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Some Decision Problems of Enormous Complexity
Harvey M. Friedman
Department of Mathematics
The Ohio State University
friedman@math.ohiostate.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
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
[cx]
. 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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 Fall '08
 JOSHUA
 Math

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