Handbook of Knowledge Representation - Chapter 2: Satisfiability Solvers

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1. Handbook of Knowledge Representation Chapter 2: Satisfiability Solvers Junya Yamaguchi May 2, 2018 Tokyo Institute of Technology, Inoue Lab yamaguchi.j.ab@m.titehc.ac.jp
2. SAT Problem Example • An assignment is a function ϕ : V → {0, 1} F = ( x literal ∨ y ∨ z ) ∧ (¬x ∨ ¬y) clause ∧ (¬y ∨ ¬z) ∧ (¬z ∨ ¬x) CNF formula assignment ϕ(x, y, z) = (0, 0, 1) derives F = 1. • A formula is satisfiable (SAT) iff there exists an assignment that evaluates it TRUE, which is called model. • Otherwise, we say a formula is unsatisfiable (UNSAT). 1
3. SAT ιϧόͷ׆༂ • ࠷ѱܭࢉྔ͕ࢦ਺ؔ਺Ͱ͋Δ͜ͱͰ༗໊͕ͩɺSAT ιϧό͸ଟ͘ͷྖҬͰޭ ੷Λ࢒͍ͯ͠Δ • ιϑτ΢ΣΞ/ϋʔυ΢ΣΞݕূ • ࣗಈςετύλʔϯੜ੒ • ϓϥϯχϯά • εέδϡʔϦϯά • ೥ʹҰճ SAT ͷίϯϖ͕͋Δ (SAT Competition) • ͦ͜Ͱ਺ଟ͘ͷૉ੖Β͍͠ SAT ιϧό͕஀ੜͨ͠ • Ϟμϯͳ SAT ιϧόʹͳΔͱɺ਺ສม਺ɾ਺ઍ੍໿͔Β੒Δඇৗʹ೉͍͠໰ ୊Ͱ΋ղ͚Δ 2
4. SAT ͱ஌ֶࣝशͷؔ܎ • ࠓͰ͸ͨ͘͞ΜͷԠ༻͕͋Δ͕ɺͦͷىݯ͸ ஌ֶࣝश • ஌ࣝͷදݱྗͱܭࢉྔͷτϨʔυΦϑʹؔ͢Δݚڀ (ୈ 3 ষͰѻ͏) ͕ϝΠϯ • Ծఆ: ࠷ѱܭࢉྔ͕ଟ߲ࣜ࣌ؒͰ͋Γͳ͕Β΋ɺΊͪΌͪ͘ΌΤϨΨϯτͰදݱ ྗͷߴ͍දݱݴޠΛΈ͚͍ͭͨ • 90 ೥୅ͷ͸͡Ίʹɺ2 ͭͷ࿦จ͕͜ͷԾఆʹ௅Μͩ • ͋Δಛघͳ໰୊Λআ͚͹ɺ΄ͱΜͲͷϥϯμϜ SAT ໰୊͸ͱͯ΋؆୯ʹͱ͚Δ ͜ͱΛࣔͨ͠࿦จ • ͦͷಛघͳ೉͍͠໰୊Ͱ͑͞΋ɺϩʔΧϧαʔνͷςΫχοΫΛ࢖͑͹؆୯ʹղ ͘͜ͱ͕Ͱ͖Δ͜ͱΛࣔͨ͠࿦จ 3
5. ࢲͨͪ͸࠷ѱܭࢉྔʹͱΒΘΕͯͳ͍͔ʁ • ਺ඦສ΋ͷม਺Λ΋ͭݱ࣮ͷ SAT ໰୊Ͱͷ੒ޭ • యܕతͳ SAT ໰୊΍ݱ࣮తͳ NP ׬શͳ໰୊ (ͷܭࢉྔ) ʹରͯ͠͸ɺޮ཰Α͘ ղ͘͜ͱ͕Ͱ͖ΔҰൠղ๏͕͋Δ • ࠷ѱܭࢉྔʹϏϏΓ͗ͯ͢͸͍͚ͳ͍ • ͜ͷ·· SAT ιϧό͕੒௕͢Ε͹ɺ΋ͬͱෳࡶͳ஌ࣝදݱͷݴޠΛѻ͑ΔΑ ͏ʹͳΔ͸ͣ • NO: ࠷ѱ࣌ܭࢉྔ͕ଟ߲ࣜͳΞϧΰϦζϜͰͷදݱ • YES: SAT ιϧόͰղ͚Δൣғͷදݱ 4
6. ষͷߏ੒ 1. ࣍ͷ 2 छྨͷιϧόʹ͓͍ͯ࢖ΘΕ͍ͯΔओͳςΫχοΫ • ׬શ (complete) SAT ιϧό • ෆ׬શ (incomplete) SAT ιϧό 2. ࣮ફతͳ SAT ූ߸Խʹର͢Δ্هςΫχοΫͷ༗ޮੑ 3. SAT ιϧόͷকདྷ΁ͷల๬ 5
7. ఆٛ • ໋୊࿦ཧࣜ (propositional or Boolean formula) ͸ɺม਺ͷू߹্ʹఆٛ ͞ΕΔ࿦ཧࣜ • ֤ม਺͸ {FALSE, TRUE} ͷͲͪΒ͔ͷ஋ΛऔΔ • ศ্ٓ {0, 1} Ͱද͢ࣄ͕ଟ͍ • ม਺ͷू߹ V ʹର͢Δ ׬શׂ౰ͯ (truth assignment)1͸ɺࣸ૾ σ : V → {0, 1} ͷ͜ͱɻ • ಛʹɺ໋୊࿦ཧࣜΛ 1 ʹධՁ͢ΔΑ͏ͳ׬શׂ౰ͯΛ satisfying assignment ΍ Ϟσϧ (model) ͱݺͿɻ • SAT ιϧόͰѻ͏ SAT ໰୊͸ CNF (conjunctive normal form) ͱݺ͹Ε Δಛผͳܗࣜͷ໋୊࿦ཧ͚ࣜͩʹ੍ݶ 1 ׬શׂ౰ͯ͸୯ʹʮׂ౰ͯʯͱݺ͹ΕΔ͜ͱ΋͋Γ·͕͢ɺຊεϥΠυͰ͸෦෼ׂ౰ͯͷ͜ͱΛׂ౰ͯͱݺͼɺ׬શׂ౰ͯͱ۠ผ͠·͢ɻ 6
8. ॆ଍Մೳੑ൑ఆ໰୊ Boolean satisfiability testing (SAT) ໰୊ ೖྗ CNF ܗࣜͷ໋୊࿦ཧࣜ (CNF ࣜ) F ࣭໰ F ʹ͸Ϟσϧ͕͋Δ͔ʁ CNF ࣜ અͷબݴ F = C1 ∧ C2 · · · ∧ Cm ·ͨ͸ F = {Ci }m i=1 અ Ϧςϥϧͷ࿈ݴ C = l1 ∨ l2 · · · ∨ ln ·ͨ͸ C = {li }n i=1 Ϧςϥϧ ม਺ x ͔ͦͷ൱ఆ ¬x. ม਺ x ∈ {0, 1} 7
9. ఆٛ • અʹؚ·ΕΔϦςϥϧ਺ΛɺઅͷαΠζͱݺͿɻ • e.g. (x ∨ ¬y ∨ z) ͳΒαΠζ͸ 3 • αΠζ͕ 0 ͷઅΛۭઅ (empty clause)ɺαΠζ͕ 1 ͷઅΛ୯Ґઅ (unit clause)ɺͦͯ͠αΠζ͕ 2 ͷઅΛόΠφϦઅ (binary clause) ͱͦΕͧΕ ݺͿɻ • ͢΂ͯͷઅͷαΠζ͕ͪΐ͏Ͳ k Ͱ͋Δ CNF ࣜΛ k-SAT ͱݺͿɻ • 2-SAT ͸ଟ߲ࣜ࣌ؒͰٻղՄೳ • 3-SAT Ҏ্ʹͳΔͱ NP ׬શ (x1 ∨ x2 ∨ x3 ) ∧ (x4 ∨ x5 ∨ x6 ) ∧ (x7 ∨ x8 ∨ x9 ) 8
10. ఆٛ • ෦෼ׂ౰ͯ (partial assignment) ͸ɺม਺ू߹ͷ෦෼ू߹ʹର͢Δ׬શׂ ౰ͯͷ͜ͱɻ • CNF ࣜ F ΁ͷ෦෼ׂ౰ͯ ρ ʹରͯ͠ɺρ Λ୅ೖͯ͠ಘΒΕͨࣜͷ͜ͱΛ simplified ͞ΕͨࣜͱݺͼɺF|ρ Ͱද͢ɻ • 1 ͭҎ্ͷϦςϥϧ͕ 1 ʹධՁ͞Εͨ͢΂ͯͷઅΛ࡟আ • 0 ʹධՁ͞Εͨ͢΂ͯͷϦςϥϧΛ࡟আ ࠓޙɺೖྗͱͯ͠༩͑Δ໋୊࿦ཧࣜʹ͸ CNF ࣜΛ҉໧ͷ͏ͪʹԾఆ͢Δ͕ɺ ଟ͘ͷ৔߹͜ͷԾఆ͸໰୊ʹͳΒͳ͍ɻ 9
11. 2.2 SAT Solver TechnologyŠ Complete Methods
12. Complete Methods • A complete SAT solver, given the input formula F, either produces a satisfying assignment for F or proves that F is unsatisfiable. • Recent complete methods remain variants of a process introduced several decades ago, DPLL • DPLL procedure: • Was introduced in the early 1960`s. • Can prune of the search space based on falsified clauses. • Performs a backtrack search in the space of partial truth assignments. • Main improvements to DPLL: • Smart branch selection heuristics • Extensions like clause learning and randomized restarts • Well-crafted data structures such as lazy implementations and watched literals 10
13. 2.2.1 The DPLL Procedure
14. DPLL Procedure 11
15. DPLL Procedure • Repeatedly select an unassigned literal l. • The step to choose l is called branching step. • Setting l to TRUE or FALSE is called a decision • decision level is used to reffer the recursion depth at that stage • Recursively search for a satisfying assignment for F|l and F|¬l . 12
16. DPLL Procedure • Repeatedly select an unassigned literal l. • The step to choose l is called branching step. • Setting l to TRUE or FALSE is called a decision • decision level is used to reffer the recursion depth at that stage • Recursively search for a satisfying assignment for F|l and F|¬l . 12
17. DPLL Procedure • Repeatedly select an unassigned literal l. • The step to choose l is called branching step. • Setting l to TRUE or FALSE is called a decision • decision level is used to reffer the recursion depth at that stage • Recursively search for a satisfying assignment for F|l and F|¬l . 12
18. DPLL Procedure • Repeatedly select an unassigned literal l. • The step to choose l is called branching step. • Setting l to TRUE or FALSE is called a decision • decision level is used to reffer the recursion depth at that stage • Recursively search for a satisfying assignment for F|l and F|¬l . 12
19. DPLL Procedure 13
20. DPLL Procedure – Unit Propagation • Initial state: F = (x) ∧ (¬x ∨ y) ∧ (¬x ∨ ¬y ∨ z) ρ = ∅ • There exists an unit clause (x). Execute unit propagation. F = (x)∧(¬x∨y) ∧ (¬x∨¬y ∨ z) ρ = {x} • Now, another unit clause (y) found, let’s more simplify the formula. F = (x) ∧ (¬x ∨ y)∧(¬x ∨ ¬y∨z) ρ = {x, y} • Execute unit propagation in the same way. F = (x) ∧ (¬x ∨ y) ∧ (¬x ∨ ¬y ∨ z) ρ = {x, y, z} • UnitPropagate() ends since there are no unit clauses. 14
21. Key Features of Modern DPLL-Based SAT Solvers • Variable selection heuristic • Clause learning • The watched literals scheme • Conflict-directed Backjump • Fast backjump • Assignment stack shrinking • Cnflict clause minimization • Randomized restrts • a.k.a. decision strategy • MOMS, BOHM maximize a moderately complex function of the dcurrent var. and cls. state. • DLIS selects and fix the literal occuring most frequently in the yet unsatisfied clauses • VSIDS chooses a literal based on its weight which preiodically decays but is boosted if a clause in which it appears is used in deriving a conflict. 15
22. Key Features of Modern DPLL-Based SAT Solvers • Variable selection heuristic • Clause learning • The watched literals scheme • Conflict-directed Backjump • Fast backjump • Assignment stack shrinking • Cnflict clause minimization • Randomized restrts • Critical role in the success of modern complete SAT solvers. • The idea here is to • cache lcauses of conflictz as learned clauses. • utilize this information to prune the search in a different part of the search space encountered later. 16
23. Key Features of Modern DPLL-Based SAT Solvers • Variable selection heuristic • Clause learning • The watched literals scheme • Conflict-directed Backjump • Fast backjump • Assignment stack shrinking • Cnflict clause minimization • Randomized restrts • is a implementaion technique to accelerate unit propagation, introduced in zChaff. • which key idea is to maintain and lwatchztwo special literals for each not yet satisfied clause. • {U, U} : Not yet statisfied • {0, U} : Unit clause • {0, 0} : Empty clause • has high compatibility with clause learning. 17
24. Key Features of Modern DPLL-Based SAT Solvers • Variable selection heuristic • Clause learning • The watched literals scheme • Conflict-directed Backjump • Fast backjump • Assignment stack shrinking • Cnflict clause minimization • Randomized restrts • Backtrack just returns to previous branching point. • Backjump repeats backtrack safely as long as possible. 18
25. Key Features of Modern DPLL-Based SAT Solvers • Variable selection heuristic • Clause learning • The watched literals scheme • Conflict-directed Backjump • Fast backjump • Assignment stack shrinking • Cnflict clause minimization • Randomized restrts • It lets a solver jump directly to a lower decision level d where; • even one branch leads to a conflict involving variables at levels d or lower only. • for completeness, the level d is not marked as unsatisfiable. • While conflict-directed backjumping is always beneficial, fast backjumping may not be so. 19
26. Key Features of Modern DPLL-Based SAT Solvers • Variable selection heuristic • Clause learning • The watched literals scheme • Conflict-directed Backjump • Fast backjump • Assignment stack shrinking • Cnflict clause minimization • Randomized restrts • It’s a technique to learn smaller and more pertinent learnt clauses. • When a conflict occurs because of a clause C′, and the size of learnt clause C exceeds a certain threshold length; • the solver backtracks to almost the highest decision level of the literals in C, • it then starts assigning to FALSE the unassigned literals of C′ until a new conflict is encountered. 20
27. Key Features of Modern DPLL-Based SAT Solvers • Variable selection heuristic • Clause learning • The watched literals scheme • Conflict-directed Backjump • Fast backjump • Assignment stack shrinking • Cnflict clause minimization • Randomized restrts • The idea is to try to reduce the size of a learned conflict clause C by repeatedly identifying and removing any literals of C that are implied to be FALSE when the rest of the literals in C are set to FALSE. a ∨ b, ¬a ∨ c b ∨ c 21
28. Key Features of Modern DPLL-Based SAT Solvers • Variable selection heuristic • Clause learning • The watched literals scheme • Conflict-directed Backjump • Fast backjump • Assignment stack shrinking • Cnflict clause minimization • Randomized restrts • It allows a SAT solver to arbitrarily stop the search and restart their branching process from decision level zero. • Most of the current SAT solvers, employ aggressive restart strategies, sometimes restarting after as few as 20 to 50 backtracks. 22
29. Clause Learning and Iterative DPLL 23
30. Clause Learning and Iterative DPLL 23
31. Clause Learning and Iterative DPLL 23
32. Clause Learning and Iterative DPLL 23
33. Clause Learning and Iterative DPLL 24
34. Clause Learning and Iterative DPLL 24
35. Clause Learning and Iterative DPLL 24
36. Clause Learning and Iterative DPLL 24
37. Clause Learning and Iterative DPLL 24
38. Cleause Learning • The idea of clause learning came from Artificial Intelligence; • seeking to improve backtrack search algorithms by generating explanations for failure points, • For general constraint satisfaction problems the explanations are called lconflictszor lno-goodsz. • In early works, clause learning could only obtain unusable clauses, but various studies have led to success. 25
39. Implication graph Implication graph The implication graph G at a given stage of DPLL is a directed acyclic graph with edges labeled with sets of clauses. 26
40. How to build implication graph 1. Create a node for each decision literal, labeled with that literal. 2. If there is an unit clause C when regarding nodes as the partial assingment; C = k ∨ i=1 li ∨ l, where {li } are assigned to FALSE • Add a node l (if not in G) • Add edges (li , l) for all i = 1, . . . , k (if not in G) • These edges are labeld with C to specify the cause of implication. 3. Repeat step 2 while no such clause found. 4. Add a special ’conflict’ node ¯ Λ. For any variable x s.t. x, ¬x are both in G, add edges (x, ¯ Λ) and (¬x, ¯ Λ). 27
41. Conflict on Implication Graph • An implication graph may not contain a conflict at all, or it may contain many conflict variables and several ways of deriving any single literal. 28
42. Conflict Graph A conflict graph H = (VH , EH ) is a subgraph of the implication graph with the following propeties: • H contains ¯ Λ and exactly one conflict variable. • ∀l ∈ VH have a path to ¯ Λ • ∀l ∈ VH \ { ¯ Λ } satisfies either of the following: 1. l is a decision literal 2. ∃¬l1 , . . . , ¬lk ∈ VH s.t. ∀¬li holds (¬li , l) ∈ EH and there exists a known clause (l1 ∨ · · · ∨ lk ∨ l) 29
43. Conflict Graph A conflict graph H = (VH , EH ) is a subgraph of the implication graph with the following propeties: • H contains ¯ Λ and exactly one conflict variable. • ∀l ∈ VH have a path to ¯ Λ • ∀l ∈ VH \ { ¯ Λ } satisfies either of the following: 1. l is a decision literal 2. ∃¬l1 , . . . , ¬lk ∈ VH s.t. ∀¬li holds (¬li , l) ∈ EH and there exists a known clause (l1 ∨ · · · ∨ lk ∨ l) 29
44. Reason/Conflict Side 30
45. Learning Scheme - Relsat • Relsat uses the cut whose conflict side consists of all implied variables at the current decision level. • The learnt clause (¬a ∨ ¬b) has exactly one variable from the current decision level (i.e. b). • After learning it and backtraking until unassigning b (to decision level 2), filpping the value b to ¬b. 31
46. Learning Scheme - Relsat • Relsat uses the cut whose conflict side consists of all implied variables at the current decision level. • The learnt clause (¬a ∨ ¬b) has exactly one variable from the current decision level (i.e. b). • After learning it and backtraking until unassigning b (to decision level 2), filpping the value b to ¬b. 32
47. Unique Implication Points (UIPs) • An UIP (unique implication point) is a node at the latest decision level s.t. • every path from the lastest decision variable to conflict literal must go through the node. • Intuitively, an UIP is single reason at the current decisioin level. • There can be more than one UIP , but not 0. • The decision variabe is the obvious UIP. 33
48. Application of UIPs • Learning scheme with UIP: • relsat uses the decision variable as UIP • Grasp and zChaff use FirstUIP, which is the UIP closest to the conflict variable. • Grasp also uses all UIPs to learn multiple clauses. • kUIP – Generalized UIP to k previous decision levels. • However, the study showed 1UIP is quite robust and outperforms all other schemes they consider on most of the benchmarks. 34
49. A Proof Complexity Perspective
50. Propositional proof complexity • Propositional proof complexity is the study of the proofs of validity of mathematical statements expressed in a Boolean form. • A propositional proof system is an algorithm A Input A propositional statement S and a purported proof π Returns Rejection / Acceptance • The crucial property of A is that: • For all invalid S, A rejects the pair (S, π) for all π • For all valid S, A accepts the pair (S, π) for some π 35
51. Resolution • Resolution is a very simple proof system with only one rule. a ∨ b, ¬a ∨ c b ∨ c • Repeated application of this rule can derive an empty disjunction iff the initial formula is unsatisfiable; • Such a derivation serves as a proof of unsatisfiability of the formula. • There are several resolution types due to the implementaion restriction. 36
52. General Resolution 1 Paul Beame University of Washington Satisfiability and Unsatisfiability: Proof Complexity and Algorithms. http://slideplayer.com/slide/4978670/ 37
53. Tree-like Resolution 1 Paul Beame University of Washington Satisfiability and Unsatisfiability: Proof Complexity and Algorithms. http://slideplayer.com/slide/4978670/ 38
54. Variation of Resolution • Tree-like resolution uses non-empty derived clauses exactly once in the proof and is equivalent to an optimal DPLL procedure. • Regular resolution allows any variable to be resolved upon at most once along any “path” in the proof from an initial clause to Λ, allowing (restricted) reuse of derived clauses. • While these and other refinements are sound and complete as proof systems, they differ vastly in efficiency. 39
55. SAT Solver as Proof System • Most of today`s complete SAT solvers implement a subset of the resolution proof system. • Till recently, it was not clear: • Where do they fit in the proof system hierarchy? • How do we compare them to refinements of resolution, e.g. regular resolution? • Clause learning and random restarts was considered to be important. • Despite overwhelming empirical evidence, for many years not much was known of the ultimate strengths and weaknesses of the two. 40
56. FIrstNewCut • Beame, Kautz, and Sabharwal answered several of these questions in a formal proof complexity framework. • They characterized clause learning as a proof system called CL, and proposed a new learning scheme called FirstNewCut • FirstNewCut can provide exponentially shorter proofs than refinement of general resolution satisfying a natural self-reduction property. • These include regular and ordered resolution. • These are already known to be much stronger than the ordinary DPLL procedure. • They also showed that a slight variant of clause learning with unlimited restarts is as powerful as general resolution itself. 41
57. Ralation Between SAT/UNSAT Proof Complexity • INTEREST is a familiy of unsatisfiable formulas, BECAUSE only proofs of unsatisfiability can be large. • The minimum proofs of satisfiability are O(n). • However, in practice, many formulas are satisfiable. • Achlioptas, Beame, and Molloy have shown • Negative proof complexity results for unsatisfiable formulas can be used; • for the run-time lower bounds of specific inference algorithms, especially DPLL, running on satisfiable formulas as well. • The key observation in their work is that before hitting a satisfying assignment, an algorithm is very likely to explore a large unsatisfiable part of the search space due to uninformative initial assignments. 42
58. Branching Heuristic for SAT/UNSAT proofs. • Proof complexity does not capture everything we intuitively mean by the power of a reasoning system because it says nothing about how difficult it is to find shortest proofs. • However, it is a good notion with which to begin our analysis because the size of proofs provides a lower bound on the running time of any implementation of the system. Negative Positive Proofs Size Must be large Small proofs exist Perfect Branching Hopeless Possible 43
59. DPLL and Tree-like Resolution For a CNF formula F, the size of the smallest DPLL refutation of F is equal to the size of the smallest tree-like resolution refutation of F. • The best performances are the same. • Considering a propositional proof system CL is exponentially stronger than tree-like resolution, clause learning is very effective. 44
60. Natural Refinement / Proper • CS (F) : the length of a shortest refutation of a formula F under a proof system S • tree-like, regular, linear, positive, negative, semantic, and ordered resolution are natural refinements. • tree-like, reglar, and ordered resolution are exponentially-proper. 45
61. The Power of CL • (pros) There are some formulas that CL is more effective than some exponentially-propered resolutions. • (cons) CL may not be able to simulate all regular resolution proofs! 46
62. CL– with non-redundant vs. RES • CL– – : CL allowed to branch on a literal whose value is already assigned. • Such a branch can: • lead to an immediate conflict; • allow one to learn a key conflict clause that would otherwise have not been learned. • A clause learning scheme will be called non-redundant if on a conflict, it always learns a clause not already known. • Most of the practical clause learning schemes are non-redundant. 47
63. Symmetry Breaking
64. Symmetry • Symmetry in real-world problems: • in FPGA routing, all available wires or channels used for connecting two switch boxes are equivalent; • in circuit modeling, all gates of the same ltypezare interchangeable; • in planning, all identical boxes that need to be moved from city A to city B are equivalent; • in multi-processor scheduling, all available processors are equivalent; • in cache coherency protocols in distributed computing, all available identical caches are equivalent. • A key property of such objects is that when selecting k of them, we can choose any k, without loss of generality . • We would like to treat without-loss-of-generality reasoning in an automatic fashion. 48
65. Semantic Meaning • A CNF formula consists of constraints over different kinds of variables that typically represent tuples of these high level objects. • During the problem modeling phase, we could have a Boolean variable zw,c that is TRUE iff the first end of wire w is attached to connector c. • When the formula is converted into DIMACS format, the semantic meaning of the variables is discarded. 49
66. Symmetry Breaking Predicates (SBPs) • Symmetry Breaking Predicates (SBPs) is a method to deal with this problem: • Symmetries can be broken by adding one lexicographic ordering constraint, called llex-constraintsz. • This constraint removes all redundant solutions but the lexically-first solutions. • The idea is to identify the set of permutations of variables {πi }, where πi keeps the formula unchanged. • Let σ be a lexically-first solution, πi (σ) is also a satisfying assignment. 50
67. Symmetry in SAT – Graph Isomorphism • Shatter by Aloul et al. improves the number of lex-constraints from O(n2) to O(n), where n is the number of variables. • They use techniques of graph isomorphism to generate SBPs. • Since there is no polynomial-time algorithms to praph isomorphism, the number os SBPs to break all symetries can be exponential. • Shatter handled this by discarding “large” symmetries. 51
68. Symmetry in SAT – Non-CNF Formulation • Utilizing non-CNF formulations , known as pseudo-Boolean inequalities, is another approach for symmetry problem. • PBS by Aloul et al. • pbChaff by Dixon et al. • Galena by Chai and Kuehlmann • Instead of resolution system, they use Cutting Planes proof system. • It is difficult to implement, so pseudo-Boolean solvers often implement only a subset of it. 52
69. 2.3 SAT Solver Technology - Incomplete Methods
70. Incomplete SAT Solver • What about an incomplete method? • It can provide the model of satisfiable formula, but it is not always possible to solve; • It can’t say anything about unsatisfiability. • Incomplete methods significantly outperform DPLL-base methods for some problems. • They are generally based on stochastic local search, while typical complete methods are based on exhaustive branching and backtracking search. 53
71. Local Search on Satisfiability • Original impetus of local search on satisfiability problems was the successful application for finding solutions to N-queens problems. • Researchers regarded N-queens was an easy problem. • They felt that it would fail in practice for SAT. 54
72. GSAT • GSAT is based on randomized local search technique. • Initialize truth assignment randomlly. • Then, greedily flip the assignment of the variable that leads to the greatest decrease in the total number of UNSAT clauses. • Finish the search if a model is found, otherwise repeat fliping. 55
73. GSAT • GSAT outperformed DPLL-based solvers on various SAT problem. • From randomly generated problems to SAT encodings of graph coloring problems. • Most time is spent searching plateaus • Plateaus are possible sideways movements that don’t increase/ decrease the # of UNSAT clauses too much. 56
74. Walksat 57
75. Walksat • Walksat is a refinement of GSAT, employing random walk moves of a standard Metropolis search. • For the purpose to increase randomness. • Always select the variable to flip from unsatisfied clauses. • If there is a variable that can be flipped without affecting satisfied clauses, pick it up. • Otherwise, perform a p-greedy strategy. • This idea is inpired by the O(n2) randomized algorithm for 2-SAT. • It guarantees that one can reach a model by selecting only from variables on unsatified clauses. 58
76. Walksat vs. GSAT http://ai.cs.unibas.ch/_files/teaching/fs16/ai/slides/ai32.pdf 59
77. Phase Transition Phenomenon
78. Clause-to-Variable Ratio • Random k-SAT with n variables and m clauses. • For each clause, select k unique variables uniformaly. • For each variable, negate it with probablity 0.5 • clause-to-variable ratio α = m n • The difficulity of random k-SAT problems is characterized by α. • For random 3-SAT, the run-time of problems with α > 4.26 tends to be much larger than the opposite one. 60
79. Phase Transition • Left: The run-time • Right: The ratio of unsatisfiable problem • As n grows, the phenomenon becomes shaper and shaper. 61
80. 2.3.2 Survey Propagation
81. Survey Propagation • Survey Propagation (SP) is a new algorithm for solving hard combinational problems. • It showed good results on random 3-SAT instances with one million variables. • SP has a connection to Blief Propagation (BP): • BF is a method to calculate marginal probability on graphical models (factor-graph). • Both using an iterative process of local “message” update. • Unfortunately, the success of SP is limited to random SAT instances. 62
82. Survey Propagation http://slideplayer.com/slide/4559400/ 63
83. 2.4 Runtime Variance and Problem Structure
84. Exceptionally Hard Problem • The performance of backtrack-style modhods can vary: • The way to select the next variable to branch on. • In what order the possible values are assigned to the variable. • There are exceptionally hard problems. • They ignore the “easy-hard-easy” phenomenon. • They occur in the under-constrained area. • However, subsequent research showed that such instances are not inherently difficult. • Simply renaming the variables or by considering a different search heuristic such can easily solved such problems. • The “hardness” is in the combination of the instance with the search method, rather than in the instances per se2. • This is why the median is used to measure the “hardness” of problems. 2Adverb: by or in itself or themselves; intrinsically. 64
85. Fat and Heavy Tailed Behavior • Runtime distributions of search methos provides a better characterization. • Quite often complete backtrack search methods exhibit fat and heavy-tailed behavior. • Looks like lognormal, Wibull dist. • Such distributions can be observed: • when running a deterministic backtracking procedure on a distribution of random instances; • by repeated runs of a randomized backtracking procedure on a single instance. 65
86. Runtime of CSP Summarizing CSP Hardness with Continuous Probability Distributions. Proceedings of the National Conference on Artificial Intelligence. 327-333. 66
87. Heavy Tail • fat-tailedness – • kurtosis = µ4 µ2 2 , where µ2 , µ4 are 2nd and 4th moment resp. • A distribution with igh central peak and long tails has generally large kurtosis. • A heavy-taild distribution is “heavier” than fat-tailed one. • It have some infinite moments, e.g. mean, variance, etc. • DPLL style complete backtrack search methods have been shown to exhibit heavy-tailed behavior. • They can solve some problems so quickly, meanwhile do some problems super slowly. 67
88. 2.4.2 Backdoors
89. Backdoors • The idea of heavy-tailed behavior comes from backdoor variables . • Variables which gives us a polynomial subproblem when set. • This is for explaining how a backtrack search method can get “lucky” on certain runs. • When backdoor variables are identified early on in the search and set the right way, we say it’s lucky. • The definition of a backdoor depends on a particular algorithm, reffered to as sub-solver. • It solves a tractable sub-case of the general constraint satisfaction problem. 68
90. Sub-Solver and Backdoor • Intuitively, the backdoor are variables such that when set correctly, the sub-solver can solve the remaining problem. • A stronger notion of backdoors, strong backdoor, considers both satisfiable and unsatisfiable problems. 69
91. Backdoor Detection Complexity • Nishimura et al. provided that detecting backdoor sets where the sub-solver solves Horn or 2-CNF formulas, both of which are linear time problems. • They proved that the detection of such a strong backdoor set is fixed-parameter tractable; • while the detection of a weak backdoor set is not. • Dilkina et al. proved that the complexity of backdoor detection jumps from NP to NP-hard/coNP-hard when adding certain obvious inconsistency checks to the underlying class. 70
92. Backdoor Size • The backdoor size of random formulas is large. • For random 3-SAT near the phase transition, about 30% of variables are backdoors. • That’s why DPLL-based search algorithm can’t solve these problems • Structured (real-world) problems have very few backdoor variables. 71
93. 2.4.3 Restarts
94. Restart Strategy • Another way to exploit heavy-tailed behavior is to add restarts to backtracking. • Gomes et al. proposed randomized rapid restarts (RRR) to take advantage of heavy-tailed behavior and boost the efficiency of complete backtrack search. • They also showed that restart strategy with a fixed cutoff eliminates heavy-tail behavior and has finite moments. 72
95. Luby’s Restarts • Luby et al. showed that: • When the distriution is known, the optimal restart policy is the fixed cutoff. • When there is no a priori to the distribution, a universal strategy minimizes the expected cost. 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 4, 8, . . . • Theoretically the universal strategy is within a log factor of the optimal one, but the convergence is too slowly in practice. • State-of-the-art SAT solver’s restart: • Default cutoff value, which is increased linearly 73
96. Bayesian Framework • Horvitz et al. introduced a Bayesian framework for learning predictive models of randomized backtrack solvers based on “knowledgeless” situation. • Kautz et al. extended it, enabling restart policies to use information of real-time observations about a solver. • They also showed the optimal policy for dynamic restarts. • It can be implemented using dynamic programming technique. 74