On the Expansion (n,+, 2) of Presburger Arithmetic

This is based on a preprint ([9]) which appeared in the Proceedings of the fourth Easter Conference on model theory, Gross Köris, 1986, 17-34, Seminarberichte 86, Humboldt University, Berlin, where, with G. Cherlin, we gave a detailed proof of a result of Alexei L. Semenov that the theory of (N,+, 2) is decidable and admits quantifier elimination in an expansion of the language containing the Presburger congruence predicates and a logarithmic function. Expansions of Presburger arithmetic have been (and are still) extensively studied (see, for instance [5]). Let us give a quick review on the expansions of (N,+, P2), where P2 is the set of powers of 2. J. Richard Büchi showed that this expansion is decidable using the fact that the definable subsets are recognizable by a finite 2-automaton (and Kleene’s theorem that the empty problem for finite automata is decidable). (In his article, a stronger result is claimed, namely that Thω(N, S), the weak monadic secondorder theory of N with the successor function S, is bi-interpretable with Th(N,+, P2), which is incorrect, as later pointed out by R. McNaughton ([19])). In his review, McNaughton suggested to replace the predicate P2 by the binary predicate 2(x, y) interpreted by ”x is a power of 2 and appears in the binary expansion of y”. It is easily seen that this predicate is inter-definable with the unary function V2(y) sending y to the highest power of 2 dividing it. Since then, several proofs of the fact that Th(N,+, V2) is bi-interpretable with Thω(N, S) and that Def(N,+, V2) are exactly the 2-recognizable sets (in powers of N) appeared (see [6], [7]), where Def(N,+, V2) are the definable sets in the structure (N,+, V2). A.L. Semenov exhibited a family of 2-recognizable subsets which are not definable in (N,+, P2) (see [24] Corollary 4 page 418). Another way to show that this last theory has less expressive power than Th(N,+, V2) is to use a result of C. Elgot and M. Rabin ([16] Theorem 2) that if g is a function from P2 to P2 with the property that g skips at least one value, namely that ∀n > 1 ∀m (m > n → (∃y ∈ P2 g(m) > y > g(n))), then Th(N,+, V2, n → g(n)) is undecidable and so Th(N,+, V2, 2) is undecidable (another proof was given by G. Cherlin (see [9]). Consequences are that neither the graph of 2 is definable in (N,+, V2), nor the graph of V2 in (N,+, 2) and that Th(N,+, P2) has less expressive power than Th(N,+, 2). Which unary predicate can we add to the structure (N,+, V2) and retain decidability? Let us mention two kinds of results. On one hand, R. Villemaire showed that