Presburger arithmetic

Presburger arithmetic is a weak arithmetic theory. It is consistent, complete, and decidable, but is not strong enough to form statements about multiplication. The decision problem of determining whether a given statement is a theorem of Presburger arithmetic is in a doubly exponential complexity class.

Definition
The language of Presburger arithmetic extends predicate calculus with constants 0 and 1, unary operator S, binary operator +, and relation =. Its axioms are:


 * \(\forall n: S(n) \neq 0\)
 * \(\forall n \forall m: S(n) = S(m) \Rightarrow n = m\)
 * \(\forall n: n + 0 = n\)
 * \(\forall n \forall m: n + S(m) = S(n + m)\)
 * For every first-order formula \(\varphi(n)\): \((\varphi(0) \wedge (\forall n: \varphi(n) \Rightarrow \varphi(S(n)))) \Rightarrow \forall m: \varphi(m)\).