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Unformatted text preview: Combining decision procedures for the reals Jeremy Avigad and Harvey Friedman * January 31, 2006 Abstract We address the general problem of determining the validity of boolean combinations of equalities and inequalities between realvalued expres sions. 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. Let T add [ Q ] be the firstorder theory of the real numbers in the lan guage with symbols 0 , 1 , + , ,<,...,f a ,... where for each a ∈ Q , f a de notes 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 al though 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 valid ity 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 [18] 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 [8] method of cylindrical algebraic decomposition made this procedure feasible in practice, and ongoing research in computational real geometry has resulted in various optimizations and alterna tives (see e.g. [11, 5, 4]). Recently, a proofproducing version of an elimination * This is a preliminary version. Both authors are supported by NSF grant DMS0401042. 1 procedure due to Paul Cohen has even been implemented in the framework of a theorem prover for higherorder logic [16]. There are two reasons, however, that one might be interested in alterna tives to q.e. procedures for real closed fields. The first is that their generality means that they can be inefficient in restricted settings. For example, one might encounter an inference like < x < y → (1 + x 2 ) / (2 + y ) 17 < (1 + y 2 ) / (2 + x ) 10 , in an ordinary mathematical proof. Such an inference is easily verified, by noticing that all the subterms are positive and then chaining through the obvious inferences. Computing sequences of partial derivatives, which is necessary forinferences....
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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, Inequalities

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