79 Deductive Reasoning in Propositional Logic For a CNF we distribute ∨ over ∧ and simplify: ≡ (¬p ∨ r ∨ ¬p ∨ ¬q ) ∧ (¬p ∨ r ∨ q ∨ p) ≡ (¬p ∨ r ∨ ¬q ) ∧ ( ∨ r ∨ q ) ≡ (¬p ∨ r ∨ ¬q ) ∧ ≡ ¬p ∨ r ∨ ¬q As can be seen, in this case the CNF also turns out to be a DNF, even simpler than the one we obtained above The problem of minimization of normal forms, which is of practical importance, will not be discussed here The second method constructs the normal forms directly from the truth table of the given formula I outline it for a DNF Given the truth table of the formula A we consider all rows (i.e., all assignments of truth values to the occurring variables) where the truth value of A is true If there are no such rows, the formula is a contradiction and a DNF for it is, for example, p ∧ ¬p Otherwise, with every such row we associate an elementary conjunction in which all variables assigned value T occur positively, while those assigned value F occur negated For instance, the assignment F, T, F to the variables p, q, r is associated with the elementary conjunction ¬p ∧ q ∧ ¬r Note that such an elementary conjunction is true only for the assignment with which it is associated As an exercise, show that the disjunction of all elementary conjunctions associated with the rows in the truth table of a formula A is logically equivalent to A Example 53 The formula p ↔ ¬q has a truth table p T T F F q T F T F p ↔ ¬q F T T F The corresponding DNF is (p ∧ ¬q ) ∨ (¬p ∧ q ) Check this! As an exercise, outline a similar method for construction of a CNF from the truth table of a formula and prove your claim 2.5.2 Clausal Resolution Definition 54 A clause is essentially an elementary disjunction l1 ∨ ∨ ln but written as a set of literals {l1 , , ln } The empty clause {} is a clause containing no literals; a unit clause is a clause containing only one literal A clausal form is a (possibly empty) set of clauses, written as a list: C1 Ck It represents the conjunction of these clauses