Computability and Logic

296 DECIDABILITY OF ARITHMETIC WITHOUT MULTIPLICATION
Given a sentence S of L without ·, replace ′ everywhere in it by +1, and replace
every quantification ∀x or ∃x by a relativized quantification
∀x((x = 0 ∨ 0 < x) →···) ∃x((x = 0 ∨ 0 < x)& ...).
to obtain a sentence S* ofK . Then S will be true in N if and only if S* is true in
M. Thus to prove Presburger’s theorem, it will be sufficient to describe an effective
procedure for determining whether or not a given sentence of K is true in M.
For the remainder of this chapter, therefore, term or formula or sentence will always
mean term or formula or sentence of K, while denotation or satisfaction or truth
will always mean denotation or satisfaction or truth in M. We call two terms r and
s coextensive if ∀v 1 ...∀v n r = s is true, where the v i are all the variables occurring
in r or s. We call two formulas F and G coextensive if ∀v 1 ...∀v n (F ↔ G) is true,
where the v i are all the free variables occurring in F or G.
Given any closed term, we can effectively calculate its denotation. Given any
atomic sentence, we can effectively determine its truth value; and we can therefore
do the same for any quantifier-free sentence. We are going to show how one can
effectively decide whether a given sentence S is true by showing how one can effectively
associate to S a coextensive quantifier-free sentence T : once T is found, its
truth value, which is also the truth value of S, can be effectively determined.
The method to be used for finding T ,givenS, is called elimination of quantifiers.
It consists in showing how one can effectively associate to a quantifier-free
formula F(x), which may contain other free variables besides x, and quantifier-free
G such that G is coextensive with ∃xF(x) and G contains no additional free variables
beyond the free variables in ∃xF(x). This shown, given S, we put it in prenex form,
then replace each quantification ∀x by ∼∃x ∼, and work from the inside outward,
successively replacing existential quantifications of quantifier-free formulas by coextensive
quantifier-free formulas with no additional free variables, until at last a
sentence with no free variables, which is to say, a quantifier-free sentence T ,is
obtained.
So let F(x) be a quantifier-free formula. We obtain G, coextensive with ∃xF(x)
and containing no addition free variables beyond those in ∃xF(x), by performing, in
order, a sequence of 30 operations, each of which replaces a formula by a coextensive
formula with no additional free variables.
In describing the operations to be gone through, we make use of certain notational
conventions. When writing of a positive integer k and a term t we allow ourselves to
write
−t instead of 0 --- t
k instead of 1+1+···+ 1 (k times)
kt instead of t + t + ···+ t (k times)
for instance. With such notation, the 30 operations are as follows:
(1) Put F into disjunctive normal form. (See section 19.1.) Thus we get a disjunction
of conjunctions of atomic formulas of the forms r = s or r < s or D m s (where r
and s are terms) and negations of such.