CLAUSE LEARNING Level Dec. Unit Prop. ∅ 0 a ∨ ¯ (¯ b ) (¯ z ∨ b ) (¯ x ∨ ¯ z ∨ a ) 1 x (¯ a ∨ ¯ z ) 2 y ⊥ (¯ x ∨ ¯ 3 z a 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 14
CLAUSE LEARNING Level Dec. Unit Prop. ∅ 0 a ∨ ¯ (¯ b ) (¯ z ∨ b ) (¯ x ∨ ¯ z ∨ a ) 1 x (¯ a ∨ ¯ z ) 2 y ⊥ (¯ x ∨ ¯ 3 z a 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 14
CLAUSE LEARNING Level Dec. Unit Prop. ∅ 0 a ∨ ¯ (¯ b ) (¯ z ∨ b ) (¯ x ∨ ¯ z ∨ a ) 1 x (¯ a ∨ ¯ z ) 2 y ⊥ (¯ x ∨ ¯ 3 z a 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 14
CLAUSE LEARNING Level Dec. Unit Prop. ∅ 0 a ∨ ¯ (¯ b ) (¯ z ∨ b ) (¯ x ∨ ¯ z ∨ a ) 1 x (¯ a ∨ ¯ z ) 2 y ⊥ (¯ x ∨ ¯ 3 z a 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 14
CLAUSE LEARNING – AFTER BRACKTRACKING Level Dec. Unit Prop. ∅ 0 1 x z 2 y ⊥ 3 z z a b 15
CLAUSE LEARNING – AFTER BRACKTRACKING Level Dec. Unit Prop. ∅ 0 1 x z 2 y ⊥ 3 z a b ◮ Clause (¯ x ∨ ¯ z ) is asserting at decision level 1 15
CLAUSE LEARNING – AFTER BRACKTRACKING Level Dec. Unit Prop. Level Dec. Unit Prop. ∅ ∅ 0 0 ¯ 1 x z 1 x z 2 y ⊥ 3 z a b ◮ Clause (¯ x ∨ ¯ z ) is asserting at decision level 1 15
CLAUSE LEARNING – AFTER BRACKTRACKING Level Dec. Unit Prop. Level Dec. Unit Prop. ∅ ∅ 0 0 ¯ 1 x z 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] 15
UNIQUE IMPLICATION POINTS (UIPS) Level Dec. Unit Prop. ∅ 0 1 w w 2 x 3 y y 4 z z a c ⊥ b 16
UNIQUE IMPLICATION POINTS (UIPS) (¯ b ∨ ¯ c ) (¯ w ∨ c ) (¯ x ∨ ¯ a ∨ b ) (¯ y ∨ ¯ z ∨ a ) Level Dec. Unit Prop. ∅ 0 w ∨ ¯ (¯ b ) 1 w 2 x x (¯ (¯ w ∨ ¯ w ∨ ¯ x ∨ ¯ x ∨ ¯ a ) a ) 3 y y (¯ w ∨ ¯ x ∨ ¯ y ∨ ¯ z ) 4 z z a c ⊥ b ◮ Learn clause (¯ w ∨ ¯ x ∨ ¯ y ∨ ¯ z ) 16
UNIQUE IMPLICATION POINTS (UIPS) (¯ b ∨ ¯ c ) (¯ w ∨ c ) (¯ x ∨ ¯ a ∨ b ) (¯ y ∨ ¯ z ∨ a ) Level Dec. Unit Prop. ∅ 0 w ∨ ¯ (¯ b ) 1 w 2 x x (¯ (¯ w ∨ ¯ w ∨ ¯ x ∨ ¯ x ∨ ¯ a ) a ) 3 y y (¯ w ∨ ¯ x ∨ ¯ y ∨ ¯ z ) 4 z z a a c ⊥ b ◮ Learn clause (¯ w ∨ ¯ x ∨ ¯ y ∨ ¯ z ) ◮ But a is a UIP 16
UNIQUE IMPLICATION POINTS (UIPS) (¯ b ∨ ¯ c ) (¯ w ∨ c ) (¯ x ∨ ¯ a ∨ b ) (¯ y ∨ ¯ z ∨ a ) Level Dec. Unit Prop. ∅ 0 w ∨ ¯ (¯ b ) 1 w 2 x (¯ (¯ w ∨ ¯ w ∨ ¯ x ∨ ¯ x ∨ ¯ a ) a ) 3 y (¯ w ∨ ¯ x ∨ ¯ y ∨ ¯ z ) 4 z a c ⊥ b ◮ Learn clause (¯ w ∨ ¯ x ∨ ¯ y ∨ ¯ z ) ◮ But a is a UIP ◮ Learn clause (¯ w ∨ ¯ x ∨ ¯ a ) 16
MULTIPLE UIPS Level Dec. Unit Prop. ∅ 0 1 w w w 2 x 3 y y 4 z r a a a c ⊥ s b 17
MULTIPLE UIPS ◮ First UIP: Level Dec. Unit Prop. ◮ Learn clause (¯ w ∨ ¯ y ∨ ¯ ∅ a ) 0 1 w 2 x 3 y y 4 z r a c ⊥ s b 17
MULTIPLE UIPS ◮ First UIP: Level Dec. Unit Prop. ◮ Learn clause (¯ w ∨ ¯ y ∨ ¯ ∅ a ) 0 ◮ But there can be more than 1 UIP 1 w 2 x x 3 y y 4 z r a c ⊥ s b 17
MULTIPLE UIPS ◮ First UIP: Level Dec. Unit Prop. ◮ Learn clause (¯ w ∨ ¯ y ∨ ¯ ∅ a ) 0 ◮ But there can be more than 1 UIP 1 w ◮ Second UIP: ◮ Learn clause (¯ x ∨ ¯ z ∨ a ) 2 x 3 y 4 z r a c ⊥ s b 17
MULTIPLE UIPS ◮ First UIP: Level Dec. Unit Prop. ◮ Learn clause (¯ w ∨ ¯ y ∨ ¯ ∅ a ) 0 ◮ But there can be more than 1 UIP 1 w ◮ Second UIP: ◮ Learn clause (¯ x ∨ ¯ z ∨ a ) 2 x ◮ In practice smaller clauses more effective 3 y ◮ Compare with (¯ w ∨ ¯ x ∨ ¯ y ∨ ¯ z ) 4 z r a c ⊥ s b 17
MULTIPLE UIPS ◮ First UIP: Level Dec. Unit Prop. ◮ Learn clause (¯ w ∨ ¯ y ∨ ¯ ∅ a ) 0 ◮ But there can be more than 1 UIP 1 w ◮ Second UIP: ◮ Learn clause (¯ x ∨ ¯ z ∨ a ) 2 x ◮ In practice smaller clauses more effective 3 y ◮ Compare with (¯ w ∨ ¯ x ∨ ¯ y ∨ ¯ z ) 4 z r a c ⊥ 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 17
MULTIPLE UIPS ◮ First UIP: Level Dec. Unit Prop. ◮ Learn clause (¯ w ∨ ¯ y ∨ ¯ ∅ a ) 0 ◮ But there can be more than 1 UIP 1 w ◮ Second UIP: ◮ Learn clause (¯ x ∨ ¯ z ∨ a ) 2 x ◮ In practice smaller clauses more effective 3 y ◮ Compare with (¯ w ∨ ¯ x ∨ ¯ y ∨ ¯ z ) 4 z r a c ⊥ 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] 17
CLAUSE MINIMIZATION I Level Dec. Unit Prop. ∅ 0 1 x x b b b 2 y ⊥ 3 z c a 18
CLAUSE MINIMIZATION I z ∨ ¯ (¯ a ∨ ¯ c ) (¯ b ∨ c ) (¯ x ∨ ¯ y ∨ ¯ z ∨ a ) Level Dec. Unit Prop. ∅ 0 z ∨ ¯ (¯ b ∨ ¯ a ) 1 x b 2 y y z ∨ ¯ (¯ x ∨ ¯ y ∨ ¯ b ) ⊥ 3 z z c a z ∨ ¯ ◮ Learn clause (¯ x ∨ ¯ y ∨ ¯ b ) 18
CLAUSE MINIMIZATION I z ∨ ¯ (¯ a ∨ ¯ c ) (¯ b ∨ c ) (¯ x ∨ ¯ y ∨ ¯ z ∨ a ) (¯ x ∨ b ) Level Dec. Unit Prop. ∅ 0 z ∨ ¯ (¯ b ∨ ¯ a ) 1 x b 2 y y z ∨ ¯ (¯ x ∨ ¯ y ∨ ¯ b ) ⊥ 3 z z c a z ∨ ¯ ◮ Learn clause (¯ x ∨ ¯ y ∨ ¯ b ) ◮ Apply self-subsuming resolution (i.e. local minimization) [SB09] 18
CLAUSE MINIMIZATION I z ∨ ¯ (¯ a ∨ ¯ c ) (¯ b ∨ c ) (¯ x ∨ ¯ y ∨ ¯ z ∨ a ) (¯ x ∨ b ) Level Dec. Unit Prop. ∅ 0 z ∨ ¯ (¯ b ∨ ¯ a ) 1 x b 2 y z ∨ ¯ (¯ x ∨ ¯ y ∨ ¯ b ) ⊥ 3 z c (¯ x ∨ ¯ y ∨ ¯ z ) a z ∨ ¯ ◮ Learn clause (¯ x ∨ ¯ y ∨ ¯ b ) ◮ Apply self-subsuming resolution (i.e. local minimization) 18
CLAUSE MINIMIZATION I z ∨ ¯ (¯ a ∨ ¯ c ) (¯ b ∨ c ) (¯ x ∨ ¯ y ∨ ¯ z ∨ a ) (¯ x ∨ b ) Level Dec. Unit Prop. ∅ 0 z ∨ ¯ (¯ b ∨ ¯ a ) 1 x b 2 y z ∨ ¯ (¯ x ∨ ¯ y ∨ ¯ b ) ⊥ 3 z c (¯ x ∨ ¯ y ∨ ¯ z ) a z ∨ ¯ ◮ Learn clause (¯ x ∨ ¯ y ∨ ¯ b ) ◮ Apply self-subsuming resolution (i.e. local minimization) ◮ Learn clause (¯ x ∨ ¯ y ∨ ¯ z ) 18
CLAUSE MINIMIZATION II Level Dec. Unit Prop. ∅ 0 1 w w a c c b 2 x e ⊥ d 19
CLAUSE MINIMIZATION II Level Dec. Unit Prop. ◮ Learn clause (¯ w ∨ ¯ x ∨ ¯ c ) ∅ 0 1 w a c b 2 x x e ⊥ d 19
CLAUSE MINIMIZATION II Level Dec. Unit Prop. ◮ Learn clause (¯ w ∨ ¯ x ∨ ¯ c ) ∅ ◮ Cannot apply self-subsuming 0 resolution 1 w a c ◮ Resolving with reason of c yields a ∨ ¯ (¯ w ∨ ¯ x ∨ ¯ b ) b 2 x x e ⊥ d 19
CLAUSE MINIMIZATION II Level Dec. Unit Prop. ◮ Learn clause (¯ w ∨ ¯ x ∨ ¯ c ) ∅ ◮ Cannot apply self-subsuming 0 resolution 1 w a c ◮ Resolving with reason of c yields a ∨ ¯ (¯ w ∨ ¯ x ∨ ¯ b ) b ◮ Can apply recursive minimization 2 x x e ⊥ d 19
CLAUSE MINIMIZATION II Level Dec. Unit Prop. ◮ Learn clause (¯ w ∨ ¯ x ∨ ¯ c ) ∅ ◮ Cannot apply self-subsuming 0 resolution 1 w a c ◮ Resolving with reason of c yields a ∨ ¯ (¯ w ∨ ¯ x ∨ ¯ b ) b ◮ Can apply recursive minimization 2 x x e ⊥ d ◮ Marked nodes: literals in learned clause [SB09] 19
CLAUSE MINIMIZATION II Level Dec. Unit Prop. ◮ Learn clause (¯ w ∨ ¯ x ∨ ¯ c ) ∅ ◮ Cannot apply self-subsuming 0 resolution 1 w a c ◮ Resolving with reason of c yields a ∨ ¯ (¯ w ∨ ¯ x ∨ ¯ b ) b ◮ Can apply recursive minimization 2 x e ⊥ d ◮ Marked nodes: literals in learned clause [SB09] ◮ Trace back from c until marked nodes or new nodes / decisions ◮ Learn clause if only marked nodes visited 19
CLAUSE MINIMIZATION II Level Dec. Unit Prop. ◮ Learn clause (¯ w ∨ ¯ x ∨ ¯ c ) ∅ ◮ Cannot apply self-subsuming 0 resolution 1 w a c ◮ Resolving with reason of c yields a ∨ ¯ (¯ w ∨ ¯ x ∨ ¯ b ) b ◮ Can apply recursive minimization ◮ Learn clause (¯ w ∨ ¯ x ) 2 x e ⊥ d ◮ Marked nodes: literals in learned clause [SB09] ◮ Trace back from c until marked nodes or new nodes / decisions ◮ Learn clause if only marked nodes visited 19
SEARCH RESTARTS I ◮ Heavy-tail behavior: [GSK98] ◮ 10000 runs, branching randomization on industrial instance ◮ Use rapid randomized restarts (search restarts) 20
SEARCH RESTARTS II ◮ Restart search after a number of conflicts cutoff cutoff solution 21
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 21
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 ◮ Works for SAT & UNSAT instances. Why? 21
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) 21
DATA STRUCTURES BASICS ◮ Each literal l should access clauses containing l ◮ Why? 22
DATA STRUCTURES BASICS ◮ Each literal l should access clauses containing l ◮ Why? Unit propagation 22
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 22
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 22
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 ) 22
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 ) 22
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 ! 22
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: 22
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 22
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 22
WATCHED LITERALS [MMZZM01] ◮ Important states of a clause 23
WATCHED LITERALS [MMZZM01] ◮ Important states of a clause ◮ Associate 2 references with each clause 23
WATCHED LITERALS [MMZZM01] ◮ Important states of a clause ◮ Associate 2 references with each clause ◮ Deciding unit requires traversing all literals 23
WATCHED LITERALS [MMZZM01] ◮ Important states of a clause ◮ Associate 2 references with each clause ◮ Deciding unit requires traversing all literals ◮ References unchanged when backtracking 23
ADDITIONAL KEY TECHNIQUES ◮ Lightweight branching [e.g. MMZZM01] ◮ Use conflict to bias variables to branch on, associate score with each variable ◮ Prefer recent bias by regularly decreasing variable scores 24
ADDITIONAL KEY TECHNIQUES ◮ Lightweight branching [e.g. MMZZM01] ◮ Use conflict to bias variables to branch on, associate score with each variable ◮ Prefer recent bias by regularly decreasing variable scores ◮ Clause deletion policies ◮ Not practical to keep all learned clauses ◮ Delete less used clauses [e.g. MSS96,GN02,ES03] 24
ADDITIONAL KEY TECHNIQUES ◮ Lightweight branching [e.g. MMZZM01] ◮ Use conflict to bias variables to branch on, associate score with each variable ◮ Prefer recent bias by regularly decreasing variable scores ◮ Clause deletion policies ◮ Not practical to keep all learned clauses ◮ Delete less used clauses [e.g. MSS96,GN02,ES03] ◮ Proven recent techniques: ◮ Phase saving [PD07] ◮ Literal blocks distance [AS09] 24
What’s hot in SAT 25
WHAT’S HOT IN SAT? ◮ Clause learning techniques [e.g. ABHJS08,AS09] ◮ Clause learning is the key technique in CDCL SAT solvers ◮ Many recent papers propose improvements to the basic clause learning approach ◮ Preprocessing & inprocessing ◮ Many recent papers [e.g. JHB12,HJB11] ◮ Lots of recent work on symmetry exploitation (static/dynamic) [e.g. DBB17,JKKK17] ◮ Essential in some applications 26
WHAT’S HOT IN SAT? ◮ Proofs ◮ Proof logging (RUP , RAT, DRAT) [HHKW17] ◮ Proof complexity [VEGGN18] ◮ Other Inference Methods ◮ (Probabilistic) Model counting [e.g. AHT18] ◮ Optimisation (E.g., MAXSAT – more later) [e.g. LM09] ◮ Enumeration ◮ MUSes / MCSes ◮ Applications ◮ In various domains 27
Tentacles of CDCL 28
SOME TENTACLES OF CDCL ◮ Lazy Clause Generation for Constraint solving or SAT modulo theories ◮ Conflict-driven pseudo-Boolean solving ◮ Incremental SAT solving for MAXSAT & QBF. 29
SAT ENCODINGS ◮ Many different problems can easily be encoded into SAT ◮ For instance, finite-domain Constraint Solving ◮ Various encoding options: ◮ Equality: encode variable X ∈ [ − 100 , 100 ] by Boolean variables � X = − 100 � , � X = − 99 � , ...with uniqueness constraints ◮ Bound: encode variable X ∈ [ − 100 , 100 ] by Boolean variables � X ≤− 100 � , � X ≤− 99 � , ...with constraints � X ≤− 100 � ∨ � X ≤− 99 � , � X ≤− 99 � ∨ � X ≤− 98 � , . . . ◮ Log: encode variable X ∈ [ − 100 , 100 ] by means of bitvectors ◮ This talk assumes the Bound encoding. ◮ For each type of constraints, an encoding has to be invented 30
SAT ENCODINGS – EXAMPLE X , Y , Z , U , V ∈ [ − 100 , 100 ] (1) 4 U − X − Y ≥ 4 (2) V ≥ U (3) Z ≥ 5 V (4) Y + Z ≤ 24 (5) ( � X ≤− 100 � ∨ � X ≤− 99 � ) ∧ ( � X ≤− 99 � ∨ � X ≤− 98 � ) ∧ · · · ∧ ( � Y ≤− 100 � ∨ � Y ≤− 99 � ) ∧ . . . · · · ∧ ( � X ≤− 3 � ∨ � Y ≤ 9 � ∨ � U ≤ 2 � ) ∧ · · · ∧ ( � X ≤ 9 � ∨ � Y ≤ 9 � ∨ � U ≤ 5 � ) ∧ . . . ( � V ≤ 100 � ∨ � U ≤ 100 � ) ∧ ( � V ≤ 99 � ∨ � U ≤ 99 � ) ∧ · · · ∧ ( � V ≤ 5 � ∨ � U ≤ 5 � ) ∧ . . . · · · ∧ ( � V ≤ 0 � ∨ � Z ≤ 4 � ) ∧ · · · ∧ ( � V ≤ 2 � ∨ � Z ≤ 14 � ) ∧ . . . · · · ∧ ( � Y ≤ 9 � ∨ � Z ≤ 14 � ) ∧ . . . 31
CONSTRAINT PROGRAMMING USING SAT ◮ If the SAT encoding of a CP program is not too large (at least: fits in memory), we can create it eagerly and use a CDCL solver to solve it. ◮ But... we can also generate it lazily = Lazy Clause Generation (LCG) ◮ Many constraint propagators work by search + domain propagation ◮ Idea: generate parts of the encoding only when CDCL solvers needs it: ◮ During propagation ◮ During explanation 32
CONSTRAINT PROGRAMMING USING SAT ◮ If the SAT encoding of a CP program is not too large (at least: fits in memory), we can create it eagerly and use a CDCL solver to solve it. ◮ But... we can also generate it lazily = Lazy Clause Generation (LCG) ◮ Many constraint propagators work by search + domain propagation ◮ Idea: generate parts of the encoding only when CDCL solvers needs it: ◮ During propagation ◮ During explanation Can use structure in constraints to learn better clauses ! Example on Blackboard 32
CONSTRAINT PROGRAMMING USING SAT ◮ If the SAT encoding of a CP program is not too large (at least: fits in memory), we can create it eagerly and use a CDCL solver to solve it. ◮ But... we can also generate it lazily = Lazy Clause Generation (LCG) ◮ Many constraint propagators work by search + domain propagation ◮ Idea: generate parts of the encoding only when CDCL solvers needs it: ◮ During propagation ◮ During explanation Can use structure in constraints to learn better clauses ! Example on Blackboard ◮ Many more interesting phenomena going on in LCG (see you next week!) 32
PSEUDO-BOOLEAN SOLVING Observations: ◮ Resolution proof system is weak (cfr Pigeonhole) ◮ Stronger proof systems exist, for instance cutting planes makes use of (linear) pseudo-Boolean constraints (linear constraints over literals) [CCT87] ◮ A clause a ∨ ¯ b ∨ c corresponds to a PB constraint a + ¯ b + c ≥ 1 ◮ A PB constraint a + ¯ b + 2 · c + d ≥ 2 cannot be translated into a single clause 33
CUTTING PLANE PROOF SYSTEM (literal axiom) l ≥ 0 � i a i l i ≥ A � i b i l i ≥ B (linear combination) i ( ca i + db i ) l i ≥ cA + dB � � i a i l i ≥ A (division) � i ⌈ a i / c ⌉ l i ≥ ⌈ A / c ⌉ 34
CUTTING PLANES VS RESOLUTION ◮ In theory, learning cutting planes could allow to derive unsat proofs much faster ◮ In practice, CDCL solvers seem to outperform cutting plane solvers ◮ Very recently, new cutting-plane solvers, inspired by CDCL are arising [GNY19] ◮ Various issues show up: generalizing CDCL, 1UIP , ... far from obvious! 35
INCREMENTAL SAT SOLVING & SAT ORACLES ◮ Incremental SAT Solving: [ES01] ◮ Allow calling a solver with a set of assumptions ◮ Variables whose value is set before the search start (never backtrack over them!) ◮ Often used: replace each clause C i with C i ∨ ¬ a i ◮ a i = 1 to activate clause C i ◮ a i = 0 to deactivate clause C i ◮ Enables clause reuse 36
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