Satisfiability Boolean in Electronic Design Automation
Joo Marques-Silva Informatics Department Technical University of Lisbon IST/INESC, CEL Karem A. Sakallah EECS Department University of Michigan
Context
SAT is the quintessential NP-complete problem Theoretically well-studied Practical algorithms for large problem instances started emerging in the last five years Has many applications in EDA and other fields Can potentially have similar impact on EDA as BDDs EDA professionals should have good working knowledge of SAT formulations and algorithms
Outline
Boolean Satisfiability (SAT) Basic Algorithms Representative EDA Applications Taxonomy of Modern SAT Algorithms Advanced Backtrack Search Techniques Experimental Evidence Conclusions
Boolean Satisfiability
Given a suitable representation for a Boolean function f(X):
Find an assignment X* such that f(X*) = 1 Or prove that such an assignment does not exist (i.e. f(X) = 0 for all possible assignments)
In the "classical" SAT problem, f(X) is represented in product-of-sums (POS) or conjunctive normal form (CNF) Many decision (yes/no) problems can be formulated either directly or indirectly in terms of Boolean Satisfiability
Conjunctive Normal Form (CNF)
= ( a + c ) ( b + c ) (a + b + c )
Clause
Positive Literal
Negative Literal
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 )
= (a + c)(b + c)(a + b + c) (a + c) (a + c)
Consensus
General technique for deriving new clauses Example: 1 = (a + b + c), 2 = (a + b + d) Consensus: con( 1, 2, a) = (b + c + d)
Complete procedure for satisfiability [Davis, JACM'60] Impractical for real-world problem instances Application of restricted forms has been successful!
E.g., always apply restricted consensus con((a + ), (a + ), a) = () is a disjunction of literals
Literal & Clause Classification
violated satisfied unresolved satisfied
= (a + b)(a + b + c )(a + c + d )(a + b + c )
a as...
lec08