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jointAvigadfinal091506 - Combining decision procedures for...

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Combining decision procedures for the reals Jeremy Avigad and Harvey Friedman * September 15, 2006 Abstract We address the general problem of determining the validity of boolean combinations of equalities and inequalities between real-valued 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 first-order 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 quantifier-free 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 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 Tarski’s proof [23] that the theory of the real numbers as an or- dered 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 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 proof-producing version of an * Work by both authors has been supported by NSF grant DMS-0401042. We are grateful to three anonymous referees for numerous comments, suggestions, and corrections. 1
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elimination procedure due to Paul Cohen has even been implemented in the framework of a theorem prover for higher-order logic [20]. 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 0 < 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 for the full decision procedure, seems misguided in this instance. A second, more
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