CDCL SAT Solvers & SAT-Based Problem Solving Joao Marques-Silva 1 - PowerPoint PPT Presentation

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, Espoo, Finland

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

Part I CDCL SAT Solvers

Outline Basic Definitions DPLL Solvers CDCL Solvers What Next in CDCL Solvers?

Outline Basic Definitions DPLL Solvers CDCL Solvers What Next in CDCL Solvers?

Preliminaries • Variables: w , x , y , z , a , b , c , . . . • Literals: w , ¯ x , ¯ y , 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 } • Formula can be SAT/UNSAT

Preliminaries • Variables: w , x , y , z , a , b , c , . . . • Literals: w , ¯ x , ¯ y , 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 } • Formula can be SAT/UNSAT • Example: z ∨ c ) ∧ (¯ F , ( r ) ∧ (¯ r ∨ s ) ∧ (¯ w ∨ a ) ∧ (¯ x ∨ b ) ∧ (¯ y ∨ ¯ b ∨ ¯ c ∨ d ) – Example models: I { r , s , a , b , c , d } I { r , s , ¯ x , y , ¯ w , z , ¯ a , b , c , d }

Resolution • Resolution rule: [DP60,R65] ( α ∨ x ) ( β ∨ ¯ x ) ( α ∨ β ) – Complete proof system for propositional logic

Resolution • Resolution rule: [DP60,R65] ( α ∨ x ) ( β ∨ ¯ x ) ( α ∨ β ) – Complete proof system for propositional logic ( x ∨ a ) (¯ x ∨ a ) (¯ y ∨ ¯ a ) ( y ∨ ¯ a ) ( a ) (¯ a ) ⊥ – Extensively used with (CDCL) SAT solvers

Resolution • Resolution rule: [DP60,R65] ( α ∨ x ) ( β ∨ ¯ x ) ( α ∨ β ) – Complete proof system for propositional logic ( x ∨ a ) (¯ x ∨ a ) (¯ y ∨ ¯ a ) ( y ∨ ¯ a ) ( a ) (¯ a ) ⊥ – Extensively used with (CDCL) SAT solvers • Self-subsuming resolution (with α 0 ⊆ α ): [e.g. SP04,EB05] ( α 0 ∨ ¯ ( α ∨ x ) x ) ( α ) – ( α ) subsumes ( α ∨ x )

Unit Propagation ( r ) ∧ (¯ r ∨ s ) ∧ F = w ∨ a ) ∧ (¯ x ∨ ¯ a ∨ b ) (¯ z ∨ c ) ∧ (¯ y ∨ ¯ b ∨ ¯ c ∨ d ) (¯

Unit Propagation ( r ) ∧ (¯ r ∨ s ) ∧ F = w ∨ a ) ∧ (¯ x ∨ ¯ a ∨ b ) (¯ z ∨ c ) ∧ (¯ y ∨ ¯ b ∨ ¯ c ∨ d ) (¯ • Decisions / Variable Branchings: w = 1 , x = 1 , y = 1 , z = 1

Unit Propagation Level Dec. Unit Prop. 0 ∅ r s ( r ) ∧ (¯ r ∨ s ) ∧ F = 1 w a w ∨ a ) ∧ (¯ x ∨ ¯ a ∨ b ) (¯ z ∨ c ) ∧ (¯ y ∨ ¯ b ∨ ¯ c ∨ d ) (¯ 2 x b 3 y • Decisions / Variable Branchings: w = 1 , x = 1 , y = 1 , z = 1 4 z c d

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

Outline Basic Definitions DPLL Solvers CDCL Solvers What Next in CDCL Solvers?

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

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

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

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

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

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

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

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

Outline Basic Definitions DPLL Solvers CDCL Solvers What Next in CDCL Solvers?

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

How Significant are CDCL SAT Solvers? Results of the SAT competition/race winners on the SAT 2009 application benchmarks, 20mn timeout 1200 Limmat (2002) Zchaff (2002) Berkmin (2002) Forklift (2003) Siege (2003) 1000 Zchaff (2004) SatELite (2005) Minisat 2 (2006) Picosat (2007) Rsat (2007) Minisat 2.1 (2008) 800 Precosat (2009) Glucose (2009) CPU Time (in seconds) Clasp (2009) Cryptominisat (2010) Lingeling (2010) Minisat 2.2 (2010) 600 Glucose 2 (2011) Glueminisat (2011) Contrasat (2011) Lingeling 587f (2011) 400 GRASP 200 DPLL 0 0 20 40 60 80 100 120 140 160 180 200 ? ? Number of problems solved

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

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

Clause Learning Level Dec. Unit Prop. 0 ∅ a ∨ ¯ z ∨ b ) x ∨ ¯ z ∨ a ) (¯ b ) (¯ (¯ 1 x (¯ a ∨ ¯ z ) 2 y ab -> false = !(ab) + false = !a + !b z -> b = !z + b 3 ⊥ z a x ∨ ¯ (¯ z ) xz -> a. =. !(xz) + a =. !x + !z + a b • Analyze conflict – Reasons: x and z I Decision variable & literals assigned at lower decision levels x ∨ ¯ – Create new clause: (¯ z ) • Can relate clause learning with resolution

Clause Learning – After Bracktracking Level Dec. Unit Prop. ∅ 0 1 x z 2 y ⊥ 3 z z a b

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

Clause Learning – After Bracktracking Level Dec. Unit Prop. Level Dec. Unit Prop. ∅ ∅ 0 0 1 1 ¯ x z x z 2 y ⊥ 3 z a b • Clause (¯ x ∨ ¯ z ) is asserting at decision level 1