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Satisfiability problem of Boolean expressions in

Conjunctive Normal Form

- Alodeep Sanyal
- Electrical and Computer Engineering
- University of Massachusetts at Amherst

Source K. A. Sakallah (University of Michigan),

S. Malik (Princeton University) and M.

Ciesielski (University of Massachusetts)

Boolean Satisfiability (SAT)

- Given a suitable representation for a Boolean

function - Find a truth assignment T appropriate to

such that , or - prove that such an assignment does not exist
- (i.e. for all possible

assignments) - Definitions A Boolean expression is
- satisfiable if there is a truth assignment T

such that - valid or tautology if for all T
- unsatisfiable if for any T

Figure adopted from C. Papadimitriou,

Computational Complexity, Addison Wesley

Motivation

- SAT is a core computational engine for major

applications - VLSI CAD
- Testing and verification
- Logic synthesis
- FPGA routing
- Path delay analysis
- AI
- Knowledge based deduction
- Automatic theorem proving

Adopted from S. Malik, Princeton University

Development in SAT research a brief history

- Main contributions in SAT research
- 1960 Davis and Putnam resolution based -

dealing with 10 variables - 1962 Davis, Logemann and Loveland DFS-based

dealing with 10 variables - Basic framework for many modern SAT solvers
- 1986 R. E. Bryant BDD-based dealing with

100 variables - 1996 Silva and Sakallah GRASP (conflict-driven

learning and non-chronological backtracking)

dealing with 1k variables - 2001 Malik et al. Efficient BCP and decision

making dealing with 10k vars

Adopted from S. Malik, Princeton University

Satisfiability of Boolean expressions in

Conjunctive Normal Form (CNF)

- In a classical SAT problem, function is

represented in conjunctive normal form (CNF) - If the function involves n variables, there

are 2n possible truth assignments to be checked - As a matter of fact, SAT was the first

established NP-Complete problem - Many other decision (yes/no) problems can be

formulated in terms of Boolean Satisfiability

S. A. Cook. The complexity of theorem proving

procedures, Third Annual ACM Symp. On Theory of

Computing, pp. 151-158, 1971

Conjunctive Normal Form (CNF)

j ( a c ) ( b c ) (a b c )

CNF Formulas

- Product of Sum (POS) representation of Boolean

function - Describes solution using a set of constraints
- very handy in many applications because new

constraints can just be added to the list of

existing constraints - very common in AI community
- Example
- j ( ab c)(a b c)( abc)( a b c)

CNF Basics

- Implication
- x y x y
- (y) (x)
- y x (contra positive)
- Assignments a 0, b 1 a b
- Partial (some variables still unassigned)
- Complete (all variables assigned)
- Conflicting (imply j)
- Implications binate clauses
- SAT find any satisfying assignment
- Covering find an assignment optimizing some cost

function

CNF for Boolean gate

jd d (a b )d a b

d a bd a b

(a d)(b d)(a b d)

(a d)(b d)(a b d)

Example

Single gate

a

(ab c)(ac)(bc)

c

b

Circuit network of connected gates

(124)(14)(24) (235)(25)(35) (23

6)(26)(36) (457)(47)(57) (568)(5

8)(68) (789)(79)(89) (9)

Justify to 0

- Note
- Node 1,2,3 PI
- Node 4,5,6,7,8,9 AND gate

Circuit Satisfiability

j h d(ab) e(bc) fd gde hfg

h

(a d)(b d)(a b d)

(b e)(c e)(b c e)

(d f)(d f)

(d g)(e g)(d e g)

(f h)(g h)(f g h)

Solving SAT - Basic Case Splitting Algorithm

(Davis-Putnam-Logemann-Loveland 1962)

a

b

b

c

c

c

d

d

d

d

d

Source Karem A. Sakallah, Univ. of Michigan

Solving SAT Case Splitting with Implications

a

b

b

c

c

Source Karem A. Sakallah, Univ. of Michigan

Implementation

- Clauses are stored in array
- Track sensitivity of clauses for changes
- all literals but one assigned (unit clause) -gt

implication - all literals but two assigned -gt clause is

sensitive to a change of either literal - all other clauses are insensitive and do not need

to be observed - Learning
- learned implications are added to the CNF formula

as additional clauses - limit the size of the clause
- limit the lifetime of a clause, will be removed

after some time - Non-chronological back-tracking
- similar to circuit case

Solving SAT Conflict-based Learning(GRASP

Silva and Sakallah 1996)

a

ab j ß j (a b)

a j ß j (a)

bc j ß j (b c)

b

b

c

Source Karem A. Sakallah, Univ. of Michigan

Further Improvements

- Random restarts
- stop after a given number of backtracks
- start search again with modified ordering

heuristic - keep learned structures !!!
- very effective for satisfiable formulas but often

also effective for unsatatisfiable formulas - Learning of equivalence relations
- (a Þ b) Ù (b Þ a) Þ (a b)
- very powerful for formal equivalence checking

SAT used in our research

- Kunal Ganeshpure, Alodeep Sanyal, and Sandip

Kundu. A pattern generation technique for

maximizing power supply currents. ICCD 2006 - apply SAT to set a maximal set of nodes (say S)

to logic value 0 - apply SAT again to set S to find a maximal subset

S ( ) to set it to logic value 1 - the switching occurring in set S gives

an estimation on worst case power supply current - Alodeep Sanyal, Kunal Ganeshpure, and Sandip

Kundu. On accelerating soft error detection by

targeted pattern generation. ISQED 2007 - identify a set of soft-error susceptible nodes

using some filtering scheme - apply SAT to obtain a truth assignment in the PIs

such that the set of susceptible nodes are

excited in their vulnerable state

Acknowledgement

- This presentation liberally borrowed material

from the following tutorials - M. Ciesielski. Lecture on Boolean SAT CNF

representation. Course Synthesis and

Verification of Digital Systems - A. Kuehlmann. SAT Tutorial. UC Berkeley 2003
- J. M. Silva and K. A. Sakallah. Boolean

satisfiability in electronic design automation.

DAC 2000 - S. Malik. The quest for efficient Boolean

satisfiability solvers. CMU seminar on SAT, 2004

???

Thank you!!!

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