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Boolean Satisfiability (SAT)

Class Presentation By

Girish Paladugu

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Overview

• Introduction• Conjunctive Normal Form (CNF)• DLL Procedure for solving SAT problems• Backtrack search algorithm• Eqvivalence Checking• Summary

INTRODUCTION• SAT Given a Boolean formula (propositional logic formula), find a variable assignment such that the formula evaluates to 1, or prove that no such assignment exists.

• For a formula with n variables, there are 2n possible truth assignments to be checked.

a

b b

c c c c

0 1

0 0

00 001

1 1

1 1 1

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik3

( a +c ) ( b +c ) (¬a +¬b + ¬c )

CNF

• Conjunctive Normal Form (CNF): A conjunctive normal form (CNF) formula ϕ on n binary variables x1, …, xn is the conjunction of m clauses ω1, …, ωm.

• Literal: Variable x or its complement x’.

• Clause: Disjunction of one or more literals.

• CNF formula is satisfiable if each clause is satisfiable ( or evaluate to true) else it is unsatisfiable.

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( a +c ) ( b +c ) (¬a +¬b + ¬c )

2- input AND Gate CNF

ab d

d= [d = ¬(a b)]= ¬[d ¬(a b)]

= ¬[¬(a b)¬d + a b d]

= ¬[¬a ¬d + ¬b ¬d + a b d]

= (a +d)(b +d)(¬a +¬b + ¬d)

a b d

0 0 1

0 1 1

1 0 1

1 1 0

Truth Table

This method can be applied to any number input gate.

5Boolean Satisfiability in Electronic Design Automation - joao marques silva, Karem A Sakellah

CNF Formulas for simple gates

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Circuit - CNF conversion

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• The CNF formula for circuit is the set union of the CNF formulas of each gate.

DLL Algorithm

• Davis, Logemann and Loveland

• Basic framework for many modern SAT solvers

• Complete Backtracking-based algorithm for solving the

CNF-SAT problem

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik8

Basic DLL Procedure - DFS

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik9

Basic DLL Procedure - DFS

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

a

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik10

Basic DLL Procedure - DFS

a0

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

Decision

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik

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Basic DLL Procedure - DFS

a0

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

b

0 Decision

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik

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Basic DLL Procedure - DFS

a0

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

b

0

c

0 Decision

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik

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Basic DLL Procedure - DFS

a0

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

b

0

c

0

d=1

c=0

(a + c + d)a=0

d=0(a + c + d’)

Conflict!Implication Graph

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik

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Basic DLL Procedure - DFS

a0

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

b

0

c

0

d=1

c=0

(a + c + d)a=0

d=0(a + c + d’)

Conflict!Implication Graph

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik

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Basic DLL Procedure - DFS

a0

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

b

0

c

0

Backtrack

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik

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Basic DLL Procedure - DFS

a0

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

b

0

c

0

d=1

c=1

(a + c’ + d)a=0

d=0(a + c’ + d’)

Conflict!

1 Forced Decision

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik

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Basic DLL Procedure - DFS

a0

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

b

0

c

0 1

Backtrack

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik

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Basic DLL Procedure - DFS

a0

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

b

0

c

0 1

1 Forced Decision

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik

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Basic DLL Procedure - DFS

a0

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

b

0

c

0

d=1

c=0

(a + c’ + d)a=0

d=0(a + c’ + d’)

Conflict!

1

c

0

1

Decision

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik

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Basic DLL Procedure - DFS

a0

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

b

0

c

0 1

c

0

1

Backtrack

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik21

Basic DLL Procedure - DFS

a0

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

b

0

c

0

d=1

c=1

(a + c’ + d)a=0

d=0(a + c’ + d’)

Conflict!

1

c

0 1

1

Forced Decision

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik

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Basic DLL Procedure - DFS

a0

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

b

0

c

0 1

c

0 1

1

Backtrack

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik

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Basic DLL Procedure - DFS

a0

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

b

0

c

0 1

c

0 1

1

1 Forced Decision

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik

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Basic DLL Procedure - DFS

a0

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

b

0

c

0 1

c

0 1

1

1

b

0 Decision

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik25

Basic DLL Procedure - DFS

a0

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

b

0

c

0 1

c

0 1

1

1

b

0

c=1

b=0

(a’ + b + c)a=1

c=0(a’ + b + c’)

Conflict!

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik26

Basic DLL Procedure - DFS

a0

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

b

0

c

0 1

c

0 1

1

1

b

0

Backtrack

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik27

Basic DLL Procedure - DFS

a0

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

b

0

c

0 1

c

0 1

1

1

b

0 1

a=1

b=1

c=1(a’ + b’ + c)

Forced Decision

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik

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Basic DLL Procedure - DFS

a

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

b

0

c

0 1

c

0 1

1

1

b

0 1

a=1

b=1

c=1(a’ + b’ + c) (b’ + c’ + d)

d=1

0

Adapted From ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik29

Basic DLL Procedure - DFS

a

(a + c + d)(a + c + d’)

(a + c’ + d)(a + c’ + d’)

(a’ + b + c)

(b’ + c’ + d)(a’ + b + c’)(a’ + b’ + c)

b

0

c

0 1

c

0 1

1

1

b

0 1

a=1

b=1

c=1(a’ + b’ + c) (b’ + c’ + d)

d=1

SAT

0

Adapted from ‘The Quest for Efficient Boolean Satisfiability Solvers’ – Sharad Malik30

Backtrack Search Algorithm

• Backtrack search has proven useful for solving instances of SAT from EDA applications, in particular for applications where the objective is to prove unsatisfiability.

• The algorithm conducts a search through the space of the possible assignments to the problem instance variables.

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Backtrack Search SAT Algorithm

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Backtrack Search Algorithm

• Decide( ) : Necessary assignments are identified.

• Deduce( ): Returns a conflict whenever a clause becomes unsatisfiable.

• Diagnose( ): Analyzes the conflict and returns a decision level to which the search process is required to backtrack to.

• Erase( ): Clears implied assignments that results from each assignment selection.

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Handling Conflicts

• Assignments w=1, y3=0, x1=1, y1=0, y2=0

clearly results in a conflict.

• To prevent this conflict at least one of the assignments must be complemented.

• So the clause (¬x1 + ¬w + y3) is generated.

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Equivalence Checking

If z = 1 is unsatisfiable, thetwo circuits are equivalent !

CB

CA

z = 1 ?

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Summary

• So CNF representation of a problem is given as input to the SAT solvers• Core computation engine in many applications like verification and testing, logic synthesis, FPGA routing.• State of the art SAT solvers use DLL algorithm• The first established NP-Complete problem• If the number of variables increases its harder to solve SAT problems

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Questions

References

[1]JP Marques-Silva, “Boolean Satisfiability in Electronic Design Automation,” 2000

[2] Lintao Zhang, Sharad Malik, “The Quest for Efficient Boolean Satisfiability Solvers”

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