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Unformatted text preview: Logical Methods in Computer Science Vol. 2 (4:4) 2006, pp. 142 www.lmcs-online.org Submitted Jan. 31, 2006 Published Oct. 18, 2006 COMBINING DECISION PROCEDURES FOR THE REALS JEREMY AVIGAD a AND HARVEY FRIEDMAN b a Department of Philosophy, Carnegie Mellon University, Pittsburgh, PA 15213 e-mail address : firstname.lastname@example.org b Department of Mathematics, Ohio State University, Columbus, OH 43210 e-mail address : email@example.com Abstract. We address the general problem of determining the validity of boolean com- binations of equalities and inequalities between real-valued expressions. In particular, we consider methods of establishing such assertions using only restricted forms of distributiv- ity. 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 T add [ Q ] be the first-order theory of the real numbers in the language with symbols , 1 , + ,- , <, . . . , f a , . . . where for each a Q , f a denotes the function f a ( x ) = ax . Let T mult [ Q ] be the analogous theory for the language with symbols 0 , 1 , , , <,. . . , f a , . . . . We show that although T [ Q ] = T add [ Q ] T mult [ 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. Introduction This paper is generally concerned with the problem of determining the validity of boolean combinations of equalities and inequalities between real-valued expressions. Such computational support is important not only for the formal verification of mathematical proofs, but, more generally, for any application which depends on such reasoning about the real numbers. Alfred Tarskis proof  that the theory of the real numbers as an ordered field admits quantifier-elimination is a striking and powerful response to the problem. The result implies decidability of the full first-order theory, not just the quantifier-free fragment. George Collinss  method of cylindrical algebraic decomposition made this procedure feasible in practice, and ongoing research in computational real geometry has resulted in various optimizations and alternatives (see e.g. [14, 6, 5]). Recently, a proof-producing version of 2000 ACM Subject Classification: F.4.1, I.2.3. Key words and phrases: decision procedures, real inequalities, Nelson Oppen methods, universal sentences....
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