Satisfiability

Satisfiability:Applications and Algorithms

Jim [email protected]

Outline

• Boolean functions and expressions

• Applications and related formalisms

• Satisfiability Algorithms– Expression-Based– Assignment-Based

Boolean Functions

• Bit vector• Corners of n-

dimensional cube• Base-2 coded integer

• True/false• In/out of set• Pass/fail

1,01,0 n

Truth Table

• Read-only memory is like this: – Addr_in -> data_out– 16 address bits enough to

store a book– No way, in general, to

squeeze n bit values into any smaller space

0111

1011

0101

0001

0110

1010

1100

1000

a b c

Composing Functions

a

b

b

c

cbba

Logic Optimization

• Very difficult to find optimal expression for a given Boolean function– Gate count– Longest path from input to output

• Any function has infinite expressions

• Most functions require exponential space– Counting argument: Boolean functions

with n variables

n22

2-Level Expressions

• DNF: Disjunctive Normal Form

• CNF: Conjunctive Normal Form

01

00

cbbacbba

cb

bacbba

Decision Tree

a

b b

c c c c

1 1 1 0 0 0 1 0

0

0

0 0

0

0 0

1

1 1

1 1 1 1

Pruning Unnecessary Nodes

a

b b

c c1

1 0

0

1 0

0

0

0

0

0

1

1 1

1 1

Reusing Duplicate Nodes

a

b b

c1

1 0

0

0

0

0

0

1

1 1

1

Tree becomes Diagram, or BDD

Binary Decision Diagrams

• Breakthrough paper

Randy Bryant,Graph-Based Algorithms for Boolean Function Manipulation. IEEE Transaction on Computers, 1986

http://vlsi.colorado.edu/~fabio/CUDD/

• High quality public package

BDD Manipulation

• BDDs are canonical, given variable order– Trivial to check function equality, satisfiability

• Often compact– Optimizing variable order is difficult

• Interleave bits of an adder!

• BDD for (f&g) can be computed efficiently, from BDD for f and BDD for g

Formalized Decisions

• Boolean Functions– Set of fixed length strings

• State Machines– Regular languages

• Turing Machines– Universal computability

Computational Problems• Optimizing expressions

– Language in which machine expressed– Optimality criteria

• Language Emptiness– Boolean Functions

• NP-Complete

– Finite State Machines• P-Space Complete

– Turing Machines• Undecidable

Applications for Satisfiability

• Given a Boolean function, find an input bit vector for which the function evaluates to 1– Artificial Intelligence

• E.g. medical diagnosis

– Operations Research• Optimizing over discrete domains

– Electronics Engineering• Detecting faults

– Computer Science• Check resource contention

Bounded Emptiness

• Convert infinite set to finite set– Bound length of string– Discretize space

• Use SAT to attack harder problems– E.g. Bounded Model Checking

Satisfiability Community

• International Conference onTheory and Applications of Satisfiability Testing– http://ie.technion.ac.il/SAT10/ (Edinburgh)

• Journal on Satisfiability, Boolean Modeling and Computation – http://www.isa.ewi.tudelft.nl/Jsat/

• Up-to-date links concerning research – http://www.satlive.org/

Dimacs Formatp cnf 59056 3237001 2 01 3 01 4 01 -5 01 6 01 -7 01 -8 01 -9 01 -10 0-2 -3 -4 5 -6 7 8 9 10 -1 0-11 -12 -13 14 0-14 11 0-14 12 0-14 13 0

Random CNF

P(Sat)

1

0

# Clauses / # Variables

10 2 3 4 5 6 7

Limit for large #V

Small #V

Algorithms

• CNF Resolution

• BDD variable elimination

• Local Search

• Circuit-based value assignment

• Conflict-based Learning

Resolution

• Exhaustive application will yield empty clause if problem is unsatisfiable– 3n potential clauses!

aa

cbcbacba

ecbdcba

edbaecbdcba

BDD Variable Elimination

• Construction of BDD from circuit often infeasible– Intermediate blow-up even if ultimate answer is trivial

• Interleave existential quantification– Eliminating variables often reduces BDD sizes– Finding an order of variables to eliminate is difficult

• Keep the support sizes small of intermediate results• Equivalent to elimination with sparse matrices

cbgcbafabcbgbafcba ,.,..,,.

10 |,|,,. aa bafbafbafa

Local Search

• Generally implemented with CNF• Start by assigning arbitrary value to each

variable• Flip values of variables one by one

– Any variable that appears in a false clause is a candidate

• Gradually reduce number of false clauses• Need mechanism to escape local minima• No proof of unsatisfiability

Stuck-at Fault Model

1

Circuit as designed

Circuit as manufactured

Test generation: find a input bit vector thatwill produce different outputs in correct versus faulty circuits

Circuit-Based SAT

1

Can we findinput assignments todrive the circuit outputto 1?

Circuit-Based SAT

1

1

1

Inputs to AND mustbe 1 for output to be 1

Circuit-Based SAT

1

1

1

0

At least one input of NANDmust be 0 for output to be 1,so we make a choice.

Circuit-Based SAT

1

1

1

1

0

1

Inputs to NAND must be 1for output to be 0

Circuit-Based SAT

1

1

1

1

0

1

1

1

Inputs to AND must be 1 for output to be 1

Circuit-Based SAT

1

1

1

1

0

1

1

1

Conflict! need to backtrack andmake different choices

Loosely Coupled Subproblems

• Subproblems revisited many times• Need to save and reuse learning• CNF is simple & incremental

…

ABCDE

UVW

Landmark Papers in CNF SAT

• Joao Marques-Silva and Karem Sakallah, “GRASP: A Search Algorithm for Propositional Satisfiability,” ICCAD 1996

• Matthew W. Moskewicz,  Conor F. Madigan,  Ying Zhao,  Lintao Zhang,  Sharad Malik, “Chaff: Engineering an Efficient SAT Solver,” DAC 2001

CNF Decision & Implication

1aDecide: iga

hgb

fe

eba

dca

ba

CNF Decision & Implication

1

1

b

aDecide:

Imply:

iga

hgb

fe

eba

dca

ba

CNF Decision & Implication

0

1

1

e

b

aDecide:

Imply:

Imply:

iga

hgb

fe

eba

dca

ba

CNF Decision & Implication

1

0

1

1

f

e

b

aDecide:

Imply:

Imply:

Imply:

iga

hgb

fe

eba

dca

ba

CNF Decision & Implication

0

1

0

1

1

g

f

e

b

aDecide:

Imply:

Imply:

Imply:

Decide:

iga

hgb

fe

eba

dca

ba

CNF Decision & Implication

1

0

1

0

1

1

h

g

f

e

b

aDecide:

Imply:

Imply:

Imply:

Decide:

Imply:

iga

hgb

fe

eba

dca

ba

Implication Graph

iga

hgb

fe

eba

dca

ba

Chains of implication linked by pairs of clauses with common variable but opposite signs:opportunities for resolution

Learned Clauses

Cut in implication graphcorresponds to learned clausederivable by resolution

Look for small cuts – short clauses are tighter constraints

Leveraging Structure

• SAT is very hard in the worst case

• But practical instances often manageable– Test Generation, Equivalence Checking

• similarity of two halves of problem

– Bounded Model Checking• Repetitive structure

– Arithmetic • Bit order