Büchi arithmetic

Büchi arithmetic of base k is the first-order theory of the natural numbers with addition and the function $V_{k}(x)$ which is defined as the largest power of k dividing x, named in honor of the Swiss mathematician Julius Richard Büchi. The signature of Büchi arithmetic contains only the addition operation, $V_{k}$ and equality, omitting the multiplication operation entirely.

Unlike Peano arithmetic, Büchi arithmetic is a decidable theory. This means it is possible to effectively determine, for any sentence in the language of Büchi arithmetic, whether that sentence is provable from the axioms of Büchi arithmetic.

Büchi arithmetic and automata

A subset $X\subseteq \mathbb {N} ^{n}$ is definable in Büchi arithmetic of base k if and only if it is k-recognisable.

If $n=1$ this means that the set of integers of X in base k is accepted by an automaton. Similarly if $n>1$ there exists an automaton that reads the first digits, then the second digits, and so on, of n integers in base k, and accepts the words if the n integers are in the relation X.

Properties of Büchi arithmetic

If k and l are multiplicatively dependent, then the Büchi arithmetics of base k and l have the same expressivity. Indeed $V_{l}$ can be defined in ${\text{FO}}(V_{k},+)$ , the first-order theory of $V_{k}$ and $+$ .

Otherwise, an arithmetic theory with both $V_{k}$ and $V_{l}$ functions is equivalent to Peano arithmetic, which has both addition and multiplication, since multiplication is definable in ${\text{FO}}(V_{k},V_{l},+)$ .

Further, by the Cobham–Semënov theorem, if a relation is definable in both k and l Büchi arithmetics, then it is definable in Presburger arithmetic.^[1]^[2]

References

^ Cobham, Alan (1969). "On the base-dependence of sets of numbers recognizable by finite automata". Math. Systems Theory. 3 (2): 186–192. doi:10.1007/BF01746527. S2CID 19792434.
^ Semenov, A. L. (1977). "Presburgerness of predicates regular in two number systems". Sibirsk. Mat. Zh. (in Russian). 18: 403–418.

Bès, Alexis. "A survey of Arithmetical Definability". Archived from the original on 2012-11-28. Retrieved 27 June 2012.