Complexity of Deciding First-Order Theory of Modules Over Noetherian Rings

<p dir="ltr">In this thesis, we consider the decidability of first-order theory of modules. More specifically, we consider this problem under the assumption that the structure is an <i>ω</i>-categorical module over a Noetherian ring. It is known that checking validity of a first-order sentence is reduced to checking solvability of a system of linear equations on the module which is equivalent to checking the solvability of constraint satisfaction problem with the module as its domain. We show this can be done in polynomial time. We end by providing an algorithm that decides the first-order theory of an <i>ω</i>-categorical module over a Noetherian ring that is solvable in deterministic logspace.</p>