Büchi arithmetic of base "k" is the first-order theory of the natural numbers with addition and the function formula_1 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, formula_2 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 formula_3 is definable in Büchi arithmetic of base "k" if and only if it is "k"-recognisable.
If formula_4 this means that the set of integers of "X" in base "k" is accepted by an automaton. Similarly if formula_5 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 formula_6 can be defined in formula_7, the first-order theory of formula_2 and formula_9.
Otherwise, an arithmetic theory with "both" formula_2 and formula_6 functions is equivalent to Peano arithmetic, which has both addition and multiplication, since multiplication is definable in formula_12.
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.
References.
 ;templatestyles src="Reflist/styles.css" / ;