Decidable Theories

This chapter presents the method of elimination of quantifiers, model theoretic methods, the method of interpretations (semantic interpretations, decidable second-order theories, tree theorem, Presburger's arithmetic revisited, etc.), and complexity of decision procedures that includes Turing-machine computations, the theory WSlS, theories of addition and real-closed fields, and propositional calculus. The study of decidability involves trying to establish, for a given mathematical theory T, or a given problem P, the existence of a decision algorithm AL that will accomplish the following task. Given a sentence A expressed in the language of T, the algorithm AL will determine whether A is true in T—that is, whether A∈T. In the case of a problem P, given an instance Z of the problem P, the algorithm AL will produce the correct answer for this instance. Depending on the problem P, the answer may be yes, no, or an integer. If such an algorithm exists, it can be said that the decision problem of T or P is solvable, that the theory T is decidable, or that the problem P is solvable.