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Clauses and Resolvents CSE2303 Formal Methods I Lecture 21

Overview Formula Disjunctive Normal Form Conjunctive Normal Form Clauses Resolvents

Formula Using propositional variables every statement can be represented by a formula. Example. S: If the price is less than $30 and I have at least $50, then I will buy that CD. P: The price of the CD is less than $30. L: I have at least $50. B: I will buy that CD. S: (P  L)  B

Terminology (P  ¬R  Q)  (¬P  R  Q) propositional variables literals A literal is either a propositional variable, or the negation of a propositional variable.

Terminology (P  ¬R  Q)  (¬P  R  Q) conjunction of literals disjunction of literals

Logical Equivalent We say two formulae are logically equivalent if they always have the same truth table. E.g. (P  Q ) and ¬P  Q ¬(P  R  Q) and ¬P  ¬R  ¬Q ¬(P  R  Q) and ¬P  ¬R  ¬Q P  Q and (¬P  Q)  (P  ¬Q)

Special Forms A formula is in Disjunctive Normal Form (DNF) if it is a disjunction D 1  D 2  …  D n where each D i is a conjunction of literals. A formula is in Conjunctive Normal Form (CNF) if it is a conjunction C 1  C 2  …  C n where each C i is a disjunction of literals.

Example Disjunctive Normal Forms (¬P  ¬R)  (¬P  Q)  (Q  ¬R)  Q (¬P  ¬R  ¬Q)  (¬P  ¬R  Q)  (¬P  R  Q)  (P  ¬R  Q)  (P  R  Q) Conjunctive Normal Forms (¬P  ¬R  Q)  (¬P  R  Q)  (P  ¬R  Q) (¬P  Q)  (¬R  Q) P  Q

Theorem Every formula is logically equivalent to formula in disjunctive normal form. Example: (P  Q )  (R  Q) is logically equivalent to: (¬P  ¬R  ¬Q)  (¬P  ¬R  Q)  (¬P  R  Q)  (P  ¬R  Q)  (P  R  Q)

Proof of Theorem Take any formula A. Write the truth table A. Delete those rows where A gets F. If no rows are left output (P  ¬P) Else, delete the last column. Replace each T by the variable at the top of the column, each F by the negation of the variable. Put  between the literals. Put brackets around the rows, and  between the rows.

Theorem Every formula is logically equivalent to formula in conjunctive normal form. Example: (P  Q )  (R  Q) is logically equivalent to: (¬P  ¬R  Q)  (¬P  R  Q)  (P  ¬R  Q)

Proof of Theorem Take any formula A. Write the truth table A. Delete those rows where A gets T. If no rows are left output (P  ¬P) Else, delete the last column. Replace each T by the negation of the variable at the top of the column, each F by the variable. Put  between the literals. Put brackets around the rows, and  between the rows.

Clause A clause is a set of literals. Every clause corresponds to a disjunction. E.g. {¬P, Q, ¬R} (¬P  Q  ¬R) Every set of clauses correspond to a CNF. E.g. {{¬P, Q, ¬R}, {¬Q, R}} (¬P  Q  ¬R)  (¬Q  R)

Clausal Form Every formula can be represented as a set of clauses. Example: (P  Q )  (R  Q) CNF: (¬P  ¬R  Q)  (¬P  R  Q)  (P  ¬R  Q) Clausal Form: {{¬P, ¬R, Q}, {¬P, R, Q}, {P, ¬R, Q}}

Resolvents {¬P, ¬R, Q}{¬P, R, Q} {¬P, Q}R-resolvent

Definition If C 1 is a clause which contains P, and If C 2 is a clause which contains ¬P, then P-resolvent(C 1, C 2 ) is the set C 1  C 2 -{P, ¬P}

Examples {¬P, ¬R, Q}{P, R, Q} {¬P, ¬R, Q}{P, R, Q} {¬P}{P} {¬P, P, Q} {¬R, R, Q}  Empty clause

Terminology A formula is a tautology when it always gets the value T. A formula is called unsatisfiable if it never gets the value T.

Examples (P  Q)  ¬Q  P P  ¬P The Empty Clause 

Resolvent {A, ¬R}{B, R} {A, B}

(A  ¬R)  (B  R) is logically equivalent to (A  ¬R)  (B  R)  (A  B) {{A, ¬R},{B, R}} is logically equivalent to {{A, ¬R}, {B, R}, {A, B}} {{A, ¬R},{B, R}} is unsatisfiable if and only if {{A, ¬R}, {B, R}, {A, B}} is unsatisfiable.

Example {P, ¬Q}{P, Q}{¬P, Q}{¬P, ¬Q} {P}{Q} {¬P}

Revision Know the definition of the following forms: –DNF (disjunctive normal form). –CNF (conjunctive normal form). –Clausal Form. Be able to express a propositional formula in the following forms: –DNF (disjunctive normal form). –CNF (conjunctive normal form). –Clausal Form. Know what are resolvents.