On Variable-Bounded Non-Linear Expansions of Presburger Arithmetic
It provides decidability results for restricted fragments of a previously undecidable theory, offering new insights for logicians and computer scientists working on arithmetic theories.
The paper studies expansions of Presburger arithmetic with monadic polynomial predicates, proving decidability for single-variable theories of perfect fixed powers and polynomials of degree at most three, while showing limitations via open Diophantine problems.
We consider expansions of Presburger arithmetic with families of monadic polynomial predicates. (Examples of such predicates are the set of perfect squares, or the set of integers of the form $2n^3-5n+3$, etc.) Although the full attendant first-order theories are well known to be undecidable, very little is known when one restricts the number of variables. In the case of single-variable theories, we obtain positive results for the following two families of predicates: (i) for perfect fixed powers, decidability ofthe corresponding theory follows from the solvability of hyperellipticDiophantine equations; and (ii) for polynomials of degree at most three, we establish decidability by relying on the low genus of the resulting algebraic curves. Finally, we discuss limitations and hardness results (via encodings of longstanding open Diophantine problems) as soon as any of the above restrictions are lifted.