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Unformatted text preview: Logical Methods in Computer Science Vol. 2 (4:4) 2006, pp. 1–42 www.lmcsonline.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 email address : [email protected] b Department of Mathematics, Ohio State University, Columbus, OH 43210 email address : [email protected] Abstract. We address the general problem of determining the validity of boolean com binations of equalities and inequalities between realvalued 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 firstorder 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 quantifierfree 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 realvalued 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 Tarski’s proof [23] that the theory of the real numbers as an ordered field admits quantifierelimination is a striking and powerful response to the problem. The result implies decidability of the full firstorder theory, not just the quantifierfree fragment. George Collins’s [10] 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 proofproducing 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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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.
 Fall '08
 JOSHUA
 Math, Logic

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