80 Logic as a Tool Every CNF can therefore be rewritten in a clausal form, and every propositional formula is therefore equivalent to a formula in a clausal form As an example, the clausal form of the CNF formula (p ∨ ¬q ∨ ¬r) ∧ ¬p ∧ (¬q ∨ r ) is {p, ¬q, ¬r}{¬p}{¬q, r} The resolution rule can be rewritten for clauses as follows: {A1 , , C, , Am }{B1 , , ¬C, , Bn } {A1 , , Am , B1 , , Bn } The rule above is called Clausal Resolution The clause {A1 , , Am , B1 , , Bn } is a resolvent of the clauses {A1 , , C, , Am } and {B1 , , ¬C, , Bn } The soundness of the Clausal Resolution rule is again left as an easy exercise Example 55 Here are some examples of applying Clausal Resolution: {p, q, ¬r }{¬q, ¬r} , {p, ¬r } {¬p, q, ¬r}{r } , {¬p, q } {¬p}{p} {} Note that two clauses can have more than one resolvent, for example: {p, ¬q }{¬p, q } , {p, ¬p} {p, ¬q }{¬p, q } {¬q, q } However, it is wrong to apply the resolution rule for both pairs of complementary literals concurrently and obtain {p, ¬q }{¬p, q } {} Why? Because the conclusion here is the empty clause, which is false, does not follow logically from the premises 2.5.3 Resolution-based derivations The method of resolution works similarly to Semantic Tableaux In order to determine whether a logical consequence A1 , , An |= B holds using the method of resolution, we negate the conclusion B and transform each of the formulae A1 , , An , ¬B into clausal form We then test whether the resulting set of clauses is unsatisfiable by looking for a resolution-based proof of the empty clause from that set of clauses Formally: Definition 56 A resolution-based derivation of a formula B from a list of formulae A1 , , An is a derivation of the empty clause {} from the set of clauses obtained from A1 , , An , ¬B by successive applications of the rule of Propositional (Clausal) Resolution We will denote the system of Propositional Resolution by RES and the claim that there is a resolution-based derivation of B from A1 , , An by A1 , , An RES B Example 57 Check whether p → q, q → r RES p → r First, transform p → q, q → r, ¬(p → r) to clausal form: C1 = {¬p, q }, C2 = {¬q, r }, C3 = {p}, C4 = {¬r }