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From: masjhd@bath.ac.uk (James Davenport)
Subject: Re: arbitrary-precision real arithmetic
Message-ID: <DDzExB.KGv@bath.ac.uk>
Summary: Progress is being made
Organization: School of Mathematical Sciences, University of Bath, UK
References: <41cuio$n6q@cantaloupe.srv.cs.cmu.edu> <STEVEW.95Aug26174514@debretts.comp.vuw.ac.nz> <41mttlINNd9i@topdog.cs.umbc.edu> <hbaker-2608950928260001@192.0.2.1>
Date: Sun, 27 Aug 1995 18:19:59 GMT
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Xref: glinda.oz.cs.cmu.edu sci.math.symbolic:18904 comp.lang.functional:6283 comp.lang.lisp:18966

In the referenced article, hbaker@netcom.com (Henry Baker) writes:
>Yes and no.  There are 'logical theories' such as the theory of
>'real closed fields' that have a decision procedure, thanks to Tarski
>circa the late 1940's.  This allows most of classical geometry, for
>example, to be shown to be amenable to a decision procedure.  Such a
>theory can handle algebraic numbers such as sqrt(2).  This decision
>procedure is extraordinarily expensive, however, so it isn't particularly
>useful right now.
The procedure is getting better, though (also machines are faster). In
particular there are some (quite large) constant speedups being made.
So, although the worst case remains doubly exponential
(Davenport,J.H. & Heintz,J.,
Real Quantifier Elimination is Doubly Exponential.
J. Symbolic Comp. 5(1988) pp. 29-35.)
uses in many cases can be much beter than this.
>However, it has also been shown that putting complex functions -- especially
>periodic functions -- into the theory makes it undecidable, because one
>can now talk about integers which Goedel proved undecidable in the 1930's.
But there has recently been some work in putting exponentials in
(Richardson,D.,
A Zero Structure Theorem for Exponential Polynomials.
Proc. ISSAC 1993 (ed. M. Bronstein, ACM, 1993) pp. 144-151.)
and there is a semi-decision procedure involving periodic functions
(Richardson,D. & Fitch,J.,
The identity problem for elementary function and constants.
Proc. ISSAC 94 (ACM, New York, 1994), pp. 285-290.)
(you have to read the paper to find out in what sense it is "semi")
which for all practical purposes is as good as a decision procedure.
James Davenport
jhd@maths.bath.ac.uk
