A Decision Procedure for the First Order Theory of Real Addition with Order

Abstract

Consider the first order theory of the real numbers with the predicates + (plus) and < (less than). Let S be the set of true sentences. We first present an elimination of quantifiers decision procedure for S, and then analyse it to show that it takes at most time 2^2^cn, c a constant, to decide sentences of length n. Looking more closely at this procedure, we arrive at a second procedure by showing that a given sentence doesn't change in truth value when each of the quantifiers is limited to range over an appropriately chosen finite set of rationals. This fact leads to a decision procedure for S which takes space2^cn. We also remark that our methods lead to a decision procedure for Presburger arithmetic which operates in space 2^2^cn. These upper bounds should be compared with the results of Fischer and Rabin (Proceedings of AMS Symp. on Complexity of Real Computation Processes, to appear) that for some constant c, tim 2^cn for real addition, and time 2^2^cn for Presburger arithmetic, is required to decide some sentences of length n for infitely many n.