Presburger arithmetic is the first-order theory of the natural numbers with addition. It is not as powerful as the Peano axioms because multiplication is omitted. In fact, Presburger proved in 1929 that there is an algorithm which decides for any given statement in Presburger arithmetic whether it is true or not. No such algorithm exists for general arithmetic as a consequence of the negative answer to the Entscheidungsproblem. Furthermore, Presburger proved that his arithmetic is consistent (does not contain contradictions) and complete (every statement can either be proven or disproven). Again, this is false for general arithmetic; this is the content of Gödel's incompleteness theorem.
Presburger arithmetic is an interesting example in complexity theory and computation because Fischer and Rabin proved in 1974 that every algorithm which decides the truth of Presburger statements has at least a runtime of 2^(2^(cn)) for some constant c. Here, n is the length of the Presburger statement. Hence, the problem is one of the few that provably need more than polynomial run time.
In the formal description of the theory, we use the object constants 0 and 1, the function constant +, and the predicate constant =. The axioms are:
- ∀ x : ¬ (0 = x + 1)
- ∀ x ∀ y : ¬ (x = y) ⇒ ¬ (x + 1 = y + 1)
- ∀ x : x + 0 = x
- ∀ x ∀ y : (x + y) + 1 = x + (y + 1)
- This is an axiom scheme consisting of infinitely many axioms. If P(x) is any formula involving the constants 0, 1, +, = and a single free variable x, then the following formula is an axiom: ( P(0) ∧ ∀ x : P(x) ⇒ P(x + 1) ) ⇒ ∀ x : P(x)
Concepts such as divisibility or prime number cannot be formalized in Presburger arithmetic. Here is a typical theorem that can be proven from the above axioms:
- ∀ x ∀ y : ( (∃ z : x + z = y + 1) ⇒ (∀ z : ¬ (((1 + y) + 1) + z = x) ) )
It says that if x ≤ y + 1, then y + 2 > x.
- M. Presburger. Ueber die Vollstaendigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. C.R. du I Congr. des Math. des pays Slaves, Warszawa, 1929, pp.92-101
- M.J. Fischer, M.O.Rabin. Super-Exponential Complexity of Presburger Arithmetic. "Proceedings of the SIAM-AMS Symposium in Applied Mathematics", 1974, vol. 7, pp.27-41