Combining decision procedures for the reals

TL;DR

This paper explores combining decision procedures for real-valued expressions, focusing on restricted distributivity and local-to-global methods, to improve validity checking in the theory of the reals.

Contribution

It demonstrates that the universal fragment of the combined theories is decidable and introduces normal forms for terms, extending results to subfields of the reals.

Findings

Universal fragment of T[Q] is decidable

Terms can be normalized into a standard form

Practical methods approximate theoretical decision procedures

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 "local" 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>