Avigad, Jeremy and Friedman, Harvey - Combining decision procedures for the reals

lmcs:2240 - Logical Methods in Computer Science, October 18, 2006, Volume 2, Issue 4
Combining decision procedures for the reals

Authors: Avigad, Jeremy and Friedman, Harvey

<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>


Source : oai:arXiv.org:cs/0601134
DOI : 10.2168/LMCS-2(4:4)2006
Volume: Volume 2, Issue 4
Published on: October 18, 2006
Submitted on: January 31, 2006
Keywords: Computer Science - Logic in Computer Science,F.4.1,I.2.3


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