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Hilb10Geom082310 - DECISION PROBLEMS IN EUCLIDEAN GEOMETRY...

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DECISION PROBLEMS IN EUCLIDEAN GEOMETRY Harvey M. Friedman* Ohio State University August 23, 2010 ADVANCED DRAFT MAIN ISSUE: SHOULD I CONTINUE AND DEAL WITH ARBITRARY OR NEAR ARBITRARY RINGS, OR GET OTHERS TO DO THIS, OR WRITE A SEPARATE PAPER? COMMENTS PLEASE. Abstract. We show the algorithmic unsolvability of a number of decision procedures in ordinary two dimensional Euclidean geometry, involving lines and integer points. We also consider formulations involving points in subrings of . The main tool is the solution to Hilbert's Tenth Problem. The limited number of facts used from recursion theory are isolated. 0. PRELIMINARIES. 1. REALIZING COLINEARITY. 2. REALIZING BETWEENNESS. 3. REALIZING PARALLELS. 4. REALIZING EQUIDISTANCE. 0. Preliminaries. We show the algorithmic unsolvability of a number of decision procedures in ordinary two dimensional Euclidean geometry. The main tool is the solution to Hilbert's Tenth Problem. More generally, we demonstrate algorithmic reductions between various decision problems in two dimensional Euclidean geometry over subrings of the reals, with (variants of) Hilbert's 10th Problem for subrings of the ring of real numbers. We begin with a brief account of the relevant background from recursion theory. It is convenient to work within a fixed "universal" space that is rich enough to naturally support the kind of finitary objects that our algorithms operate on. Accordingly, we use the least set T that contains the integers and the 52 lower case and upper case alphabetic
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characters, and where every finite sequence from T (possibly empty) is an element of T. We use the basic notion of partial recursive f:T T from recursion theory. Informally, this is a partial function from T into T such that the following holds. There exists an algorithm such that at each input x T, if f(x) = y then the algorithm yields the output y; if f(x) is undefined, then the algorithm yields no output. A recursive f:T T is a partial recursive f:T T whose domain is T. A recursive subset of T is a subset of T whose characteristic function is recursive. An r.e. (recursively enumerable) subset of T is the domain of a partial recursive f:T T. Let A,B T. We say that A is reducible to B if and only if there is a recursive f:T T such that for all x T, x A f(x) B. This is written A B. We write A B if and only if B A. It is obvious that is a reflexive and transitive relation. We say that A T is complete r.e. if and only if A is r.e. and for all r.e. B T, B A. Let A,B T. We write A B if and only if there exists a recursive bijection f:A B such that for all x T, x A f(x) B. This is the strongest notion of equivalence that is normally studied in recursion theory.
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