Some Topics Related to Propositional Logic Xishun Zhao [email protected] Institute of Logic and Cognition, Sun Yat-sen University Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 1/42 Outline Propositional Logic The Satisfiability Problem Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 Outline Propositional Logic The Satisfiability Problem Why SAT is so important? PversusNP Applications Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 Outline Propositional Logic The Satisfiability Problem Why SAT is so important? PversusNP Applications Challenges for SAT Solving and Its Applications Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 Outline Propositional Logic The Satisfiability Problem Why SAT is so important? PversusNP Applications Challenges for SAT Solving and Its Applications Minimal Unsatisfiability Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 Outline Propositional Logic The Satisfiability Problem Why SAT is so important? PversusNP Applications Challenges for SAT Solving and Its Applications Minimal Unsatisfiability Quantified Boolean Formulas (QBF) Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 Outline Propositional Logic The Satisfiability Problem Why SAT is so important? PversusNP Applications Challenges for SAT Solving and Its Applications Minimal Unsatisfiability Quantified Boolean Formulas (QBF) Expressive Power and Complexity Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 Outline Propositional Logic The Satisfiability Problem Why SAT is so important? PversusNP Applications Challenges for SAT Solving and Its Applications Minimal Unsatisfiability Quantified Boolean Formulas (QBF) Expressive Power and Complexity Future Work Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 1. Propositional Logic Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 3/42 Propositional Formulas Propositional Variables: x1,x2, ···. Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 4/42 Propositional Formulas Propositional Variables: x1,x2, ···. Proposition formulas (PF): constructed from variables by applying ¬, ∧, ∨, →. Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 4/42 Propositional Formulas Propositional Variables: x1,x2, ···. Proposition formulas (PF): constructed from variables by applying ¬, ∧, ∨, →. Literal: either a variable x or its negation ¬x. Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 4/42 Propositional Formulas Propositional Variables: x1,x2, ···. Proposition formulas (PF): constructed from variables by applying ¬, ∧, ∨, →. Literal: either a variable x or its negation ¬x. Clauses: disjunction of finite literals. Example (x1 ∨¬x2 ∨¬x3). Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 4/42 Propositional Formulas Propositional Variables: x1,x2, ···. Proposition formulas (PF): constructed from variables by applying ¬, ∧, ∨, →. Literal: either a variable x or its negation ¬x. Clauses: disjunction of finite literals. Example (x1 ∨¬x2 ∨¬x3). CNF: conjunction of finite clauses. Example: (x1 ∨¬x2 ∨¬x3) ∧ (x1 ∨ x2) ∧ (¬x1 ∨ x3). Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 4/42 Semantics We consider propositional variable x as a Boolean variable which can take value 0 or 1. Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 5/42 Semantics We consider propositional variable x as a Boolean variable which can take value 0 or 1. We interpret connectives as Boolean functions: ¬x 1 − x x ∧ y min(x, y) x ∨ y max(x, y) x → y max(1 − x, y) Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 5/42 Semantics We consider propositional variable x as a Boolean variable which can take value 0 or 1. We interpret connectives as Boolean functions: ¬x 1 − x x ∧ y min(x, y) x ∨ y max(x, y) x → y max(1 − x, y) Then For any formula F ,foranytruthassignment v : var(F ) →{0, 1}, F has a unique truth value under v,we denote the value by v(F ). x y (x →¬y) ∧ (¬x → y) 1 0 1 Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 5/42 Satisfiability We say a formula F is satisfiable if there is an assignment v : var(F ) →{0, 1} such that v(F )=1. Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 6/42 Satisfiability SAT Problem: determining whether a given propositional formula is satisfiable. Instance: APropositionalformulaF , Query: Is F satisfiable? Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 6/42 Satisfiability SAT Problem: determining whether a given propositional formula is satisfiable. Instance: APropositionalformulaF , Query: Is F satisfiable? SAT is the first problem shown to be NP-complete (Cook’s Theorem). Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 6/42 Splitting Rule Let F be a CNF formula, x be a literal of F .DefineF [x =0] (resp F [x =1])istheformulaobtainedfromF by deleting all clauses containing ¬x (resp. x), and removing all occurrences of x (resp. ¬x). Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 7/42 Splitting Rule Let F be a CNF formula, x be a literal of F .DefineF [x =0] (resp F [x =1])istheformulaobtainedfromF by deleting all clauses containing ¬x (resp. x), and removing all occurrences of x (resp. ¬x). Example: Let F be (x1 ∨¬x2 ∨¬x3) ∧ (x1 ∨ x2) ∧ (¬x1 ∨ x3). F [x1 =0]is (¬x2 ∨¬x3) ∧ (x2) F [x1 =1]is (x3) Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 7/42 Splitting Rule Let F be a CNF formula, x be a literal of F .DefineF [x =0] (resp F [x =1])istheformulaobtainedfromF by deleting all clauses containing ¬x (resp. x), and removing all occurrences of x (resp. ¬x). Example: Let F be (x1 ∨¬x2 ∨¬x3) ∧ (x1 ∨ x2) ∧ (¬x1 ∨ x3). F [x1 =0]is (¬x2 ∨¬x3) ∧ (x2) F [x1 =1]is (x3) F is satisfiable ⇐⇒ either F [x =0]or F [x =1]is satisfiable. Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 7/42 Unit Clause Rule Unit clause: consists of only one literal. Example: (x), (¬x) Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 8/42 Unit Clause Rule Unit clause: consists of only one literal. Example: (x), (¬x) Let F := (¬x2 ∨¬x3) ∧ (x2 ∨¬x3) ∧ (x2).Then(x2) is a unit clause. Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 8/42 Unit Clause Rule Unit clause: consists of only one literal. Example: (x), (¬x) Let F := (¬x2 ∨¬x3) ∧ (x2 ∨¬x3) ∧ (x2).Then(x2) is a unit clause. F [x2 =1]=(¬x3) Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 8/42 Unit Clause Rule Unit clause: consists of only one literal. Example: (x), (¬x) Let F := (¬x2 ∨¬x3) ∧ (x2 ∨¬x3) ∧ (x2).Then(x2) is a unit clause. F [x2 =1]=(¬x3) If F contains a unit clause (x),then F is satisfiable ⇐⇒ F [x =1]is satisfiable. Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 8/42 Pure literal rule Pure literal: Aliteralx is said pure in F if x occurs in F but ¬x does not occur in F . Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 9/42 Pure literal rule Pure literal: Aliteralx is said pure in F if x occurs in F but ¬x does not occur in F . Let F := (¬x2 ∨¬x3) ∧ (x2 ∨¬x3) ∧ (x2).Then¬x3 is a pure literal. Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 9/42 Pure literal rule Pure literal: Aliteralx is said pure in F if x occurs in F but ¬x does not occur in F . Let F := (¬x2 ∨¬x3) ∧ (x2 ∨¬x3) ∧ (x2).Then¬x3 is a pure literal. F [¬x3 =1]=(x2) Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 9/42 Pure literal rule Pure literal: Aliteralx is said pure in F if x occurs in F but ¬x does not occur in F . Let F := (¬x2 ∨¬x3) ∧ (x2 ∨¬x3) ∧ (x2).Then¬x3 is a pure literal. F [¬x3 =1]=(x2) If x is pure in F ,then F is satisfiable ⇐⇒ F [x =1]is satisfiable. Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 9/42 DPLL Algorithm F Apply unit-clause-rule Apply pure-literal-rule 0 1 0 1 0 1 continue ⊥ ⊥ continue Whenever we obtain the empty formula ", we know F is satisfiable. Otherwise F is unsatisfiable. Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 10/42 2. Why SAT Is Important? Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 11/42 First-order Logic and SAT Let A, B be two first-order formulas, (A ' B) ⇐⇒ (A ∧¬B) has no model. Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 12/42 First-order Logic and SAT Let A, B be two first-order formulas, (A ' B) ⇐⇒ (A ∧¬B) has no model. 0 Skolem Normal Form (Π1): ∀x1∀x2 ···∀xnA ,hereA is quantifier-free. Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 12/42 First-order Logic and SAT Let A, B be two first-order formulas, (A ' B) ⇐⇒ (A ∧¬B) has no model. 0 Skolem Normal Form (Π1): ∀x1∀x2 ···∀xnA ,hereA is quantifier-free. 0 For any first-order formula A,thereisaΠ1-formula B such that A has a model ⇐⇒ B has a model.