cdcl sat solvers & sat-based problem solvinghaozheng/teach/psv/slides/...cdcl sat solvers &...

CDCL SAT Solvers & SAT-Based Problem Solving

Joao Marques-Silva1,2 & Mikolas Janota2

1University College Dublin, Ireland

2IST/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?

Preliminaries

• Variables: w , x , y , z , a, b, c , . . .

• Literals: w , x̄ , 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 fromvariables to {0, 1}

• Formula can be SAT/UNSAT

• Example:

F , (r) ^ (r̄ _ s) ^ (w̄ _ a) ^ (x̄ _ b) ^ (ȳ _ z̄ _ c) ^ (b̄ _ c̄ _ d)

– Example models:I {r , s, a, b, c, d}I {r , s, x̄ , y , w̄ , z , ā, b, c, d}

Resolution

• Resolution rule: [DP60,R65]

(↵ _ x) (� _ x̄)(↵ _ �)

– Complete proof system for propositional logic

– Extensively used with (CDCL) SAT solvers

• Self-subsuming resolution (with ↵0 ✓ ↵): [e.g. SP04,EB05]

(↵ _ x) (↵0 _ x̄)(↵)

– (↵) subsumes (↵ _ x)

Resolution

• Resolution rule: [DP60,R65]

(↵ _ x) (� _ x̄)(↵ _ �)

– Complete proof system for propositional logic

(x _ a) (x̄ _ a) (ȳ _ ā) (y _ ā)

(a) (ā)

?

– Extensively used with (CDCL) SAT solvers

• Self-subsuming resolution (with ↵0 ✓ ↵): [e.g. SP04,EB05]

(↵ _ x) (↵0 _ x̄)(↵)

– (↵) subsumes (↵ _ x)

Unit Propagation

F = (r) ^ (r̄ _ s)^

(w̄ _ a) ^ (x̄ _ ā _ b)

(ȳ _ z̄ _ c) ^ (b̄ _ c̄ _ d)

• Decisions / Variable Branchings:w = 1, x = 1, y = 1, z = 1

• Additional definitions:– Antecedent (or reason) of an implied assignment

I (b̄ _ c̄ _ d) for d

– Associate assignment with decision levelsI w = 1@1, x = 1@2, y = 1@3, z = 1@4I r = 1@0, d = 1@4, ...

Unit Propagation

F = (r) ^ (r̄ _ s)^

(w̄ _ a) ^ (x̄ _ ā _ b)

(ȳ _ z̄ _ c) ^ (b̄ _ c̄ _ d)

• Decisions / Variable Branchings:w = 1, x = 1, y = 1, z = 1

Level Dec. Unit Prop.

0

1

2

3

4

;

w

x

y

z

a

b

c d

r s

• Additional definitions:– Antecedent (or reason) of an implied assignment

I (b̄ _ c̄ _ d) for d

– Associate assignment with decision levelsI w = 1@1, x = 1@2, y = 1@3, z = 1@4I r = 1@0, d = 1@4, ...

Outline

Basic Definitions

DPLL Solvers

CDCL Solvers


The DPLL Algorithm

Assign valueto variable

Unassignedvariables ?

Unit propagation

Conflict ?

Can undodecision ?

Backtrack &flip variable

Unsatisfiable

SatisfiableY

N

Y

N

Y

N

• Optional: pure literal rule

The DPLL Algorithm



Unit propagation

Conflict ?

Can undodecision ?


Unsatisfiable

SatisfiableY

N

Y

N

Y

N


F = (x_y)^(a_b)^(ā_b)^(a_b̄)^(ā_b̄)

The DPLL Algorithm



Unit propagation

Conflict ?

Can undodecision ?


Unsatisfiable

SatisfiableY

N

Y

N

Y

N


F = (x_y)^(a_b)^(ā_b)^(a_b̄)^(ā_b̄)


0

1

2

3

;

x

y

a b ?

a ā

y

a ā

ȳ

x

a ā

x̄

The DPLL Algorithm



Unit propagation

Conflict ?

Can undodecision ?


Unsatisfiable

SatisfiableY

N

Y

N

Y

N


F = (x_y)^(a_b)^(ā_b)^(a_b̄)^(ā_b̄)


0

1

2

3

;

x

y

ā b̄ ?

a ā

y

a ā

ȳ

x

a ā

x̄

The DPLL Algorithm



Unit propagation

Conflict ?

Can undodecision ?


Unsatisfiable

SatisfiableY

N

Y

N

Y

N


F = (x_y)^(a_b)^(ā_b)^(a_b̄)^(ā_b̄)


0

1

2

3

;

x

ȳ

a b ?

a ā

y

a ā

ȳ

x

a ā

x̄

The DPLL Algorithm



Unit propagation

Conflict ?

Can undodecision ?


Unsatisfiable

SatisfiableY

N

Y

N

Y

N


F = (x_y)^(a_b)^(ā_b)^(a_b̄)^(ā_b̄)


0

1

2

3

;

x

ȳ

ā b̄ ?

a ā

y

a ā

ȳ

x

a ā

x̄

The DPLL Algorithm



Unit propagation

Conflict ?

Can undodecision ?


Unsatisfiable

SatisfiableY

N

Y

N

Y

N


F = (x_y)^(a_b)^(ā_b)^(a_b̄)^(ā_b̄)


0

1

2

;

x̄

a

y

b ?

a ā

y

a ā

ȳ

x

a ā

x̄

The DPLL Algorithm



Unit propagation

Conflict ?

Can undodecision ?


Unsatisfiable

SatisfiableY

N

Y

N

Y

N


F = (x_y)^(a_b)^(ā_b)^(a_b̄)^(ā_b̄)


0

1

2

;

x̄

ā

y

b̄ ?

a ā

y

a ā

ȳ

x

a ā

x̄

Outline

Basic Definitions

DPLL Solvers

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 structuresI Watched literals [MMZZM01]

– Conflict-guided branchingI Lightweight branching heuristics [MMZZM01]I Phase saving [PD07]

– ...

How Significant are CDCL SAT Solvers?

0

200

400

600

800

1000

1200

0 20 40 60 80 100 120 140 160 180 200

CPU

Tim

e (in

seco

nds)

Number of problems solved

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)

GRASP

DPLL

? ?

Outline

Basic Definitions

DPLL Solvers

CDCL SolversClause Learning, UIPs & MinimizationSearch Restarts & Lazy Data Structures


Clause Learning


0

1

2

3

;

xx

y

zz a

b

?

• Analyze conflict

– Reasons: x and zI Decision variable & literals assigned at lower decision levels

– Create new clause: (x̄ _ z̄)

• Can relate clause learning with resolution

– Learned clauses result from (selected) resolution operations

Clause Learning


0

1

2

3

;

xx

y

zz a

b

?

(ā _ b̄) (z̄ _ b) (x̄ _ z̄ _ a)

(ā _ z̄)

(x̄ _ z̄)

• Analyze conflict– Reasons: x and z

I Decision variable & literals assigned at lower decision levels

– Create new clause: (x̄ _ z̄)

• Can relate clause learning with resolution

– Learned clauses result from (selected) resolution operations

ab -> false = !(ab) + false = !a + !b

z -> b = !z + b

xz -> a. =. !(xz) + a =. !x + !z + a

Clause Learning – After Bracktracking


0

1

2

3

;

x

y

zz aa

bb

??

z

• Clause (x̄ _ z̄) is asserting at decision level 1

• Learned clauses are always asserting [MSS96,MSS99]• Backtracking di↵ers from plain DPLL:

– Always bactrack after a conflict [MMZZM01]

Clause Learning – After Bracktracking


0

1

2

3

;

x

y

zz aa

bb

??

z


0

1

;

x z̄

• Clause (x̄ _ z̄) is asserting at decision level 1

• Learned clauses are always asserting [MSS96,MSS99]• Backtracking di↵ers from plain DPLL:

– Always bactrack after a conflict [MMZZM01]

cdcl sat solvers & sat-based problem solvinghaozheng/teach/psv/slides/...cdcl sat solvers &...

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