Combining decision procedures for the reals

Abstract

<p>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 &#34;local&#34; decision or heuristic procedures for fragments of the theory of the reals can be amalgamated into global ones. </p> <p>Let <em>Tadd[Q]</em> be the first-order theory of the real numbers in the language of ordered groups, with negation, a constant <em>1</em>, and function symbols for multiplication by rational constants. Let <em>Tmult[Q]</em> be the analogous theory for the multiplicative structure, and let <em>T[Q]</em> be the union of the two. We show that although <em>T[Q]</em> is undecidable, the universal fragment of <em>T[Q]</em> is decidable. We also show that terms of <em>T[Q]</em>can fruitfully be put in a normal form. We prove analogous results for theories in which <em>Q</em> is replaced, more generally, by suitable subfields <em>F</em> of the reals. Finally, we consider practical methods of establishing quantifier-free validities that approximate our (impractical) decidability results.</p>