This preview shows pages 1–2. Sign up to view the full content.
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
[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 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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
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.
 Fall '08
 JOSHUA
 Math

Click to edit the document details