Presburger arithmetic

Let n be the length of a statement in Presburger arithmetic.

Presburger arithmetic is an axiom system for the natural numbers under addition.

However, some problems provably require more time, for example Presburger arithmetic.

The problem of deciding the truth of a statement in Presburger arithmetic requires even more time.

The decision problem for Presburger arithmetic is an interesting example in computational complexity theory and computation.

Presburger arithmetic cannot formalize concepts such as divisibility or prime number.

The first order theory of the natural numbers with addition (but not multiplication), called Presburger arithmetic, is also decidable.

The first-order theory of the integers in the signature with equality and addition, also called Presburger arithmetic.

Dependent ML limits the sort of equality it can decide to Presburger arithmetic.

The language of Presburger arithmetic contains constants 0 and 1 and a binary function +, interpreted as addition.

Presburger arithmetic is much weaker than Peano arithmetic, which includes both addition and multiplication operations.

In this language, the axioms of Presburger arithmetic are the universal closures of the following:

Presburger arithmetic can be extended to include multiplication by constants, since multiplication is repeated addition.

The signature of Presburger arithmetic contains only the addition operation and equality, omitting the multiplication operation entirely.

Unlike Peano arithmetic, Presburger arithmetic is a decidable theory.

Generally, any number concept leading to multiplication cannot be defined in Presburger arithmetic, since that leads to incompleteness and undecidability.

Hence, the decision problem for Presburger arithmetic is an example of a decision problem that has been proved to require more than exponential run time.

This technique is used to show that Presburger arithmetic, i.e. the theory of the additive natural numbers, is decidable.

He was a student of Alfred Tarski and is known for, among other things, having invented Presburger arithmetic as a student in 1929.

There are even weaker axiomatic systems that are consistent and complete, for instance Presburger arithmetic which proves every true first-order statement involving only addition.

(For example, the Coq proof assistant system features a tactic for Presburger arithmetic.)

Mojżesz Presburger proved Presburger arithmetic to be:

Let P(x) be a first-order formula in the language of Presburger arithmetic with a free variable x (and possibly other free variables).

Some first-order theories are algorithmically decidable; examples of this include Presburger arithmetic, real closed fields and static type systems of many programming languages.

Abstraction-based Satisfiability Solving of Presburger Arithmetic.