Propositional logic resolution

1. Propositional logic resolution Conjunctive normal form: any formula of the predicate calculus can be transformed into a conjunctive normal form. Def. A formula is said to be in conjunctive normal form if it consists in the conjunction of clauses. A1 A2 …  An where Ai is a clause. Def. A formula is said to be a clause if it consists in a disjunction of literals. A clause has the following form: L1 v L2 v … v Lm where Li is a literal. Def. A literal is an atomic formula or the negation of an atomic formula. Def. A formula is said to be in clausal form if it can be expressed as a set of clauses: {C1 , … , Cn,} where Ci is a clause

2. Propositional logic resolution Transforming into clausal form 1. Eliminate implication symbols (), using the identity:    v  2. Introduce negation: reduce scopes of negation symbols by repeatedly applying the De Morgan rules: (i)  ( v )    (ii)  ()   v   3. Put matrix in conjunctive normal form by repeatedly applying the distributive laws: (i)  v ()  ( v )  ( v ) (ii)  ( v )  () v () 4. Eliminate conjunction () symbols separating the expression in clauses.

3. Propositional logic resolution Resolution refutation procedure In general a resolution refutation for proving an arbitrary wff  from a set of wffs,  |—, proceeds as follows: 1. Convert the wffs in  to clausal form. 2. Negate the formula  to be proved and convert the result to clausal form. 3. Combine the clauses resulting form steps 1 and 2 into a single set, . 4. Iteratively apply resolution to the clauses in  and add the results to  either until there are no more resolvents that can be added or until the empty clause is produced.

4. Propositional logic resolution Important results • Completeness of resolution refutation: the empty clause will be produced by the resolution refutation procedure if  |= thus we say that propositional resolution is refutation complete. • Decidibility of propositional calculus by resolution refutation: if  is a finite set of clauses and if  | then the resolution refutation procedure will terminate without producing the empty clause.

5. Propositional logic resolution Exercises 1. Transform into clausal form the following wff: ~[((p v ~q) r)  (p q)] 2. Prove using resolution refutation the axioms of the propositional logic. a. Implication introduction: p  (q  p) b. Implication distribution: (p  (q  r))  ((p  q)  (p  r)) c. Contradiction realization: (q  ~p)  ((q  p)  ~q)