Download
1 / 17
170 likes | 301 Vues
This document explores the fundamentals of propositional logic, predicate calculus, and various algorithms for solving satisfiability problems. It covers key concepts such as propositions, literals, clauses, and clausal formulas. Significant historical figures such as Aristotle, George Boole, and Kurt Gödel are highlighted for their contributions to logical reasoning. Additionally, the DPLL algorithm and its enhancements are discussed, including techniques for optimizing satisfiability checking, such as heuristic searches and conflict clause learning.
E N D
Effective Propositional Reasoning CSE 473 – Autumn 2003
Propositional Logic++ • Proposition := Predicate( Term, Term, …) • Term := CONSTANT || variable • Literal := Proposition || Proposition • Clause := (Literal Literal …) • Clausal Formula := Clause Clause …
Predicate Calculus (FOL)? • FOL includes function symbols that can construct an infinite number of new terms • successor(sucessor(successor(0)))
Why Logic? Enables deductive reasoning! Amazing property: X entails Y just in case X Y is unsatisfiable Is a sentence unsatisfiable? = theorem proving Can a sentence be satisfied? = constraint satisfaction • ARISTOTLE – 350 BC • GEORGE BOOLE – 1850 • LEWIS CARROL – 1880 • GRIEDRICH FREGE – 1900 • KURT GODEL – 1930 • HILARY PUTNAM – 1959 • ALAN ROBINSON – 1965
Algorithms for Satisfiability 1. Model enumeration (dumb!) for (m in truth assignments){ if (m makes F true) return “Sat!” } return “Unsat!” 2. Backtrack search through space of PARTIAL truth assignments: DPLL (Davis-Putnam) 3. Local search through space of TOTAL truth assignments: Walksat 4. Manipulate sentences: Resolution
DPLL version 1Davis – Putnam – Loveland – Logemann dpll_1(pa){ if (pa makes F false) return false; if (pa makes F true) return true; choose P in F; if (dpll_1(pa U {P=0})) return true; return dpll_1(pa U {P=1}); } Returns true if F is satisfiable, false otherwise
DPLL Version 1 a (a bc) b b (a ¬b) (a ¬c) c c (¬a c)
DPLL version 2Davis – Putnam – Loveland – Logemann dpll_2(F, literal){ remove clauses containing literal shorten clauses containing literal if (F contains no clauses)return true; if (F contains empty clause) return false; choose P in F; if (dpll(F, P))return true; return dpll_2(F, P); } Partial assignment corresponding to a node is the set of chosen literals on the path from the root to the node
DPLL Version 2 a (a bc) b b (a ¬b) (a ¬c) c c (¬a c)
DPLL (for real!)Davis – Putnam – Loveland – Logemann dpll(F, literal){ remove clauses containing literal shorten clauses containing literal if (F contains no clauses)return true; if (F contains empty clause) return false; if (F contains a unit or pure L) return dpll(F, L); choose P in F; if (dpll(F, P))return true; return dpll_2(F, P); }
DPLL (for real) a (a bc) c b (a ¬b) (a ¬c) c (¬a c)
DPLL (for real!)Davis – Putnam – Loveland – Logemann dpll(F, literal){ remove clauses containing literal shorten clauses containing literal if (F contains no clauses)return true; if (F contains a unit or pure L) return dpll(F, L); choose P in F; if (dpll(F, P))return true; return dpll_2(F, P); } Where could we use an heuristic to further improve performance?
Heuristic Search in DPLL • Heuristics are used in DPLL to select a (non-unit, non-pure) proposition for branching • Idea: identify a most constrained variable • Likely to create many unit clauses • MOM’s heuristic: • Most occurrences in clauses of minimum length
Success of DPLL • 1962 – DPLL invented • 1992 – 300 propositions • 1997 – 600 propositions (satz) • Additional techniques: • Learning conflict clauses at backtrack points • Randomized restarts • 2002 (zChaff) 1,000,000 propositions – encodings of hardware verification problems
