DecisonProbs042399 - 1 Some Decision Problems of Enormous...

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1 Some Decision Problems of Enormous Complexity Harvey M. Friedman Department of Mathematics The Ohio State University [email protected] Abstract We present some new decision and comparison problems of unusually high computational complexity. Most of the problems are strictly combinatorial in nature; others involve basic logical notions. Their complexities range from iterated exponential time completeness to 0 time completeness to q ( W w ,0) time completeness to ( ,,0) time completeness. These three ordinals are well known ordinals from proof theory, and their associated complexity classes represent new levels of computational complexity for natural decision problems. Proofs will appear in an extended version of this manuscript to be published elsewhere. 1. Iterated exponential time - universal relational sentences Let F be a function from A* into B*, where A,B are finite alphabets. We say that F is iterated exponential time computable if and only if there is a multitape Turing machine TM (which processes inputs from A* and outputs from B*) and an integer constant c > 0 such that TM computes F(x) with run time at most 2 [c|x|] . Here 2 [k] is the exponential stack of 2's of height k and |x| is the length of the string x. More generally, 2 [k] (n) is the exponential stack of k 2's with n placed on top. Define 2 [0] = 1 and 2 [0] (n) = n. Hence 2 [k] = 2 [k] (1) and 2 [1] (n) = 2 n . The iterated exponential time computable sets strictly include those sets in the more familiar class of elementary time computable sets - where the stack of 2's is of fixed height and |x| appears at the top of the stack. We say that X is iterated exponential time complete if and only if X is in iterated exponential time and every Y in iterated exponential time is polynomial time reducible to X. It is well known that for every finite alphabet A there exists an iterated exponential time complete X A*. As is customary, these definitions extended to include sets of strings in a finite alphabet using characteristic functions. A decision problem is given by a set of strings in a finite alphabet, where the “decision” is to decide membership. Suppose we are given a map G:A* w , where w is the set of all nonnegative integers. We can consider the associated equality problem: decide, given two strings x,y A*, whether G(x) = G(y). We can also consider the comparison problem: given x,y A*, compare the numbers G(x) and G(y). We find this terminology convenient and suggestive. Strictly speaking, this comparison problem is not a decision problem. It amounts to considering the function G’(x,y) = 0 if F(x) = F(y); 1 if F(x) < F(y); 2 if F(x) > F(y). It is easy to see that it is computationally equivalent, in the strongest possible sense, to the related decision problem: decide, given x,y A*, whether F(x) < F(y). A second kind of comparison problem that we consider
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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.

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DecisonProbs042399 - 1 Some Decision Problems of Enormous...

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