Abstract

We address the general problem of determining the validity of boolean combinations of equalities and inequalities between real-valued expressions. In particular, we consider methods of establishing such assertions using only restricted forms of distributivity. At the same time, we explore ways in which “local ” decision or heuristic procedures for fragments of the theory of the reals can be amalgamated into global ones. Let Tadd[Q] be the first-order theory of the real numbers in the language with symbols 0, 1, +, −, <,..., fa,... where for each a ∈ Q, fa denotes the function fa(x) = ax. Let Tmult[Q] be the analogous theory for the language with symbols 0, 1, ×, ÷, <,..., fa,.... We show that although T [Q] = Tadd[Q]∪Tmult[Q] is undecidable, the universal fragment of T [Q] is decidable. We also show that terms of T [Q] can fruitfully be put in a normal form. We prove analogous results for theories in which Q is replaced, more generally, by suitable subfields F of the reals. Finally, we consider practical methods of establishing quantifier-free validities that approximate our (impractical) decidability results. 1

Citations