasets. Our experimental results showed the robustness and efficiency of the proposed novel strategies on a wide range of instance sets. The results demonstrate that by following our proposed enhancement strategies, significant improvements could be achieved on structured and random instances.

5 Publications The results presented in this thesis have already been published in a number of conferences that are highly-ranked by the Excellence in Research for Australia (ERA) ranking system. The articles are listed below in their chronological order of publishing. 1. Duc-Nghia Pham, Thach-Thao Duong, Abdul Sattar, Trap Avoidance in Local Search Using Pseudo-Conflict Learning, In Proceeding of the 26th Conference on Artificial Intelligence (AAAI), pp , July 2012, Toronto, Canada. [85] 2. Thach-Thao Duong, Duc Nghia Pham, Abdul Sattar, A Study of Local Minimum Avoidance Heuristics for SAT, In Proceeding of the 20th European Conference on Artificial Intelligence (ECAI), pp , August 2012, Montpellier, France. [27] 3. Thach-Thao Duong, Duc Nghia Pham, Abdul Sattar, A Method to Avoid Duplicative Flipping in Local Search for SAT, In Proceeding of the 25th Australasian Joint Conference on Artificial Intelligence (AI), pp , 4-7 December 2012, Sydney, Australia[26] 4. Thach-Thao Duong, Duc Nghia Pham, M.A.Hakim Newton, Abdul Sattar, Weight-Enhanced Diversification in Stochastic Local Search for Satisfiability, In Proceeding of the 23th International Joint Conference on Artificial Intelligence (IJCAI), pp , 3-9 August 2012, Beijing, China [29] 5. Huu-Phuoc Duong, Thach-Thao Duong, Duc-Nghia Pham, Abdul Sattar, Anh-Duc Duong, Trap escape for local search by backtracking and conflict reverse, In the proceeding of the 12th Scandinavian Conference on Artificial Intelligence (SCAI ), pp 85-94, November 2013, Aalborg, Denmark [25] 6. Thach-Thao Duong, Duc-Nghia Pham, Abdul Sattar, Diversify intensification phases for local search with a new probability distribution, In the proceeding of the 25th Australasian iii

6 iv Publications Joint Conference on Artificial Intelligence (AI), pp , 1-6 December 2013, Dunedin, New Zealand (Best paper award and best student paper award) [28]

7 Statement of Originality This work has not previously been submitted for a degree or diploma to any university. To the best of my knowledge and belief, the thesis contains no material previously published or written by another person except where due reference is made in the thesis itself. Thach-Thao Nguyen Duong Febuary, 2014 v

8 Contents Abstract i Publications iii Statement of Originality v Contents vi List of Figures xi List of Tables xii List of Algorithms xv Abbreviations and Notations xvi Acknowledgements xviii 1 Introduction CSP SAT Constraint-based Search Strategies Search Strategies for SAT Research Questions Research contributions Thesis Outline Stochastic Local Search For SAT Constraint-based Stochastic Local Search Evaluation Function and Objective Function Global Optimum, Local Optimum and Plateau Iterative Improvement Randomised Iterative Improvement vi

9 Contents vii Probabilistic Iterative Improvement Dynamic Local Search Simulated Annealing Tabu Search Iterated Local Search Existing Stochastic Local Search for SAT Preliminary Notations GSAT HSAT GSAT-random-walk TSAT - Tabu GSAT GSAT Hill-climbing WalkSAT WalkSAT/SKC Novelty Adaptive WalkSAT and AdaptNovelty Clause-weighting Scheme GSAT-clause-weight and Breakout WGSAT and WGSAT-Decay DLM GLSSAT SDF ESG SAPS PAWS Some Contemporary and Hybrid Algorithms G 2 WSAT in 2005 and adaptg 2 WSAT+ in VW1 and VW2 in gnovelty + in TNM Sparrow EagleUP

10 viii Contents sattime CCASat The Categorisation and Milestones in Developing SLS-SAT Pseudo-conflict Learning Introduction Trap Avoidance in SLS-SAT gnovelty Exploration of Poor Stagnation Performance Trap Prevention Strategy Stagnation Paths and Stagnation Weights Learning and Prevention from Stagnation Pseudo-conflict Learning NoveltyPCL: Integration of PCL on Novelty gnovelty + PCL: Integration of PCL on gnovelty Evaluation of gnovelty + PCL Ternary Chains with Pseudo-Conflict Learning Verification Benchmarks SAT 2011 Benchmarks Search Enhancements for PCL Forgetting Outdated Conflicts A Study of Uncertainty PCL for Uncertainty and Forgetting Variants of PCL Evaluation of Search Enhancements Ternary Chains Verification Benchmarks Conclusion and Future Work Weight-Enhanced Diversification Introduction Clause and Variable Weighting gnovelty

11 Contents ix 4.3 Weight-Enhanced Diversification Clause-Weighting Enhancement Variable-Weighting Enhancement gnovelty + GC Experimental Results Experiments on Ternary Chains Experiments on Verification Problems Experiments on SAT2011 Benchmarks Discussions Conclusion And Future work Probability-based Evaluation Function Introduction Preliminaries Basic Scoring Function Dynamic Scoring Function Clause-weighting VW Sparrow Probability-based Dynamic Scoring Function Motivation Defining a Probability Distribution Probability Distribution on Greediness Probability Distribution on Diversification Diversification Parameter α The PCF Algorithm Experiments Experiment Setup Experiment Analysis Analysis of Parameter Configurations Conclusion and Future work

12 x Contents 6 Pseudo-Conflict Reversion Introduction Preliminaries gnovelty + PCL PCL Heuristics at Stagnation Phases Backtracking Retrieval to Reverse Pseudo-Conflicts Backtracking Retrieval Variable Weights for Tie-breaking NoveltyE for Trap Escape The NovEsc Algorithm Experiments Experiment Set up Result Analysis Discussion about Parameter Configurations Conclusion and Future Works Conclusions Summary of Contributions Future Directions

13 List of Figures 2.1 State space landscape of local search Dynamic local search fills local minima Flow chart illustrated the Novelty s mechanim of variable selection Performances of the 7 canonical SLS-SATs on Ternary chain gnovelty + PCL (vs) other SLS-SATs on Ternary chain size [-0] gnovelty + PCL (vs) SLS-SATs on Ternary chain size [0-00] gnovelty + PCL (vs) other SLS-SATs on Verification benchmarks gnovelty + PCL (vs) other SLS-SAT solvers on SAT 2011 competition dataset Runtime distribution of gnovelty + PCL on Ternary chain size [0,00] Runtime comparision of PCL: single (vs) duplication (vs) all selection Runtime comparision of PCL: Time-window (vs) No Time Window Runtime comparision of PCL: Smooth (vs) No Smooth Performance of gnovelty + PCL on Verification benchmarks Performance of gnovelty + PCL on SAT 2011 competition instances gnovelty + GC (vs) SLS-SATs on Ternary chains size[-0] gnovelty + GC (vs) SLS-SATs on Ternary chains size[0-00] Runtime Correlation between gnovelty + GC a and SLS-SATs Run-time correlation of PCF (vs) SLS-SATs Runtime Correlation of NovEsc (vs) SLS-SATs xi

14 List of Tables 3.1 Algorithmic comparision of SLS-SATs on trap escape Performance of gnovelty + PCL on Verification benchmarks Performance of gnovelty + PCL on the 2011 SAT competition datasets Average of duplication ratios of VW2, gnovelty +, Sparrow Variants of Pseudo-Conflict Learning Performance of optimised PCL variants in Verification benchmarks Performance of optimised PCL variants in SAT 2011 benchmarks Performance of gnovelty + GC (vs) SLS-SATs on cbmc, swv, and sss-sat Performance of gnovelty + GC (vs) SLS-SATs on SAT 2011 structured instances Performance of gnovelty + GC (vs) SLS-SATs on SAT 2011 random instances Optimal parameters of gnovelty + GC Performance of PCF on cbmc, SAT 2012 Crafted instances Performance of PCF on SAT 2011 Competition medium-size random instances Optimal parameters of PCF Performance of NovEsc (vs) SLS-SATs on cbmc and sss-sat Optimal parameters of NovEsc xii

15 List of Algorithms 1 Iterative Improvement Randomised Iterative Improvement Probabilistic Iterative Improvement Dynamic Local Search Simulated Annealing Tabu Iterative Improvement Iterated Local Search Random-Assignment ( Θ ) GSAT(Θ, maxt ries, maxsteps ) HSAT (Θ, maxt ries, maxsteps) LeastRecentVariable (V arset) RandomWalk (Set) RandomWalks (Θ,Ω) GSAT-random-walk (Θ,maxT ries, maxsteps, wp) GSAT-Tabu (Θ,maxT ries, maxsteps, k) GSAT Hill-climbing (Θ, maxt ries, maxsteps) WalkSAT-SKC (Θ) NoveltySelection (f, c, p) MostRecentVariable (V arset) Novelty (Θ, p) Novelty+ (Θ, p) InitNoveltyNoise (θ) AdaptNoveltyNoise(p, φ) AdaptNovelty + (Θ) xiii

16 xiv List of Algorithms 25 GSAT-clause-weight (Θ) Breakout (Θ) WGSAT-Decay (Θ, ρ) DLM-98-BASIC-SAT(Θ,θ 1,θ 2,δ o,δ d ) GLSSAT(Θ, smax, pmax, pdecay) SDF(Θ, η, δ) ESG(Θ, α sat, α unsat, η) SAPS(p smooth, wp, ρ, SAP S thresh ) PAWS(Θ, Max inc ) Novelty++ (f, p, dp) NoveltySelection( f, c, p) G 2 WSAT adaptg 2 WSAT+ (Θ, p, wp) VW1(Θ, p) VW2(Θ, p,s,b) gnovelty + (Θ, sp) adaptg 2 WSAT Selection (p2, wp) T NM Sparrow2011(Θ,a 1, a 2, a 3 ) EagleUP(Θ) NoveltySelection( f(), c, p, satt ime) sattime (Θ, wp, p) sattime (Θ, wp, p) CCA heuristic at Greedy phases (Θ) Swcca (Θ, γ, ρ) CCApscore (Θ) SLSSAT(Θ,MaxSteps) gnovelty + (Θ, MaxSteps, sp) Pseudo-Conflict Learning PCL(k,H) NoveltyPCL(p) gnovelty + PCL(k,sT,dp,sp)

17 List of Algorithms xv 56 Pseudo-Conflict Learning PCL(k,H,s,T,tp) gnovelty + (Θ, sp) WeightedNovelty(p) gnovelty + GC(β,p) gnovelty + GC (Θ, β, sp) PCF(Θ, sp) gnovelty + PCL(k,sp) Pseudo-conflict learning strategy PCL(k,H) NoveltyE(P, γ, p) NovEsc(γ, k, sp)

18 Abbreviations and Notations ABBREVIATION SAT SLS SLS-SAT SLS-SATs GLS GLS-SAT DLS DLS-SAT SAT MAX-SAT CNF DNF DLM PAWS SAPS GLSSAT Eq. Alg. Fig. FULL NAME Boolean Propositional Satisfiability Problems (i.e. Satisfiability in short) Stochastic Local Search Stochastic Local Search for Satisfiability Problems Stochastic Local Search Algorithms for Satisfiability Problems Guided Local Search Guided Local Search for Satisfiability Problems Dynamic local search Dynamic local search for Satisfiability Problems Satisfiability Problem Maximum Satisfiability Problems Conjunctive normal form Disjunctive normal form Discrete Langrangian Multipliers Pure Additive Weighting Scheme Scaling and Probabilistic smoothing Guided Local Search for Satisfiability Problems Equation Algorithm Figure xvi

19 Abbreviations and Notations xvii NOTATION Θ n m Ω Ω v C V V [c] L[c] c u c i v v i F Ω F Ω (v) W Ω W Ω (v) ft[v] cw[c] N[v] NC[v] x : max(f(x)) x : min(f(x)) DESCRIPTION SAT formula Number of variables of Θ Number of clauses of Θ Solution candidate Solution candidate Ω after variable v is flipped The set of clauses of Θ The set of variables in Θ The set of variables in clause c The set of literals in clause c A specific clause. A specific unsatisfied clause. The i-th clause (i = 1..m) A specific variable. The i-th variable. (i = 1..n) Number of unsatisfied clauses of Θ under the assignment Ω Decrease in number of unsatisfied clauses when variable v of Ω is flipped Total weights of unsatisfied clauses Θ under the assignment Ω Decrease in total weights of unsatisfied clauses when variable v of Ω is flipped The last step a variable v is flipped (i.e. ft stands for flip time) The weight of clause c Set of neighbor variables of v Set of clauses containing variable v The x maximising the function f(x) The x minimising the function f(x)

20 Acknowledgements I am deeply grateful to my supervisors Prof. Abdul Sattar and Dr. Duc Nghia Pham for their constructive discussions and brilliant suggestions. With their dedication, they always kept me in the right directions, while at the same time giving me the freedom to explore my own ideas. I gratefully acknowledge their encouragement and tireless support especially in the hard-time of my PhD candidate. I would also like to gratefully acknowledge my colleagues at National ICT Australia (NICTA) Ltd. (Dr. M.A.Hakim Newton and Dr. Charles Gretton) and my friends in the Institute of Integrated and Intelligent Systems of Griffith University for their kindness and helpful comments towards my research. I would like to acknowledge the generous financial and facility assistance from Griffith University and the Queensland Research Laboratory of National ICT Australia (NICTA) Ltd.; without their support this work would not have been possible. Special thanks also to the Research Computing Services of Griffith University for providing the high performance computing infrastructure and IT support during my PhD. And finally, I would like to express my gratitude to my parents for their unconditional support and love. xviii

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22 Chapter 1 Introduction In this chapter, we provide a brief background information on the two popular and well-related research problems: constraint satisfaction problem (CSP) and propositional satisfaction (SAT) problem. We then explore the search strategies used in solving these problems. Our focus, however, is mostly on stochastic local search approaches. Then, we present the research questions investigated in this study, and the contributions made thereby. Lastly, we outline the organization of the rest of the thesis. 1.1 CSP A constraint satisfaction problem (CSP) is defined as the problem of finding an assignment of valid properties or values to objects or variables so that the assignment satisfies all the specified constraints on those objects. If no such assignment exists, CSP in that case attempts to prove that. Many real world problems have constraints on objects. For example, planning and scheduling problems have temporal constraints or budget management problem has tangible constraints of costs on assets. These problems could be described as CSPs and then could be solved efficiently by using constraint solvers. While the exponential nature of possible variable assignments in CSPs make them computationally difficult, the idea of constraint could also be used to effectively prune the search space and thus solve many of the hard problems. In fact, the constraint-based approach has been successfully applied to real world problems in domains such as operation research (e.g. scheduling, timetabling), bioinformatics (DNA sequencing), and electrical engineering (circuit lay-outing) [4]. Definition A CSP is formally defined as a tuple (X, D, C) where 1

23 2 Chapter 1. Introduction V = {v 1, v 2,.., v n } is a finite set of variables whose values are to be determined. D = {d 1, d 2,.., d n } is a set of finite domains where each domain d i contains a finite set of values that can be assigned to the variable v i ; C = {c 1, c 2,.., c m } is a finite set of constraints that restrict the assignment of values to those variables in V. Similar to CSPs, a constraint optimisation problem (COP) can be defined as the problem of finding an assignment of valid properties to objects so that a pre-defined objective function is optimised (maximised or minimised) under that assignment. However, the solution to a COP may or may not be expected to satisfy all the specified constraints on the given objects. Not surprisingly, a CSP can be viewed and solved as a COP where the objective function is to minimise the number of unsatisfied constraints. In this case, the CSP is satisfiable if there exists an assignment under which the objective function equals to zero. 1.2 SAT The SAT problem [39] is a sub-class of the constraint satisfaction problem [8]. Boolean SAT is a special type of SAT where the domain of a variable is binary values {false,true}, and each constraint is a logical disjunction. If the formula is limited to only logic operations (e.g. and ( ), or ( ) and not ( )), then the formula is said to be a propositional boolean formula. There are two normal forms to express a SAT formula: Conjunctive Normal Form (CNF) and Disjunctive Normal Form (DNF). A CNF formula is a conjunction over disjunctions of literals where literals are negated or non-negated variables. In contrast, a DNF formula is a disjunction over conjunctions of literals. k-cnf is a CNF formula having exactly k literals in all clauses, whereas k-dnf is a DNF formula having exactly k literals in all clauses. Example of a 2-CNF is (v 1 v 2 ) ( v 2 v 3 ) where each clause has 2 literals. Similarly, example of a 2-DNF is (v 1 v 2 ) ( v 2 v 3 ) where each clause also has 2 literals. SAT instances in this dissertation are represented as propositional formula in conjunctive normal form, that is, each instance is a conjunction of disjunctions of literals. The problem format is restricted to CNF-encoded formula since this is widely-accepted as the standard input format for SAT solvers [2]. Any propositional formula can be encoded into CNF with linear overhead by adding additional variables.

24 1.3. Constraint-based Search Strategies 3 Definition A Boolean Propositional Satisfiability is formally defined under the CNF form in the following way: A set of n boolean variables V = {v 1, v 2,..., v n }. Each clause has a value of true or false. A set of m clauses C = {c 1, c 2,..., c n }. Each clause e.g. v 1 v 2 v 3 is a disjunction of literals where each literal is a variable or its negation. The notation l mn refers to the literal in the m th clause at the n th position. Each literal can be either true or false. A SAT formula is a conjunction of all given clauses F = c 1 c 2... c n. As an example, consider a propositional formula v 1 ( v 2 v 3 ) that has two clauses. The problem now is how to assign T of F (i.e. true or false) value to the variables x 1, x 2, x 3 so that the formula becomes true. The table below shows all the possible assignments and the corresponding boolean value of the formula. The formula is satisfiable because there exists at least one (in fact three) assignment that satisfies the formula. The bold text lines indicate the satisfied solutions. v 1 v 2 v 3 v 1 ( v 2 v 3 ) F * * F T T T T T F T T T T F F T F F T Under the CNF format, the target of the SAT problem is to determine a binary assignment to the variables so that all clauses are satisfied. If there exists no solution such that all clauses are satisfied the SAT problem is unsatisfiable. Solving propositional satisfiability problem is proved to be an NP-complete problem [20] and then it became a prototypical NP-hard problem [35]. 1.3 Constraint-based Search Strategies Like other finite-domain artificial intelligence (AI) problems, CSP and SAT problems are usually solved by using a search algorithm. In general, there are two main types of search strategy for solving CSPs and SATs: backtracking and stochastic local search (SLS).

25 4 Chapter 1. Introduction Typically, a backtracking (or systematic) algorithm starts with a partial assignment (i.e. only a subset of variables are assigned values and the rest of the variables remain free) and attempts to extend this assignment to a complete one while ensuring all the constraints are satisfied. In other words, it divides the search space hierarchically into a tree structure, where each node represents an assigned variable and each branch coming out of a node represents the value that is assigned to that node. Whenever the search cannot find a domain value to assign to a free variable without violating a constraint, it will backtrack to the previous node and will attempt to find a new value for the variable at that node. If there is a valid value, the search will continue to travel down the search tree or backtrack if necessary until a solution is found. As a result, backtracking algorithms are able to prove that a problem is satisfiable or not. In the optimisation case, backtracking algorithms guarantee to find the optimal solution (due to the ability to generate all solutions to a problem). Therefore, backtracking algorithms are complete. Despite many improvements obtained by using various powerful propagation techniques and different variables/values ordering heuristics, backtracking algorithms unfortunately still do not perform well on many large and complex real world problems [4]. In contrast, a stochastic local search algorithm starts with a full assignment to all variables regardless of whether all constraints are satisfied or not. It then iteratively tries to adjust the current assignment by assigning new values to a subset of variables. Normally, the selection of the next move (i.e. the combination of variables and to-be-assigned values) is heuristically guided towards the solution. In CSP, the objective function is generally to minimise the number of violated constraints. In COP, the objective function also includes optimisation of constraints to move closer to the optimal solution. Although SLS algorithms are incomplete (i.e. they cannot prove that a CSP is unsatisfiable and cannot prove the optimality of its solution for a COP), they tend to produce better results within a reasonable time-frame than backtracking algorithms when solving large and complex problems [34]. 1.4 Search Strategies for SAT SAT has been emerging as a popular constraint solving technology for industry and application problems. However, solving large-scale SAT-encoded application problems is still a big challenge for researchers in the field. Experts in other fields such as verification, planning, circuit design and bioinformatics have made attempts to encode their problems in SAT and used SAT algorithms

26 1.5. Research Questions 5 to solve their corresponding hard SAT instances. SAT competition [2] is a biennial contest to evaluate modern SAT solvers. It allows SAT instances which are constructed from real-world problems and industrial applications to be submitted for solving. The latest SAT solvers are encouraged to participate in the contest. Through these events, hard real world SAT instances have achieved assistance and have taken advantages from the state-of-the-art SAT solvers. Analogous to CSPs, there are two approaches to build a SAT solver: complete (or systematic) and incomplete (or non-systematic) search algorithms. Nowadays, both complete and incomplete algorithms have been used to solve SAT problems. Since systematic algorithms explore the search space completely, they are able to find a solution or prove that problem is unsatisfiable. In contrast, incomplete search does not systematically explore the whole search space. So they cannot determine whether a SAT problem is unsatisfiable or guarantee that a solution will be found. In the SAT competition series, systematic search algorithms showed their advantages over stochastic local search on solving structured problems. Therefore, improving the efficiency of the stochastic local search algorithms in solving structured problems is very critical. However, on the random problems, stochastic local search algorithms were recognised as the winner solvers. Most complete search SAT algorithms are based on the Davis-Putnam procedure [23, 22]. In the category of SLSs, most algorithms are developed based on the GSAT framework [97]. In the early version of GSAT, the algorithm used the objective function of the number of satisfied clauses and then local search strategies based on randomness. The WalkSAT algorithm was further developed incorporating a random walk strategy to select a variable in an unsatisfied clause [96]. Afterwards, dynamic local search algorithms were developed mostly based on clause-weighting scheme [95]. In these days, most state-of-the-art stochastic local search algorithms are hybrid and mixture of these approaches. 1.5 Research Questions The SAT problem is important in solving many practical problems in mathematical logic, constraint satisfaction, VLSI engineering, computing theory, and industrial verification problems [46]. SAT is conceptually a combinatorial problem that has a prominent role in complexity theory and artificial intelligence because it is a prototypical NP-complete problem. The advantage of systematic search compared to local search is that it is more effective in solving highly-constrained problems; however, systematic search is not very efficient in solving

27 6 Chapter 1. Introduction large-scale problems because of a big search space. In contrast, SLS heuristics, based on randomisation and perturbation, are very efficient and fast to find the solutions even with limited time and resource. Additionally, SLS is very effective for large-scale problems because of the compromise between complexity and goal-orientation via randomisation mechanism that help explore the search space widely. Through series of SAT competitions, the efficiency and robustness of SLS in solving large-scale combinatorial problems are greatly improving [2]. Because of the advantages of SLS-SATs, we focus our work on local search and improve the performance of local search for SAT. Stochastic local search is very quick in solving random instances but unfortunately not so efficient in highly-constrained structured problems. In general, this is due to the three main limitations incompleteness, re-visitation, and stagnation of stochastic local search algorithms [48]. Incompleteness: Since SLS does not explore the search space completely, it does not guarantee to find all solutions for the CSPs and the optimal solutions for COPs. In order to broaden visited areas and to explore unvisited areas, some SLS algorithms attempt to improve the diversification and trap escaping strategies. The simplest techniques to improve diversification are random walk [96] and restart [97]. Re-visitation: SLS heuristics neither explicitly store the history of the visited places nor possess a backtracking mechanism like in systematic searches. Therefore, it easily visits previous places several times. There are some memory-based techniques that were used to prevent SLS from re-visiting previous places. These methods save a partial history of search progress and avoid most recently visited places. Tabu search [76] when is applied to SAT act as a short-term memory and restricts re-visitation of recently flipped variables within a specific duration of time. There are SLS-SAT algorithms that employ long-term memory such as variable selection time [37], variable selection frequency [90] and clause unsatisfiability frequency [95]) Stagnation: Local search trajectory is easily attracted to local minima because SLS is not a backtracking search, it is rather based on a goal-oriented heuristic. Plateau and local minimum are two difficult stagnation problems for SLS. Plateau is one of the common problems related to stagnation in SLS. It is defined as a search area where all neighbours have the

28 1.5. Research Questions 7 same value of the objective function. When the search procedure falls in a wide plateau, the algorithm gets stuck in a flat landscape. In that situation, moving to any neighbour does not improve the objective function. In case of large plateaus, if the SLS performs flat (sideways) moves to the neighbours, it still remains in the large plateau. More over, a local minimum is a search point where all neighbours have worse-cost. If SLS falls into a deep local minimum, it will eventually fall back to the deep valley even though the algorithm performs many worse-cost move. In short, it is time-consuming for the algorithm to escape a trap if it falls into large plateaus and deep local minima. The cure from this trapped circumstance is to restart at another position in the search space or to jump to another search area. There are sophisticated methods that help the search escape local minima and improve diversification without resorting to any restart mechanism. Such methods include flat move [114], dynamic adjustment on objective function [95, 80], and probability-based random walk [96]. In reality, SLS encounters more stagnation situation in structured instances than in random instances. There are two reasons for this phenomenon. Firstly, structured instances are from real-world problem; hence they are highly constrained and they additionally contain hidden constraints. These structured constraints and hidden constraints create local minima for SLS. Once local search falls into a trap, it is hard for them to escape because of the large number of violated constraints. SLS needs to solve the conflicts in violated constraints to escape the traps. Secondly, most SLSs prefer to explore the search space greedily. Thus, it is extremely hard for SLS to escape from a deep local minimum which happens often for greedy heuristics. It needs a restart mechanism or strong perturbation to escape the local minimum because a few number of minor perturbations is not sufficient for escaping local minima. In order to improve the performance of SLS-SAT, it is necessary to target these three issues by balancing between intensification and diversification capabilities of the search algorithm. Intensification and diversification are two important aspects of SLS-SAT [48]. Intensification aims to greedily improve the solution quality by exploiting the evaluation function. In contrast, diversification aims to prevent the search stagnation by keeping the search process away from traps in the confined regions. For this reason, we targeted our research on the enhancement of diversification strategies for stochastic local search to intelligently prevent and escape stagnation. From this point of view, our research aim is to improve the diversification of SLS-SAT by focusing on the following research

29 8 Chapter 1. Introduction problems: Objective function: We aim to integrate diversification capability into the objective function in order to decrease the greediness in exploring the search space. Long-term learning mechanism: We aim to improve SLS-SATs capability to learn the reason of stagnation during the search trail. For example, we aim to learn the conflicts between the assignment and the constraints between variables and clauses. Stagnation escape: Our purpose is to construct a trap escaping mechanism to escape from current stagnation and to predict future stagnation situation. This mechanism needs to exploit the information learned over a long-term especially about local minima in order to effectively prevent future stagnation. 1.6 Research contributions The main contributions of this dissertation are listed below: Stagnation prevention strategy by exploiting pseudo-conflict learning: Firstly, we proposed pseudo-conflict as a new variable property for stagnation prevention and escaping. In this work, we introduced a long-term learning strategy focusing on local minima. Our preliminary investigation has revealed that developing efficient enhancement techniques for stagnation escape and prevention in local search is an important research issue. It also has significant practical implications. This heuristic aims to exploit long-term memory from stagnation weights in order to assist the search process to avoid and escape local minima intelligently. The approach firstly learns which variable is vulnerable with stagnation and is likely to lead the search to stagnation. A variable weighting strategy was used to compute the frequency of a variable being involved in stagnation. The approach then utilises this information to treat the variable differently in variable selection phases in order to select variable more cleverly. This new trap prevention strategies are published in AAAI-2012, ECAI-2012 and AI-2012 [85, 27, 26]. Weighting scheme in selecting clauses and variables: Weighting schemes in SLS have been recently recognised as a state-of-the-art strategy to solve structured problems. In clause weighting scheme, each clause is associated with a weight and the scoring function

30 1.7. Thesis Outline 9 is computed based on these weights. There are clause-weighting and variable-weighting schemes which provide SLS with information learnt from the search trail. In this study, we exploit the conventional clause-weighting for the new approach of trap escaping. More specifically, we proposed a new method to select clauses according to clause-weight mechanism to exploit completely the learning information accumulated in clause weights during the search progress. Additionally, we combine the variable weights with the new approach in order to boost the advantage of the weighting strategies. This new weighting strategy for stagnation phases obtains better performance over some current SLS-SAT solvers. This work is published in IJCAI-2013 [29]. The scoring formula based on a new probability distribution: One of the reasons for stagnation in local search is its greedy goal-orientation in terms of its objective function. From this point of view, we attempted to decrease the greediness of the score function by a new distribution probability formation, which is a combination of greediness and diversification instead of the maximum greediness as conventional score. The score grants a portion of probability distribution to diversification criteria in variable selection in order to balance between greediness and diversification. The significance of the result obtained from this enhancement strategies was recognised in AI-2013 [28] where the related paper won the best paper award. Back-tracking style in selecting variables to reverse the pseudo-conflicts: In the conventional method of escaping local minima, the search selects an unsatisfied clause randomly and selects variables in that clause by mechanisms such as Random walk, Novelty+ and R-Novelty. In this thesis, we developed a different approach to reverse the recent pseudo-conflicts by retrieving a variable s current stagnation path. This trap escaping strategy achieved promising results over the state-of-the-art SAT solvers. This piece of work was published in SCAI-2013 [25]. 1.7 Thesis Outline The remainder of this dissertation is organised as described below: In Chapter 2, we define the main concept of propositional satisfiability and present a general literature review on SLS solvers for SAT. The algorithm of each SLS is reviewed and its core

31 Chapter 1. Introduction techniques and contributions are analysed. For better readability of the literature, we have exhaustively provided almost all of the algorithms in pseudo-code and have explained the procedures and objective functions in a systematic way. In Chapter 3, we present our new pseudo-conflict learning strategy integrated with gnovelty +. We named the new algorithm gnovelty + PCL. This is a new contribution to heuristics based on long-term memory. The chapter introduces the new definitions of pseudo-conflict variables, stagnation paths, and stagnation weights. Additionally, the new stagnation escaping strategy based on Novelty heuristics and pseudo-conflict variables is presented. In Chapter 4, we describe the clause and variable weighting enhancement strategies that are applied at stagnation phases. Firstly, a new variable weighting scheme is integrated in order to improve the diversification. Secondly, we apply a new probability-based switching method to navigate the search between the new clause selection strategies and the conventional random walk in clause selection. In Chapter 5, we present our new dynamic objective function based on probability distribution. The new scoring function is a combination of greediness and diversification aiming to improve diversification in greedy phases of SLS-SAT. In Chapter 6, we introduce a new stagnation escaping strategy by taking advantages of the pseudo-conflict learning mechanism. The new algorithm NovEsc is developed on gnovelty + PCL. However, in order to escape the local minima efficiently, the new heuristic prefers to select the most pseudo-conflicted variable in the latest stagnation paths. Moreover, the algorithm integrated variable weights as an additional tie-breaking factor. Finally, we provide the conclusion and list the research contributions of the thesis and discuss our future work in Chapter 7.

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33 Chapter 2 Stochastic Local Search For SAT This chapter describes and discusses the key concepts and techniques of the existing constraintbased local search strategies. We then explore the general constraint-based local search and state-of-the-art local-search based SAT algorithms (SLS-SAT). We also discuss categorisation, performance and milestones of these SLS-SAT techniques. 2.1 Constraint-based Stochastic Local Search In recent decades, stochastic local search has been an emerging approach for searching large search spaces. It is among the most successful techniques for solving computationally hard problems from computing science, operation research and various application areas (e.g the traveling salesman problem, routing and scheduling problems, genome sequence assembly, winner determination in combinatorial auctions) [44]. The key concept of SLS is randomisation and perturbation. To start the search, the search procedure selects a randomised position in the search space. After this initialisation step, the algorithm iteratively improves the current candidate solution with small modifications until the termination criteria are met. In general, SLS is able to find a good solution but cannot guarantee the most optimal solution. Despite the fact that it is incomplete, it has the advantage of using less memory and capability to find acceptable solutions in a reasonable time for large and complex problems. Given a problem instance π, an SLS algorithm generally has the following components [44]: A search space that is the finite set S(π) of all candidate solutions (or search positions). 12

34 2.1. Constraint-based Stochastic Local Search 13 A neighbourhood relation N(π) S(π) S(π) that determines the criteria to generate the neighbouring candidate solutions of an observing solution. A finite set of memory state M(π) that specifies the stored knowledge during search beyond the candidate solutions (e.g. the tabu tenure in Tabu search [40, 41, 42] or the temperature in simulated annealing [65]) and can be used later as an additional search guidance for selecting a candidate solution from the neighbours. An initialisation function init : Ø:= P (S(π)) that indicates clearly an initial search position and memory states. A step function step(π) : S(π) M(π) := D(S(π) M(π)) that maps each search position and memory states onto a probability distribution over its neighbouring positions and memory states. The termination criterion terminate(π) : S(π) M(π) := D({true, f alse}) that determines the criterion to terminate the search based on the current search position and memory states. Among the above components, the neighbourhood relation and the step function are particularly important. A well-designed neighbourhood relation can restrict the local space surrounding the current solution and a smart step function can speed up the search. There are various ways to define neighbourhood relationships such as 1-exchange neighbourhood, k-exchange neighbourhood or population-based neighbourhood. Every neighbourhood relation induces a neighbourhood graph, whose vertices s correspond to the given search space and in which each pair of neighbouring search positions is connected by an edge. Therefore, under a local search strategy, the search space can be considered as a graph of search positions. The search is typically guided by an evaluation function that is used to heuristically assess or rank candidate solutions. Furthermore, some SLS methods use more than one evaluation function and some can even dynamically adjust the evaluation function while searching. Most problems in our real world can be converted to a search problem for solutions and all candidate solutions create a search space. The target of searching are formulated based on several constraints and an objective function. There are three issues that are needed to be defined to formulate a problem for local search: