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Recursion and Induction: Boolean Satisfiability; Davis-Putnam Procedure

Greg Plaxton Theory in Programming Practice, Fall 2005 Department of Computer Science University of Texas at Austin

Program Design: Boolean Satisfiability

• In this lecture and the next one,

we consider the design and implementation of a Haskell program for solving the “boolean satisfiability” problem

• Input: A propositional formula

– Example: (p ∨ q) ∧ (¬p ∨ ¬q) ∧ (p ∨ ¬q)

• Such a formula is satisfiable if there exists an assignment of (boolean)

values to the variables in the formula such that the formula evaluates to true – Example: The above formula is satisfied by setting p to true and q to false

• Goal: Determine whether the given formula is satisfiable

– The satisfiability solver that we develop produces a satisfying assignment when the given formula is satisfiable

Theory in Programming Practice, Plaxton, Fall 2005

Conjunctive Normal Form (CNF)

• A propositional formula is said to be in CNF if it is the conjunction

(AND) of a number of clauses, where each clause is the disjunction (OR) of a number of literals, and each literal is either a variable or its negation – Example: (p ∨ q) ∧ (¬p ∨ ¬q) ∧ (p ∨ ¬q)

• Theorem: Any boolean formula of length n can be converted to a

logically equivalent CNF formula of length that is polynomial (in fact, linear) in n

• Henceforth, we assume that the input formula is in CNF

Theory in Programming Practice, Plaxton, Fall 2005

Complexity of the Boolean Satisfiability Problem

• The boolean satisfiability problem is NP-complete

– Thousands of such NP-complete problems have been identified in the literature – A polynomial-time algorithm for one NP-complete problem implies polynomial-time algorithms for all of them – It is widely believed that NP-complete problems require superpolynomial (e.g., exponential) time to solve

• In practice, heuristic satisfiability solvers such as Chaff are able to solve

fairly large problem instances – Such solvers are applied in a wide variety of application domains

Theory in Programming Practice, Plaxton, Fall 2005

The Davis-Putnam Procedure

• Let f denote the input formula, and let p denote a variable in f
• Let fp denote f with all occurrences of the literal ¬p removed, and

with all clauses containing the literal p removed – There is a satisfying assignment for f with p set to true if and only if fp is satisfiable

• Let f¬p be defined similarly

– There is a satisfying assignment for f with p set to false if and only if f¬p is satisfiable

• The formula f is satisfiable if and only if fp or f¬p is satisfiable
• This suggests a recursive procedure

– Heuristics may be used to select p and to prioritize the set of remaining recursive subproblems

Theory in Programming Practice, Plaxton, Fall 2005