TITLE: What are some examples of theories stronger than Presburger Arithmetic but weaker than Peano Arithmetic?
QUESTION [22 upvotes]: What are some examples of theories stronger than Presburger Arithmetic but weaker than Peano Arithmetic?  Are all such theories decidable?  If not, by what methods other than Gödelization can undecidability be established?
EDIT: Both answers have indicated Robinson Arithmetic as an intermediate theory, but I don't understand how it can be "stronger" than Presburger Arithmetic in any reasonable way, because Robinson Arithmetic cannot prove commutativity of addition.  I have attempted to define "stronger" in the comments.  I would appreciate any suggestions for how to clarify the question and this definition.

REPLY [6 votes]: The first-order theory of $(\mathbb{N},+,|_2)$ (here $x |_2 y$ means that $x$ is a power of $2$ and that $x$ divides $y$) is decidable. This can be proved using automata. Note that the set of powers of 2 is obviously definable as $x |_2 x$.
Moreover, there are many concrete decidable expansions. One can see this as follows: 1) this first-order theory is bi-interpretable with the weak monadic-second order theory of $(\mathbb{N},n \mapsto n+1)$. Thus decidable expansions of one induce decidable expansions of the other; 2) there are many concrete decidable expansions of the monadic-second order theory of $(\mathbb{N},n \mapsto n+1)$ by unary predicates. For instance, every morphic predicate (such as the Thue-Morse word).
For a characterisation of the decidable expansions see:
Alexander Rabinovich On decidability of monadic logic of order over the naturals extended by monadic predicates. Inf. Comput. 205(6): 870-889 (2007)