Computability and Logic

246 NORMAL FORMS
19.3 Theorem (Full disjunctive normal form). Every truth-functional compound of
given formulas is logically equivalent to one in full disjunctive normal form.
The theorem on negation-normal forms can be elaborated in another direction. A
formula A is said to be in prenex form if it is of the form
Q 1 x 1 Q 2 x 2 ...Q n x n B
where each Q is either ∃ or ∀, and where B contains no quantifiers. The sequence of
quantifiers and variables at the beginning is called the prefix, and the quantifier-free
formula that follows the matrix.
19.4 Example (Finding a prenex equivalent for a given formula). Consider (∀x Fx↔ Ga),
where F and G are one-place predicates. This is officially an abbreviation for
Let us first put this in negation-normal form
(∼∀xFx ∨ Ga)&(∼Ga ∨∀xFx).
(∃x∼Fx ∨ Ga)&(∼Ga ∨∀xFx).
The problem now is to ‘push junctions in’. This may be done by noting that the displayed
negation-normal form is equivalent successively to
∃x(∼Fx ∨ Ga)&(∼Ga ∨∀xFx)
∃x(∼Fx ∨ Ga)&∀x(∼Ga ∨ Fx)
∃y(∼Fy ∨ Ga)&∀x(∼Ga ∨ Fx)
∀x(∃y(∼Fy ∨ Ga)&(∼Ga ∨ Fx))
∀x∃y((∼Fy ∨ Ga)&(∼Ga ∨ Fx)).
If we had ‘pulled quantifiers out’ in a different order, a different prenex equivalent would
have been obtained.
19.5 Theorem (Prenex normal form). Every formula is logically equivalent to one in
prenex normal form.
Proof: By induction on complexity. Atomic formulas are trivially prenex. The result
of applying a quantifier to a prenex formula is prenex (and hence the result of
applying a quantifier to a formula equivalent to a prenex formula is equivalent to a
prenex formula). The equivalence of the negation of a prenex formula (or a formula
equivalent to one) to a prenex formula follows by repeated application of the equivalence
of ∼∀x and ∼∃x to ∃x∼ and ∀x∼, respectively. The equivalence of a conjunction
(or disjunction) of prenex formulas to a prenex formula follows on first relettering
bound variables as in Problem 10.13, so the conjuncts or disjuncts have no variables
in common, and then repeatedly applying the equivalence of QxA(x) § B, where x
does not occur in B,toQx(A(x) § B), where Q may be ∀ or ∃ and § may be & or ∨.
In the remainder of this chapter our concern is less with finding a logical equivalent
of a special kind for a given sentence or formula than with finding equivalents for
satisfiability of a special kind for a given sentence or set of sentences. Two sets of
sentences Ɣ and Ɣ* are equivalent for satisfiability if and only if they are either