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Concept
Presburger arithmetic
Summary
Presburger arithmetic is the first-order theory of the natural numbers with addition, named in honor of Mojżesz Presburger, who introduced it in 1929. The signature of Presburger arithmetic contains only the addition operation and equality, omitting the multiplication operation entirely. The axioms include a schema of induction.
Presburger arithmetic is much weaker than Peano arithmetic, which includes both addition and multiplication operations. Unlike Peano arithmetic, Presburger arithmetic is a 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 computational complexity of this algorithm is at least doubly exponential, however, as shown by .
The language of Presburger arithmetic contains constants 0 and 1 and a binary function +, interpreted as addition.
In this language, the axioms of Presburger arithmetic are the universal closures of the following:
¬(0 = x + 1)
x + 1 = y + 1 → x = y
x + 0 = x
x + (y + 1) = (x + y) + 1
Let P(x) be a first-order formula in the language of Presburger arithmetic with a free variable x (and possibly other free variables). Then the following formula is an axiom:(P(0) ∧ ∀x(P(x) → P(x + 1))) → ∀y P(y).
(5) is an axiom schema of induction, representing infinitely many axioms. These cannot be replaced by any finite number of axioms, that is, Presburger arithmetic is not finitely axiomatizable in first-order logic.
Presburger arithmetic can be viewed as a first-order theory with equality containing precisely all consequences of the above axioms. Alternatively, it can be defined as the set of those sentences that are true in the intended interpretation: the structure of non-negative integers with constants 0, 1, and the addition of non-negative integers.
Presburger arithmetic is designed to be complete and decidable. Therefore, it cannot formalize concepts such as divisibility or primality, or, more generally, any number concept leading to multiplication of variables.
Official source
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