CDCL SAT Solvers. Joao Marques-Silva. Theory and Practice of SAT Solving Dagstuhl Workshop. April INESC-ID, IST, ULisbon, Portugal

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1 CDCL SAT Solvers Joao Marques-Silva INESC-ID, IST, ULisbon, Portugal Theory and Practice of SAT Solving Dagstuhl Workshop April 2015

2 The Success of SAT Well-known NP-complete decision problem [C71]

3 The Success of SAT Well-known NP-complete decision problem In practice, SAT is a success story of Computer Science Hundreds (even more?) of practical applications [C71]

4 The Success of SAT Well-known NP-complete decision problem In practice, SAT is a success story of Computer Science Hundreds (even more?) of practical applications [C71]

5 SAT Solver Improvement [Source: Le Berre&Biere 2011] CPU Time (in seconds) Results of the SAT competition/race winners on the SAT 2009 application benchmarks, 20mn timeout Limmat (2002) Zchaff (2002) Berkmin (2002) Forklift (2003) Siege (2003) Zchaff (2004) SatELite (2005) Minisat 2 (2006) Picosat (2007) Rsat (2007) Minisat 2.1 (2008) Precosat (2009) Glucose (2009) Clasp (2009) Cryptominisat (2010) Lingeling (2010) Minisat 2.2 (2010) Glucose 2 (2011) Glueminisat (2011) Contrasat (2011) Lingeling 587f (2011) Number of problems solved

6 SAT-Based Problem Solving MBD PBO Eager SMT B&B Search Planning Enumeration BMC Encodings Problem Solving with SAT Embeddings OPT SAT MC: ic3 MaxSAT Lazy SMT LCG CEGAR QBF Oracles MUS CEGAR SMT MCS Enumeration Backbones Min. Models

7 This Talk Review key ideas in implementing CDCL SAT solvers Review standard concepts Unit propagation Resolution Proof traces... Outline organization of DPLL/CDCL SAT solvers Survey most effective techniques Clause learning & non-chronological bakctracking UIPs Clause minimization Search restarts Several heuristics...

8 Outline Basic Definitions DPLL Solvers CDCL Solvers What Next in CDCL Solvers? CNF Encodings

9 Outline Basic Definitions DPLL Solvers CDCL Solvers What Next in CDCL Solvers? CNF Encodings

10 Preliminaries Variables: w, x, y, z, a, b, c,... Literals: w, x, ȳ, a,..., but also w, y,... Clauses: disjunction of literals or set of literals Formula: conjunction of clauses or set of clauses Model (satisfying assignment): partial/total mapping from variables to {0, 1} that satisfies formula Formula can be SAT/UNSAT

11 Preliminaries Variables: w, x, y, z, a, b, c,... Literals: w, x, ȳ, a,..., but also w, y,... Clauses: disjunction of literals or set of literals Formula: conjunction of clauses or set of clauses Model (satisfying assignment): partial/total mapping from variables to {0, 1} that satisfies formula Formula can be SAT/UNSAT Example: F (r) ( r s) ( w a) ( x b) (ȳ z c) ( b c d) Example models: {r, s, a, b, c, d} {r, s, x, y, w, z, ā, b, c, d}

12 Resolution Resolution rule: [DP60,R65] (α x) (β x) (α β) Complete proof system for propositional logic

13 Resolution Resolution rule: [DP60,R65] (α x) (β x) (α β) Complete proof system for propositional logic (x a) ( x a) (ȳ ā) (y ā) (a) (ā) Extensively used with (CDCL) SAT solvers

14 Resolution Resolution rule: [DP60,R65] (α x) (β x) (α β) Complete proof system for propositional logic (x a) ( x a) (ȳ ā) (y ā) (a) (ā) Extensively used with (CDCL) SAT solvers Self-subsuming resolution (with α α): [e.g. SP04,EB05] (α x) (α x) (α) (α) subsumes (α x)

15 Unit Propagation F = (r) ( r s) ( w a) ( x ā b) (ȳ z c) ( b c d)

16 Unit Propagation F = (r) ( r s) ( w a) ( x ā b) (ȳ z c) ( b c d) Decisions / Variable Branchings: w = 1, x = 1, y = 1, z = 1

17 Unit Propagation Level Dec. Unit Prop. F = (r) ( r s) ( w a) ( x ā b) 0 1 w r a s (ȳ z c) ( b c d) 2 x b Decisions / Variable Branchings: w = 1, x = 1, y = 1, z = y z c d

18 Unit Propagation Level Dec. Unit Prop. F = (r) ( r s) ( w a) ( x ā b) (ȳ z c) ( b c d) w x r a b s Decisions / Variable Branchings: w = 1, x = 1, y = 1, z = y z c d Additional definitions: Antecedent (or reason) of an implied assignment ( b c d) for d Associate assignment with decision levels w = 1, x = 2, y = 3, z = 4 r = 0, d = 4,...

19 Resolution Proofs Refutation of unsatisfiable formula by iterated resolution operations produces resolution proof An example: F = ( c) ( b) (ā c) (a b) (a d) (ā d) Resolution proof: (a b) (ā c) ( c) (b c) ( b) (b) A modern SAT solver can generate resolution proofs using clauses learned by the solver [ZM03]

20 Unsatisfiable Cores & Proof Traces CNF formula: F = ( c) ( b) (ā c) (a b) (a d) (ā d) Level Dec. Unit Prop. 0 b a c Implication graph with conflict

21 Unsatisfiable Cores & Proof Traces CNF formula: F = ( c) ( b) (ā c) (a b) (a d) (ā d) Level Dec. Unit Prop. 0 b a c Proof trace : (ā c) (a b) ( c) ( b)

22 Unsatisfiable Cores & Proof Traces CNF formula: F = ( c) ( b) (ā c) (a b) (a d) (ā d) Level Dec. Unit Prop. (a b) (ā c) 0 b a ( c) (b c) c ( b) (b) Resolution proof follows structure of conflicts

23 Unsatisfiable Cores & Proof Traces CNF formula: F = ( c) ( b) (ā c) (a b) (a d) (ā d) Level Dec. Unit Prop. (a b) (ā c) 0 b a ( c) (b c) c ( b) (b) Unsatisfiable subformula (core): ( c), ( b), (ā c), (a b)

24 Outline Basic Definitions DPLL Solvers CDCL Solvers What Next in CDCL Solvers? CNF Encodings

25 The DPLL Algorithm Unassigned variables? N Y Assign value to variable Satisfiable Unit propagation N Conflict? Y N Can undo decision? Unsatisfiable Y Backtrack & flip variable Optional: pure literal rule

26 The DPLL Algorithm Unassigned variables? N F = (x y) (a b) (ā b) (a b) (ā b) Y Assign value to variable Satisfiable Unit propagation N Conflict? Y N Can undo decision? Unsatisfiable Y Backtrack & flip variable Optional: pure literal rule

27 The DPLL Algorithm F = (x y) (a b) (ā b) (a b) (ā b) Unassigned variables? Y Assign value to variable N Satisfiable Level Dec. Unit Prop. 0 1 x Unit propagation 2 3 y a b N Conflict? Unsatisfiable N Y Can undo decision? Y Backtrack & flip variable a y ā a x ȳ ā x a ā Optional: pure literal rule

28 The DPLL Algorithm F = (x y) (a b) (ā b) (a b) (ā b) Unassigned variables? Y Assign value to variable N Satisfiable Level Dec. Unit Prop. 0 1 x Unit propagation 2 3 y ā b N Conflict? Unsatisfiable N Y Can undo decision? Y Backtrack & flip variable a y ā a x ȳ ā x a ā Optional: pure literal rule

29 The DPLL Algorithm F = (x y) (a b) (ā b) (a b) (ā b) Unassigned variables? Y Assign value to variable N Satisfiable Level Dec. Unit Prop. 0 1 x Unit propagation 2 3 ȳ a b N Conflict? Unsatisfiable N Y Can undo decision? Y Backtrack & flip variable a y ā a x ȳ ā x a ā Optional: pure literal rule

30 The DPLL Algorithm F = (x y) (a b) (ā b) (a b) (ā b) Unassigned variables? Y Assign value to variable N Satisfiable Level Dec. Unit Prop. 0 1 x Unit propagation 2 3 ȳ ā b N Conflict? Unsatisfiable N Y Can undo decision? Y Backtrack & flip variable a y ā a x ȳ ā x a ā Optional: pure literal rule

31 The DPLL Algorithm F = (x y) (a b) (ā b) (a b) (ā b) Unassigned variables? Y Assign value to variable N Satisfiable Level Dec. Unit Prop. 0 1 x y Unit propagation 2 a b N Conflict? Unsatisfiable N Y Can undo decision? Y Backtrack & flip variable a y ā a x ȳ ā x a ā Optional: pure literal rule

32 The DPLL Algorithm F = (x y) (a b) (ā b) (a b) (ā b) Unassigned variables? Y Assign value to variable N Satisfiable Level Dec. Unit Prop. 0 1 x y Unit propagation 2 ā b N Conflict? Unsatisfiable N Y Can undo decision? Y Backtrack & flip variable a y ā a x ȳ ā x a ā Optional: pure literal rule

33 Outline Basic Definitions DPLL Solvers CDCL Solvers What Next in CDCL Solvers? CNF Encodings

34 What is a CDCL SAT Solver? Extend DPLL SAT solver with: [DP60,DLL62] Clause learning & non-chronological backtracking [MSS96,BS97,Z97] Search restarts [GSK98,BMS00,H07,B08] Lazy data structures Conflict-guided branching...

35 What is a CDCL SAT Solver? Extend DPLL SAT solver with: [DP60,DLL62] Clause learning & non-chronological backtracking [MSS96,BS97,Z97] Exploit UIPs [MSS96,SSS12] Minimize learned clauses [SB09,VG09] Opportunistically delete clauses [MSS96,MSS99,GN02] Search restarts [GSK98,BMS00,H07,B08] Lazy data structures Watched literals [MMZZM01] Conflict-guided branching Lightweight branching heuristics [MMZZM01] Phase saving [PD07]...

36 How Significant are CDCL SAT Solvers? CPU Time (in seconds) GRASP DPLL Results of the SAT competition/race winners on the SAT 2009 application benchmarks, 20mn timeout Limmat (2002) Zchaff (2002) Berkmin (2002) Forklift (2003) Siege (2003) Zchaff (2004) SatELite (2005) Minisat 2 (2006) Picosat (2007) Rsat (2007) Minisat 2.1 (2008) Precosat (2009) Glucose (2009) Clasp (2009) Cryptominisat (2010) Lingeling (2010) Minisat 2.2 (2010) Glucose 2 (2011) Glueminisat (2011) Contrasat (2011) Lingeling 587f (2011) Number of problems solved??

37 Outline Basic Definitions DPLL Solvers CDCL Solvers Clause Learning, UIPs & Minimization Search Restarts & Lazy Data Structures What Next in CDCL Solvers? CNF Encodings

38 Clause Learning Level Dec. Unit Prop. 0 1 x 2 y 3 z a b

39 Clause Learning Level Dec. Unit Prop. 0 1 x 2 y 3 z a b Analyze conflict

40 Clause Learning Level Dec. Unit Prop. 0 1 x 2 y 3 z a b Analyze conflict Reasons: x and z Decision variable & literals assigned at lower decision levels

41 Clause Learning Level Dec. Unit Prop. 0 1 x 2 y 3 z a b Analyze conflict Reasons: x and z Decision variable & literals assigned at lower decision levels Create new clause: ( x z)

42 Clause Learning Level Dec. Unit Prop. 0 1 x (ā b) ( z b) ( x z a) 2 y (ā z) 3 z a ( x z) b Analyze conflict Reasons: x and z Decision variable & literals assigned at lower decision levels Create new clause: ( x z) Can relate clause learning with resolution

43 Clause Learning Level Dec. Unit Prop. 0 1 x (ā b) ( z b) ( x z a) 2 y (ā z) 3 z a ( x z) b Analyze conflict Reasons: x and z Decision variable & literals assigned at lower decision levels Create new clause: ( x z) Can relate clause learning with resolution

44 Clause Learning Level Dec. Unit Prop. 0 1 x (ā b) ( z b) ( x z a) 2 y (ā z) 3 z a ( x z) b Analyze conflict Reasons: x and z Decision variable & literals assigned at lower decision levels Create new clause: ( x z) Can relate clause learning with resolution

45 Clause Learning Level Dec. Unit Prop. 0 1 x (ā b) ( z b) ( x z a) 2 y (ā z) 3 z a ( x z) b Analyze conflict Reasons: x and z Decision variable & literals assigned at lower decision levels Create new clause: ( x z) Can relate clause learning with resolution Learned clauses result from (selected) resolution operations

46 Clause Learning After Backtracking Level Dec. Unit Prop. 0 1 x z 2 y 3 z a b

47 Clause Learning After Backtracking Level Dec. Unit Prop. 0 1 x z 2 y 3 z a b Clause ( x z) is asserting at decision level 1

48 Clause Learning After Backtracking Level Dec. Unit Prop. 0 1 x z Level Dec. Unit Prop. 0 1 x z 2 y 3 z a b Clause ( x z) is asserting at decision level 1

49 Clause Learning After Backtracking Level Dec. Unit Prop. 0 1 x z Level Dec. Unit Prop. 0 1 x z 2 y 3 z a b Clause ( x z) is asserting at decision level 1 Learned clauses are always asserting [MSS96,MSS99] Backtracking differs from plain DPLL: Always bactrack after a conflict [MMZZM01]

50 Unique Implication Points (UIPs) Level Dec. Unit Prop. 0 1 w 2 x 3 y 4 z a c b

51 Unique Implication Points (UIPs) Level Dec. Unit Prop. 0 1 w ( b c) ( w ā c) ( x ā b) (ȳ z a) ( w ā b) 2 x ( w x ā) 3 y ( w x ȳ z) 4 z a c b Learn clause ( w x ȳ z)

52 Unique Implication Points (UIPs) Level Dec. Unit Prop. 0 1 w ( b c) ( w ā c) ( x ā b) (ȳ z a) ( w ā b) 2 x ( w x ā) 3 y ( w x ȳ z) 4 z a c b Learn clause ( w x ȳ z) But a is an UIP Dominator in DAG for level 4

53 Unique Implication Points (UIPs) Level Dec. Unit Prop. 0 1 w ( b c) ( w ā c) ( x ā b) (ȳ z a) ( w ā b) 2 x ( w x ā) 3 y ( w x ȳ z) 4 z a c b Learn clause ( w x ȳ z) But a is an UIP Dominator in DAG for level 4 Learn clause ( w x ā)

54 Multiple UIPs Level Dec. Unit Prop. 0 1 w 2 x 3 y 4 z r a c s b

55 Multiple UIPs Level Dec. Unit Prop. 0 1 w First UIP: Learn clause ( w x ā) 2 x 3 y 4 z r a c s b

56 Multiple UIPs Level Dec. Unit Prop w x First UIP: Learn clause ( w x ā) But there can be more than 1 UIP 3 y 4 z r a c s b

57 Multiple UIPs Level Dec. Unit Prop w x y First UIP: Learn clause ( w x ā) But there can be more than 1 UIP Second UIP: Learn clause ( x z a) 4 z r a c s b

58 Multiple UIPs Level Dec. Unit Prop w x y z r a c First UIP: Learn clause ( w x ā) But there can be more than 1 UIP Second UIP: Learn clause ( x z a) In practice smaller clauses more effective Compare with ( w x ȳ z) s b

59 Multiple UIPs Level Dec. Unit Prop w x y z r a c First UIP: Learn clause ( w x ā) But there can be more than 1 UIP Second UIP: Learn clause ( x z a) In practice smaller clauses more effective Compare with ( w x ȳ z) s b Multiple UIPs proposed in GRASP [MSS96] First UIP learning proposed in Chaff [MMZZM01] Not used in recent state of the art CDCL SAT solvers

60 Multiple UIPs Level Dec. Unit Prop w x y z r a c First UIP: Learn clause ( w x ā) But there can be more than 1 UIP Second UIP: Learn clause ( x z a) In practice smaller clauses more effective Compare with ( w x ȳ z) s b Multiple UIPs proposed in GRASP [MSS96] First UIP learning proposed in Chaff [MMZZM01] Not used in recent state of the art CDCL SAT solvers Recent results show it can be beneficial on current instances [SSS12]

61 Clause Minimization I Level Dec. Unit Prop. 0 1 x b 2 y 3 z c a

62 Clause Minimization I Level Dec. Unit Prop. 0 1 x b (ā c) ( z b c) ( x ȳ z a) ( z b ā) 2 y ( x ȳ z b) 3 z c a Learn clause ( x ȳ z b)

63 Clause Minimization I Level Dec. Unit Prop. 0 1 x b (ā c) ( z b c) ( x ȳ z a) ( z b ā) ( x b) 2 y ( x ȳ z b) 3 z c a Learn clause ( x ȳ z b) Apply self-subsuming resolution (i.e. local minimization) [SB09]

64 Clause Minimization I Level Dec. Unit Prop. 0 1 x b (ā c) ( z b c) ( x ȳ z a) ( z b ā) ( x b) 2 y ( x ȳ z b) 3 z c ( x ȳ z) a Learn clause ( x ȳ z b) Apply self-subsuming resolution (i.e. local minimization)

65 Clause Minimization I Level Dec. Unit Prop. 0 1 x b (ā c) ( z b c) ( x ȳ z a) ( z b ā) ( x b) 2 y ( x ȳ z b) 3 z c ( x ȳ z) a Learn clause ( x ȳ z b) Apply self-subsuming resolution (i.e. local minimization) Learn clause ( x ȳ z)

66 Clause Minimization II Level Dec. Unit Prop. 0 1 w a c b 2 x e d

67 Clause Minimization II Level Dec. Unit Prop. 0 1 w a c Learn clause ( w x c) b 2 x e d

68 Clause Minimization II Level Dec. Unit Prop. 0 1 w a b c Learn clause ( w x c) Cannot apply self-subsuming resolution Resolving with reason of c yields ( w x ā b) 2 x e d

69 Clause Minimization II Level Dec. Unit Prop. 0 1 w a b 2 x e c Learn clause ( w x c) Cannot apply self-subsuming resolution Resolving with reason of c yields ( w x ā b) Can apply recursive minimization d

70 Clause Minimization II Level Dec. Unit Prop. 0 1 w a b 2 x e c Learn clause ( w x c) Cannot apply self-subsuming resolution Resolving with reason of c yields ( w x ā b) Can apply recursive minimization d Marked nodes: literals in learned clause [SB09]

71 Clause Minimization II Level Dec. Unit Prop. 0 1 w a b 2 x e c Learn clause ( w x c) Cannot apply self-subsuming resolution Resolving with reason of c yields ( w x ā b) Can apply recursive minimization d Marked nodes: literals in learned clause Trace back from c until marked nodes or new decision nodes Drop literal c if only marked nodes visited [SB09]

72 Clause Minimization II Level Dec. Unit Prop. 0 1 w a b 2 x e d c Learn clause ( w x c) Cannot apply self-subsuming resolution Resolving with reason of c yields ( w x ā b) Can apply recursive minimization Learn clause ( w x) Marked nodes: literals in learned clause Trace back from c until marked nodes or new decision nodes Drop literal c if only marked nodes visited [SB09]

73 Clause Minimization II Level Dec. Unit Prop. 0 1 w a b 2 x e d c Learn clause ( w x c) Cannot apply self-subsuming resolution Resolving with reason of c yields ( w x ā b) Can apply recursive minimization Learn clause ( w x) Marked nodes: literals in learned clause Trace back from c until marked nodes or new decision nodes Drop literal c if only marked nodes visited Complexity of recursive minimization? [SB09]

74 Outline Basic Definitions DPLL Solvers CDCL Solvers Clause Learning, UIPs & Minimization Search Restarts & Lazy Data Structures What Next in CDCL Solvers? CNF Encodings

Search Restarts I Heavy-tail behavior: [GSK98] 10000 runs, branching

75 Search Restarts I Heavy-tail behavior: [GSK98] runs, branching randomization on industrial instance Use rapid randomized restarts (search restarts)

76 Search Restarts II Restart search after a number of conflicts cutoff cutoff solution

77 Search Restarts II Restart search after a number of conflicts Increase cutoff after each restart Guarantees completeness Different policies exist (see refs) cutoff cutoff solution

78 Search Restarts II Restart search after a number of conflicts Increase cutoff after each restart Guarantees completeness Different policies exist (see refs) Works for SAT & UNSAT instances. Why? cutoff cutoff solution

79 Search Restarts II Restart search after a number of conflicts Increase cutoff after each restart Guarantees completeness Different policies exist (see refs) Works for SAT & UNSAT instances. Why? Learned clauses effective after restart(s)

80 Data Structures Basics Each literal l should access clauses containing l Why?

81 Data Structures Basics Each literal l should access clauses containing l Why? Unit propagation

82 Data Structures Basics Each literal l should access clauses containing l Why? Unit propagation Clause with k literals results in k references, from literals to the clause

83 Data Structures Basics Each literal l should access clauses containing l Why? Unit propagation Clause with k literals results in k references, from literals to the clause Number of clause references equals number of literals, L

84 Data Structures Basics Each literal l should access clauses containing l Why? Unit propagation Clause with k literals results in k references, from literals to the clause Number of clause references equals number of literals, L Clause learning can generate large clauses Worst-case size: O(n)

85 Data Structures Basics Each literal l should access clauses containing l Why? Unit propagation Clause with k literals results in k references, from literals to the clause Number of clause references equals number of literals, L Clause learning can generate large clauses Worst-case size: O(n) Worst-case number of literals: O(m n)

86 Data Structures Basics Each literal l should access clauses containing l Why? Unit propagation Clause with k literals results in k references, from literals to the clause Number of clause references equals number of literals, L Clause learning can generate large clauses Worst-case size: O(n) Worst-case number of literals: O(m n) In practice, Unit propagation slow-down worse than linear as clauses are learned!

87 Data Structures Basics Each literal l should access clauses containing l Why? Unit propagation Clause with k literals results in k references, from literals to the clause Number of clause references equals number of literals, L Clause learning can generate large clauses Worst-case size: O(n) Worst-case number of literals: O(m n) In practice, Unit propagation slow-down worse than linear as clauses are learned! Clause learning to be effective requires a more efficient representation:

88 Data Structures Basics Each literal l should access clauses containing l Why? Unit propagation Clause with k literals results in k references, from literals to the clause Number of clause references equals number of literals, L Clause learning can generate large clauses Worst-case size: O(n) Worst-case number of literals: O(m n) In practice, Unit propagation slow-down worse than linear as clauses are learned! Clause learning to be effective requires a more efficient representation: Watched Literals

89 Data Structures Basics Each literal l should access clauses containing l Why? Unit propagation Clause with k literals results in k references, from literals to the clause Number of clause references equals number of literals, L Clause learning can generate large clauses Worst-case size: O(n) Worst-case number of literals: O(m n) In practice, Unit propagation slow-down worse than linear as clauses are learned! Clause learning to be effective requires a more efficient representation: Watched Literals Watched literals are one example of lazy data structures But there are others

90 Watched Literals [MMZZM01] Important states of a clause

91 Watched Literals [MMZZM01] Important states of a clause Associate 2 references with each clause

92 Watched Literals [MMZZM01] Important states of a clause Associate 2 references with each clause Deciding unit requires traversing all literals

93 Watched Literals [MMZZM01] Important states of a clause Associate 2 references with each clause Deciding unit requires traversing all literals References unchanged when backtracking

94 Watched Literals, in Practice [ES03,G13] In practice, first two positions of clause are watched Fixed Assign watched Swap Assign watched Swap

95 Watched Literals, in Practice [ES03,G13] In practice, first two positions of clause are watched May require to traverse already assigned literals, multiple times Assign watched Fixed Swap Assign watched Swap

96 Watched Literals, in Practice [ES03,G13] In practice, first two positions of clause are watched May require to traverse already assigned literals, multiple times Worst-case time of unit propagation is quadratic on clause size and so on number of literals Assign watched Swap Fixed Assign watched Swap

97 Watched Literals, in Practice [ES03,G13] In practice, first two positions of clause are watched May require to traverse already assigned literals, multiple times Worst-case time of unit propagation is quadratic on clause size and so on number of literals In practice, no gains observed from considering alternative implementations (see previous slide) Assign watched Swap Assign watched Swap Fixed

98 Additional Key Techniques Lightweight branching Use conflict to bias variables to branch on, associate score with each variable Prefer recent bias by regularly decreasing variable scores [e.g. MMZZM01]

99 Additional Key Techniques Lightweight branching Use conflict to bias variables to branch on, associate score with each variable Prefer recent bias by regularly decreasing variable scores [e.g. MMZZM01] Clause deletion policies Not practical to keep all learned clauses Delete larger clauses [e.g. MSS96] Delete less used clauses [e.g. GN02,ES03]

100 Additional Key Techniques Lightweight branching Use conflict to bias variables to branch on, associate score with each variable Prefer recent bias by regularly decreasing variable scores [e.g. MMZZM01] Clause deletion policies Not practical to keep all learned clauses Delete larger clauses [e.g. MSS96] Delete less used clauses [e.g. GN02,ES03] Proven recent techniques: Phase saving [PD07] Luby restarts [H07] Literal blocks distance [AS09]

101 Outline Basic Definitions DPLL Solvers CDCL Solvers What Next in CDCL Solvers? CNF Encodings

102 CDCL A Glimpse of the Future Clause learning techniques Clause learning is the key technique in CDCL SAT solvers Many recent papers propose improvements to the basic clause learning approach [e.g. ABHJS08]

103 CDCL A Glimpse of the Future Clause learning techniques Clause learning is the key technique in CDCL SAT solvers Many recent papers propose improvements to the basic clause learning approach [e.g. ABHJS08] Preprocessing & inprocessing Many recent papers [e.g. JHB12,HJB11] Essential in some applications

104 CDCL A Glimpse of the Future Clause learning techniques Clause learning is the key technique in CDCL SAT solvers Many recent papers propose improvements to the basic clause learning approach [e.g. ABHJS08] Preprocessing & inprocessing Many recent papers [e.g. JHB12,HJB11] Essential in some applications Application-driven improvements Incremental SAT Handling of assumptions due to MUS extractors [LB13] Changing SAT solvers to better suit applications [AS13]

105 But Also, SAT-Based Problem Solving MBD PBO Eager SMT B&B Search Planning Enumeration BMC Encodings Problem Solving with SAT Embeddings OPT SAT MC: ic3 MaxSAT Lazy SMT LCG CEGAR QBF Oracles MUS CEGAR SMT MCS Enumeration Backbones Min. Models

106 Outline Basic Definitions DPLL Solvers CDCL Solvers What Next in CDCL Solvers? CNF Encodings

107 Outline Basic Definitions DPLL Solvers CDCL Solvers What Next in CDCL Solvers? CNF Encodings

108 Encoding to CNF What to encode? Boolean formulas Tseitin s encoding Plaisted&Greenbaum s encoding... Cardinality constraints Pseudo-Boolean (PB) constraints Can also translate to SAT: Constraint Satisfaction Problems (CSPs) Answer Set Programming (ASP) Model Finding... Key issues: Encoding size Arc-consistency?

109 Outline Basic Definitions DPLL Solvers CDCL Solvers What Next in CDCL Solvers? CNF Encodings Boolean Formulas Cardinality Constraints Pseudo-Boolean Constraints Encoding CSPs

110 Representing Boolean Formulas / Circuits I Satisfiability problems can be defined on Boolean circuits/formulas Can represent circuits/formulas as CNF formulas [T68,PG86] For each (simple) gate, CNF formula encodes the consistent assignments to the gate s inputs and output Given z = OP(x, y), represent in CNF z OP(x, y) CNF formula for the circuit is the conjunction of CNF formula for each gate F c = (a c) (b c) (ā b c) a b NAND c F t = ( r t) ( s t) (r s t) r s OR t

111 Representing Boolean Formulas / Circuits II a b NAND ab c c a b c F c (a,b,c) F c = (a c) (b c) (ā b c)

112 Representing Boolean Formulas / Circuits III CNF formula for the circuit is the conjunction of the CNF formula for each gate Can specify objectives with additional clauses a b NAND x c AND y d OR z = 1? F = (a x) (b x) (ā b x) (x ȳ) (c ȳ) ( x c y) (ȳ z) ( d z) (y d z) (z)

113 Representing Boolean Formulas / Circuits III CNF formula for the circuit is the conjunction of the CNF formula for each gate Can specify objectives with additional clauses a b NAND x c AND y d OR z = 1? F = (a x) (b x) (ā b x) (x ȳ) (c ȳ) ( x c y) (ȳ z) ( d z) (y d z) (z) Note: z = d (c ( (a b))) No distinction between Boolean circuits and formulas

114 Outline Basic Definitions DPLL Solvers CDCL Solvers What Next in CDCL Solvers? CNF Encodings Boolean Formulas Cardinality Constraints Pseudo-Boolean Constraints Encoding CSPs

115 Cardinality Constraints How to handle cardinality constraints, n j=1 x j k? How to handle AtMost1 constraints, n j=1 x j 1? General form: n j=1 x j k, with {<,, =,, >} Solution #1: Use native PB solver, e.g. BSOLO, PBS, Galena, Pueblo, etc. Difficult to keep up with advances in SAT technology For SAT/UNSAT, best solvers already encode to CNF E.g. Minisat+, WBO, etc.

116 Cardinality Constraints How to handle cardinality constraints, n j=1 x j k? How to handle AtMost1 constraints, n j=1 x j 1? General form: n j=1 x j k, with {<,, =,, >} Solution #1: Use native PB solver, e.g. BSOLO, PBS, Galena, Pueblo, etc. Difficult to keep up with advances in SAT technology For SAT/UNSAT, best solvers already encode to CNF E.g. Minisat+, WBO, etc. Solution #2: Encode cardinality constraints to CNF Use SAT solver

117 Equals1, AtLeast1 & AtMost1 Constraints n j=1 x j = 1: encode with ( n j=1 x j 1) ( n j=1 x j 1) n j=1 x j 1: encode with (x 1 x 2... x n ) n j=1 x j 1 encode with: Pairwise encoding Clauses: O(n 2 ) ; No auxiliary variables Sequential counter [S05] Clauses: O(n) ; Auxiliary variables: O(n) Bitwise encoding [P07,FP01]... Clauses: O(n log n) ; Auxiliary variables: O(log n)

118 Bitwise Encoding Encode n j=1 x j 1 with bitwise encoding: An example: x 1 + x 2 + x 3 1

119 Bitwise Encoding Encode n j=1 x j 1 with bitwise encoding: Auxiliary variables v 0,..., v r 1 ; r = log n (with n > 1) If x j = 1, then v 0... v r 1 = b 0... b r 1, the binary encoding of j 1 x j (v 0 = b 0)... (v r 1 = b r 1) ( x j (v 0 = b 0)... (v r 1 = b r 1)) An example: x 1 + x 2 + x 3 1 j 1 v 1 v 0 x x x

120 Bitwise Encoding Encode n j=1 x j 1 with bitwise encoding: Auxiliary variables v 0,..., v r 1 ; r = log n (with n > 1) If x j = 1, then v 0... v r 1 = b 0... b r 1, the binary encoding of j 1 x j (v 0 = b 0)... (v r 1 = b r 1) ( x j (v 0 = b 0)... (v r 1 = b r 1)) Clauses ( x j (v i b i )) = ( x j l i ), i = 0,..., r 1, where l i v i, if b i = 1 l i v i, otherwise An example: x 1 + x 2 + x 3 1 j 1 v 1 v 0 x x x ( x 1 v 1 ) ( x 1 v 0 ) ( x 2 v 1 ) ( x 2 v 0 ) ( x 3 v 1 ) ( x 3 v 0 )

121 Bitwise Encoding Encode n j=1 x j 1 with bitwise encoding: Auxiliary variables v 0,..., v r 1 ; r = log n (with n > 1) If x j = 1, then v 0... v r 1 = b 0... b r 1, the binary encoding of j 1 x j (v 0 = b 0)... (v r 1 = b r 1) ( x j (v 0 = b 0)... (v r 1 = b r 1)) Clauses ( x j (v i b i )) = ( x j l i ), i = 0,..., r 1, where l i v i, if b i = 1 l i v i, otherwise If x j = 1, assignment to v i variables must encode j 1 All other x variables must take value 0 If all x j = 0, any assignment to v i variables is consistent O(n log n) clauses ; O(log n) auxiliary variables An example: x 1 + x 2 + x 3 1 j 1 v 1 v 0 x x x ( x 1 v 1 ) ( x 1 v 0 ) ( x 2 v 1 ) ( x 2 v 0 ) ( x 3 v 1 ) ( x 3 v 0 )

122 General Cardinality Constraints General form: n j=1 x j k (or n j=1 x j k) Sequential counters Clauses/Variables: O(n k) BDDs Clauses/Variables: O(n k) Sorting networks [S05] [ES06] [ES06] Clauses/Variables: O(n log 2 n) Cardinality Networks: [ANORC09,ANORC11a] Clauses/Variables: O(n log 2 k) Pairwise Cardinality Networks: [CZI10]...

123 Outline Basic Definitions DPLL Solvers CDCL Solvers What Next in CDCL Solvers? CNF Encodings Boolean Formulas Cardinality Constraints Pseudo-Boolean Constraints Encoding CSPs

124 Pseudo-Boolean Constraints General form: n j=1 a j x j b Operational encoding [W98] Clauses/Variables: O(n) Does not guarantee arc-consistency BDDs [ES06] Worst-case exponential number of clauses Polynomial watchdog encoding [BBR09] Let ν(n) = log(n) log(a max) Clauses: O(n 3 ν(n)) ; Aux variables: O(n 2 ν(n)) Improved polynomial watchdog encoding [ANORC11b] Clauses & aux variables: O(n 3 log(a max))...

125 Encoding PB Constraints with BDDs I Encode 3x 1 + 3x 2 + x 3 3 Construct BDD E.g. analyze variables by decreasing coefficients Extract ITE-based circuit from BDD x x x x x

126 Encoding PB Constraints with BDDs I Encode 3x 1 + 3x 2 + x 3 3 Construct BDD E.g. analyze variables by decreasing coefficients Extract ITE-based circuit from BDD x x x x x x 2 s x 3 s s x 1 z ITE 0 1 ITE 0 1 a b 0 a 1 z ITE 1 0 b s x 2 s x 3 ITE 0 1 a b a b 0 1 z z z ITE 0 1 a b 1 1 0

127 Encoding PB Constraints with BDDs II Encode 3x 1 + 3x 2 + x 3 3 Extract ITE-based circuit from BDD Simplify and create final circuit: x 1 1 z s ITE 1 0 a b NOR NAND x 2 x 3 x 3 x 2

128 More on PB Constraints How about n j=1 a j x j = k?

129 More on PB Constraints How about n j=1 a j x j = k? Can use ( n j=1 a j x j k) ( n j=1 a j x j k), but... n j=1 a j x j = k is a subset-sum constraint (special case of a knapsack constraint)

130 More on PB Constraints How about n j=1 a j x j = k? Can use ( n j=1 a j x j k) ( n j=1 a j x j k), but... n j=1 a j x j = k is a subset-sum constraint (special case of a knapsack constraint) Cannot find all consequences in polynomial time [S03,FS02,T03]

131 More on PB Constraints How about n j=1 a j x j = k? Can use ( n j=1 a j x j k) ( n j=1 a j x j k), but... n j=1 a j x j = k is a subset-sum constraint (special case of a knapsack constraint) Cannot find all consequences in polynomial time [S03,FS02,T03] Example: 4x 1 + 4x 2 + 3x 3 + 2x 4 = 5

132 More on PB Constraints How about n j=1 a j x j = k? Can use ( n j=1 a j x j k) ( n j=1 a j x j k), but... n j=1 a j x j = k is a subset-sum constraint (special case of a knapsack constraint) Cannot find all consequences in polynomial time [S03,FS02,T03] Example: 4x 1 + 4x 2 + 3x 3 + 2x 4 = 5 Replace by (4x 1 + 4x 2 + 3x 3 + 2x 4 5) (4x 1 + 4x 2 + 3x 3 + 2x 4 5)

133 More on PB Constraints How about n j=1 a j x j = k? Can use ( n j=1 a j x j k) ( n j=1 a j x j k), but... n j=1 a j x j = k is a subset-sum constraint (special case of a knapsack constraint) Cannot find all consequences in polynomial time [S03,FS02,T03] Example: 4x 1 + 4x 2 + 3x 3 + 2x 4 = 5 Replace by (4x 1 + 4x 2 + 3x 3 + 2x 4 5) (4x 1 + 4x 2 + 3x 3 + 2x 4 5) Let x 3 = 0

134 More on PB Constraints How about n j=1 a j x j = k? Can use ( n j=1 a j x j k) ( n j=1 a j x j k), but... n j=1 a j x j = k is a subset-sum constraint (special case of a knapsack constraint) Cannot find all consequences in polynomial time [S03,FS02,T03] Example: 4x 1 + 4x 2 + 3x 3 + 2x 4 = 5 Replace by (4x 1 + 4x 2 + 3x 3 + 2x 4 5) (4x 1 + 4x 2 + 3x 3 + 2x 4 5) Let x 3 = 0 Either constraint can still be satisfied, but not both

135 Outline Basic Definitions DPLL Solvers CDCL Solvers What Next in CDCL Solvers? CNF Encodings Boolean Formulas Cardinality Constraints Pseudo-Boolean Constraints Encoding CSPs

136 CSP Constraints Many possible encodings: Direct encoding [dk89,gj96,w00] Log encoding Support encoding [W00] [K90,G02] Log-Support encoding [G07] Order encoding for finite linear CSPs [TTKB09]...

137 Direct Encoding for CSP w/ Binary Constraints Variable x i with domain D i, with m i = D i Represent values of x i with Boolean variables x i,1,..., x i,mi Require m i k=1 x i,k = 1 Suffices to require m i k=1 x i,k 1 [W00] If the pair of assignments x i = v i x j = v j is not allowed, add binary clause ( x i,vi x j,vj )

138 Thanks!

CDCL SAT Solvers & SAT-Based Problem Solving

CDCL SAT Solvers & SAT-Based Problem Solving Joao Marques-Silva 1,2 & Mikolas Janota 2 1 University College Dublin, Ireland 2 IST/INESC-ID, Lisbon, Portugal SAT/SMT Summer School 2013 Aalto University,