Editing Presburger arithmetic (section)

{{Short description|Decidable first-order theory of the natural numbers with addition}}
'''Presburger arithmetic''' is the [[first-order predicate calculus|first-order theory]] of the [[natural number]]s with [[addition]], named in honor of [[Mojżesz Presburger]], who introduced it in 1929. The [[signature (mathematical logic)|signature]] of Presburger arithmetic contains only the addition operation and [[equality (mathematics)|equality]], omitting the [[multiplication]] operation entirely. The theory is [[Computable set|computably]] axiomatizable; the axioms include a schema of [[mathematical induction|induction]].

Presburger arithmetic is much weaker than [[Peano arithmetic]], which includes both addition and multiplication operations. Unlike Peano arithmetic, Presburger arithmetic is a [[Decidability (logic)|decidable theory]]. This means it is possible to algorithmically determine, for any sentence in the language of Presburger arithmetic, whether that sentence is provable from the axioms of Presburger arithmetic. The asymptotic running-time [[Analysis of algorithms|computational complexity]] of this algorithm is at least [[Double exponential function|doubly exponential]], however, as shown by {{harvtxt|Fischer|Rabin|1974}}.