SAT J. Gonzalez. Summary Introduction Algorithms for SAT –Splitting and Resolution –Local Search...

SAT

J. Gonzalez

Summary

• Introduction

• Algorithms for SAT– Splitting and Resolution– Local Search– Integer Programming Method– Global Optimization

• SAT Solvers Features

• Final Thoughts

SAT

• Decission Problem

• Assing true and false values to make the sentence true

• NP-Complete

( ) ( )P Q S P Q R SÚ ÚØ Ù Ø Ú Ú Ú

• Literals: An atom or its negation• Clauses: Disjunction of Literals• Sentence in CNF: Conjunction of Clauses

Literal Literal

Clauses Clauses

Sentences

In CNF

.

.

.

CNF and DNF

• CNF: Conjunctive normal form

• DNF: Disjunctive normal form

( ) ( )b c b dÚØ Ù Ø Ú

( ) ( )( )( ) ( )

( ) ( )

b c b d

b c b d

b c b d

Ø ÚØ Ù Ø Ú

Ø ÚØ ÚØ Ø Ú

Ø Ù Ú ÙØ

CSP (Constraint Satisfaction Problems)

• Variables: WA, NT, Q, NSW, V, SA, T • Domains: Di = {red,green,blue}• Constraints:

– WA ≠ NT, – WA ≠ SA– NT ≠ SA– NT ≠ Q– SA ≠ Q– SA ≠ NSW– SA ≠ V– Q ≠ NSW– NSW ≠ V– V ≠ T



– WA ≠ NT, – NT ≠ SA– NT ≠ Q– SA ≠ Q– SA ≠ NSW– SA ≠ V– Q ≠ NSW– NSW ≠ V– V ≠ T

XWA,red XWA,green XWA,blue

XNT,red XNT,green XNT,blue

XWA,red → ¬XNT,red

XWA,green → ¬XNT,green

XWA,blue → ¬XNT,blue




XWA,red XWA,green XWA,blue

XNT,red XNT,green XNT,blue

¬XWA,red ¬XNT,red

¬XWA,green ¬XNT,green

¬XWA,blue ¬XNT,blue




(XWA,red XWA,green XWA,blue)

(XNT,red XNT,green XNT,blue)

(¬XWA,red ¬XNT,red)

(¬XWA,gree ¬XNT,gree)

(¬XWA,blue ¬XNT,blue) …


• SAT is an special case of CSP.• CSP to SAT • CSP

– Discrete CSP• N-queen problem• Graph coloring problem• Scheduling problem

– Binary CSP• SAT problem• Max-SAT problem

Summary

• Introduction



• Final Thoughts

Algorithms for SAT• Discrete Constrained Algorithms:

– Discrete, satisfies all clauses– Goal: satisfy all clauses (constraints )– Usually uses Splitting (search) and/or Resolution

• Discrete Unconstrained Algorithms:– Discrete, minimize a function– Minimize the number of unsatisfied clauses– Usually uses Local Search.

• Constrained Programming Algorithms:– Non discrete, satisfies all clauses– From CNF to IP (Integer Programming).– Usually uses Linear Programming relaxations

• Unconstrained Global Optimization Algorithms:– Non discrete, minimize a function (constrains included in such function)– From a formula on Boolean Space (a decision problem) to an unconstrained

problem on real Space (unconstrained global optimization problem). – Usually uses many global optimization methods.

Summary

• Introduction



• Final Thoughts

Splitting and Resolution

• Replace one formula by one or more other equivalent formulas.• Splitting:

– A variable v is replaced by True and False. – Two sub-formulas are generated. – The original formula has a satisfying truth assignment iff either sub-

formula has one satisfying truth assignment .

( )

( )

True Q S

False Q R S

Ú ÚØ Ù

Ú Ú Ú

( )

( )

False Q S

True Q R S

Ú ÚØ Ù

Ú Ú Ú

( )

( )

P Q S

P Q R S

Ú ÚØ Ù

Ø Ú Ú Ú

P=true

P=False


• Stop recursion:– Any formula with no variables is True– All subformulas are false and there is no formulas with

variables

( )

( )

True Q S

False Q R S

Ú ÚØ Ù

Ú Ú Ú

( )

( )

False True

True

Ú Ù

( )

( )

False False

False

Ú Ù ( )

( )

True False

False

Ú Ù ( )

( )

False False False

True True

Ú Ú Ù

Ú

Satisfiable

Unsatisfiable

Splitting and Resolution• Resolution: Using a resolvent to create new clauses.• Resolvent: clauses transformation based on a given variable v.

( )

( )

P Q S

M Q R S

Ú ÚØ Ù

Ø Ú Ú Ú ( )P Q M Q RÚ ÚØ Ú Ú

Resolvent


• Stop recursion:– Formulas with an empty clause have no solution– No more resolvent could be created

( ) ( )S SØ Ù ()

( )

( )

P N

M Q R

Ú Ù

Ø Ú Ú

Unsatisfiable

Satisfiable


• First methods:– DP (Davis-Putnam): Resolution– DPL (Davis-Putnam-Loveland): Splitting (+

depth-first search avoids memory explosion)

Summary

• Introduction



• Final Thoughts

Local Search

• Begins with an initial vector y0

• F(y0): Number of unsatisfiable formulas

• We define neighbourhood function N(yi)

• N(y0)=y1

• F(yi+1)>F(y0): strategies are applied to help escape from the local minima.

( ) ( ) ( )A B C A D AÚ ÚØ Ù Ú Ù Ø

True(C), False(A,B,D)

( ) ( ) ( )F F F F F TÚ Ú Ù Ú ÙF=2

( ) ( ) ( )A B C A D AÚ ÚØ Ù Ú Ù Ø


( ) ( ) ( )F F F F F TÚ Ú Ù Ú ÙF=2

( )( ) ( )A B C A D AÚ ÚØ Ù Ú Ù Ø

True(A,C), False(B,D)

( ) ( ) ( )T F F T F FÚ Ú Ù Ú ÙF=1

( ) ( ) ( )A B C A D AÚ ÚØ Ù Ú Ù Ø


( ) ( ) ( )F F F F F TÚ Ú Ù Ú ÙF=2

( )( ) ( )A B C A D AÚ ÚØ Ù Ú Ù Ø


( ) ( ) ( )T F F T F FÚ Ú Ù Ú ÙF=1

( ) ( ) ( )A B C A D AÚ ÚØ Ù Ú Ù Ø

True(A,B,C), False(D)

( ) ( ) ( )T T F T F FÚ Ú Ù Ú Ù

F=1

( ) ( ) ( )A B C A D AÚ ÚØ Ù Ú Ù Ø


( ) ( ) ( )F F F F F TÚ Ú Ù Ú ÙF=2

( )( ) ( )A B C A D AÚ ÚØ Ù Ú Ù Ø


( ) ( ) ( )T F F T F FÚ Ú Ù Ú ÙF=1

( ) ( ) ( )A B C A D AÚ ÚØ Ù Ú Ù Ø

True(A,B,C), False(D)

( ) ( ) ( )T T F T F FÚ Ú Ù Ú Ù

F=1

( )( ) ( )A B C A D AÚ ÚØ Ù Ú Ù Ø

True(A,C,D), False(B)

( ) ( ) ( )T F F T T FÚ Ú Ù Ú Ù

F=1

( ) ( ) ( )A B C A D AÚ ÚØ Ù Ú Ù Ø

True(C), False(A,D,B)

( ) ( ) ( )F F F F F TÚ Ú Ù Ú ÙF=2

( )( ) ( )A B C A D AÚ ÚØ Ù Ú Ù Ø

True(C,D), False(A,B)

( ) ( ) ( )F F F F T TÚ Ú Ù Ú ÙF=1

( ) ( ) ( )A B C A D AÚ ÚØ Ù Ú Ù Ø

True(C,D,A), False(B)

( ) ( ) ( )T F F T T FÚ Ú Ù Ú Ù

F=1

( )( ) ( )A B C A D AÚ ÚØ Ù Ú Ù Ø

True(C,D,B), False(A)

( ) ( ) ( )F T F F T TÚ Ú Ù Ú Ù

F=0

Summary

• Introduction



• Final Thoughts

Integer Programming Method (IP)

( ) ( ) ( )A A B A B CÙ Ú Ù Ø Ú Ú 1, 1, True False® ®- Ú®+


( ) ( ) ( )A A B A B CÙ Ú Ù Ø Ú Ú

( )1 1=

1, 1, True False® ®- Ú®+

( )A True= ( )1 1- = 1X ³


( ) ( ) ( )A A B A B CÙ Ú Ù Ø Ú Ú

( )1 1=


( )A True= ( )1 1- = 1X ³

( )A B TrueÚ = ( )1 1 2,

(1 1) 0,

( 1 1) 0,

( 1 1) 2

+ =

- =

- + =

- - =-

0X Y+ ³


( ) ( ) ( )A A B A B CÙ Ú Ù Ø Ú Ú

( )1 1=


( )A True= ( )1 1- = 1X ³

( )A B TrueÚ = ( )1 1 2,

(1 1) 0,

( 1 1) 0,

( 1 1) 2

+ =

- =

- + =

- - =-

0X Y+ ³

( )A B C TrueØ Ú Ú = ( )1 1 1 1- - + =- 1X Y Z- + + ³ -


( ) ( ) ( )A A B A B CÙ Ú Ù Ø Ú Ú

( )1 1=


( )A True= ( )1 1- = 1X ³

( )A B TrueÚ = ( )1 1 2,

(1 1) 0,

( 1 1) 0,

( 1 1) 2

+ =

- =

- + =

- - =-

0X Y+ ³

( )A B C TrueØ Ú Ú = ( )1 1 1 1- - + =- 1X Y Z- + + ³ -

( ),1

2n

i j j ij

q w C=

³ -å


• Solving– Linear Programs (LP): solved by the Simplex– Integer Programs (IP): no fast technique.– Integer Linear Programs (ILP): Is a LP constrained by

integrality restrictions

• Method:– Solve a LP– If solution is integer => end– If solution is non integer, round off such values

• If it is a solution => end• Else adds a new constraint and try again

Summary

• Introduction



• Final Thoughts

Global optimization

• Continuous Unconstrained Formulation

• From discrete to continuous (UniSAT)

• Try to minimize a function instead of trying to verify concrete restrictions.

Summary

• Introduction



• Final Thoughts

Which is the best method?

• Local search: – Faster for satisfiable CNF– Can not prove unsatisfiability

• DP algorithm (split, resolution):– Slower for satisfiable CNF– Can prove unsatisfiability

Algorithm categories

• Complete– Find the unique hard solution – Determine whether or not a solution exits– Give the variable settings for one solution– Find all solutions or an optimal solution– Prove that there is no solution

• Incomplete– Verify that there is no solution but could not find one.– Can not optimize solution quality

Performance evaluation

• Performance evaluation– Experimentally

• Works for random formulas• Does not work for worst-case formulas (too many formulas

for every given size)• Sometimes inconclusive

– Analytically• Works for random formulas• Works for worst-case formulas• Does not work for typical formulas (which is his mathematical

structure)• Need simplified algorithms

P-Subclasses of SAT

• Solved in polynomial time– Determine if a formula (or a portion) is polynomial-time solvable– The study of this subclasses reveals the nature of these “easy”

SAT formulas

• Subclasses:– 2-SAT: CNF formula with clauses of one or two literals only:

– Horn Formulas: CNF formula where every formula has at most one positive literal.

– Extended Horn Formulas: Rounding the results of a Linear Programming method we obtain the real solutions.

– Balanced formulas: When a Linear Programming method can be used to obtain integer solutions.

( )A B C DØ ÚØ Ú ÚØ

( ) ( ) ( )A D C A BÚ Ù Ø Ú Ù

Applications• Mathematics• Computer Science and AI• Machine Vision• Robotics• Computer-aided manufacturing• Database systems• Text processing• Computer graphics• Integrated circuit design automation• Computer architecture design• High-speed networking• Communications• Security

Summary

• Introduction



• Final Thoughts

2005 Industrial

Gold Silver Bronze

SAT and UNSAT SatELiteGTI MiniSAT 1.13zChaff_rand andHaifaSAT

SAT SatELiteGTI MiniSAT 1.13Jerusat 1.31 B and HaifaSAT

UNSAT SatELiteGTI Zchaff_rand HaifaSat

2005 Handmake

Gold Silver Bronze

SAT and UNSAT Vallst SatELiteGTI March_dl

SAT Vallst March_dl Hsat1

UNSAT SatELiteGTI MiniSAT 1.13Vallst and March-dl

2005 Random

Gold Silver Bronze

SAT and UNSAT kcnf-2004 March_dl Dew_Satz1a

SAT ranov g2wsat VW

UNSAT kcnf-2004 March_dl Dew_Satz1a

2004 Industrial handmade Random

ALL (SAT+UNSAT)

Zchaff 2004 March-eq AdaptNovelty

SAT Jerusat Satzoo AdaptNovelty

UNSAT Zchaff 2004 March-eq Kcnfs


Complete on all Forklift Satzoo Kcnfs

ALL on SAT Forklift Satzoo Unitwalk


Complete on all zChaff zChaff OKSolver

ALL on SAT Limmat Berkmin OKSolver

A few names

• SatELite: CNF minimizer (+MiniSAT)• MiniSAT: the SAT solver properly• Authors: Niklas Eén and Niklas Sörensson • Vallst by Daniel Vallstrom • Kcnfs: is a generalization of cnfs. Olivier Dubois

and Gilles Dequen• Ranov by by L&C Researcher Anbulagan and

Nghia Duc Pham • Zchaff: Boolean Satisfiability Research Group at

Princeton University

Some last ideas

• Produce each solution => exponential in the worst case (whether or not P=NP)– List an exponential number of solutions– Solutions in compressed form

• Cylinders of solutions: variables not listed could have any value.

• BDD: Binary Decision Diagrams

• SAT solvers can be faster than other non-NP systems under some circumstances.

Bibliography

• Algorithms for the Satisfiability (SAT) Problem: A Survey.

Jun Gu, Paul W. Purdom, John Franco, Benjamin W. Wah

• SAT 2004: http://www.satisfiability.org/SAT04/• SAT 2005: http://www.satisfiability.org/SAT05/• SAT 2006:

http://www.easychair.org/FLoC-06/SAT.html

http://www.satisfiability.org/SAT04/

http://www.satisfiability.org/SAT05/

http://www.easychair.org/FLoC-06/SAT.html

SAT J. Gonzalez. Summary Introduction Algorithms for SAT –Splitting and Resolution –Local Search...

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