The decision problem for standard classes
The standard classes of a first-order theory T are certain classes of prenex T-sentences defined by restrictions on prefix, number of monadic, dyadic, etc. predicate variables, and number of monadic, dyadic, etc. operation variables. In [3] it is shown that, for any theory T, (1) the decision problem for any class of prenex T-sentences specified by such restrictions reduces to that for the standard classes, and (2) there are finitely many standard classes K1, …, Kn such that any undecidable standard class contains one of K1, …, Kn. These results give direction to the study of the decision problem.Below T is predicate logic with identity and operation variables. The Main Theorem solves the decision problem for the standard classes admitting at least one operation variable.
On the computational complexity and geometry of the first-order theory of the reals. Part I: Introduction. Preliminaries. The geometry of semi-algebraic sets. The decision problem for the existential theory of the reals
