somedecproenocom5_6_99

somedecproenocom5_6_99 - Some Decision Problems of Enormous...

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Some Decision Problems of Enormous Complexity Harvey M. Friedman Department of Mathematics The Ohio State University friedman@math.ohio-state.edu 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 θ ( ω ,0) time completeness to ( ,,0) time completeness. These three ordinals are well known ordinals from proof theory, and their associated com- plexity 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 expon- ential 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* ω , where ω 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).
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somedecproenocom5_6_99 - Some Decision Problems of Enormous...

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