Berman showed that every m-complete set for EXP has an infinite P-subset and raised as an open ques-. Con whether the same holds for m-complete sets for.
On
P-Immunity
Department
of Nondeterministic Nicholas Tran of Computer and Information University of Pennsylvania Philadelphia, PA 19104
Abstract
2
Introduction
o if all n-complete sets for NP are polynomially isomorphic to one another, then P # NP [BH77];
l
if t,here exist two nonisomorphic m-complete for EXP, then P # UP [KLD87];
sets
if t,here exist two nonisomorphic m-complete for NEXP, then P # PSPACE [FKR89].
sets
Science
and in Sec-
Preliminaries
We assume the reader is familiar with the notions of Turing machines and time-bounded complexity classes such as P and NP. Let NEXP = IJ, NTIME(2nc). All languages are subsets of C”, where C = (0, l}. Tally languages are subsets of {O}*. The length of a string w is denoted by ]wj. Let < .,’ > be the standard pairing function that maps C* x C” to C*, such that I < X>Y > I = /xl+ 214. A set A is many-one reducible to a set B if there exists a polynomial-time computable function f such that for all 2, x E A ti f(z) E B. A set C is mcomplete for a class of sets C if C E C and every set in C is many-one reducible to C. Let < fz >ieN be an enumeration of polynomialtime computable functions such that f;(z) can be effectively computable from i and 2 in 2°((l’lt10~(lzl))z) time as shown in [GH92]. The following hierarchy theorem for nondeterministic time is due to Seiferas, Fischer, and Meyer [SFM78] (see also [Zbk83]):
Complete sets are widely studied in complexity theory, because they embody the structures of all sets in t.hr class t,hey represent. In particular, the structures of many-one (m-)complete sets can have profound implications on the relationships between central complexity classes. For example,
l
Sets
Section 2 we present relevant definitions, tion 3 we give the main result.
We show that every m-complete set for NEXP as well as ats complement have an infinite subset in P. Thw nnswers an open question first raised in [Ber76].
1
Complete
At present, the Berman-Hartmanis conjecture on t,he isomorphism of m-complete sets for NP and the generalized versions for EXP and NEXP have not been set*tled. One way to lend evidence to these conjectures is to show that all complete sets of NP (EXP, NEXP) share many properties of known “standard” m-complete sets, such as paddability, self-reducibility, or P-nonimmunity. In the cases of EXP and NEXP, P-nonimmunity is a desirable property since their mcomplete sets are known to be intractable, and thus having an infinite subset in P shows that a nontrivial portion of these m-complete sets can be easily recognized. Berman showed that every m-complete set for EXP has an infinite P-subset and raised as an open quesCon whether the same holds for m-complete sets for NEXP [Ber76]. Berman’s proof involves showing that n-complete sets for EXP are actually complete with respect t,o one-one length increasing reductions. Since t,his proof relies on the closure of EXP under complementation, it does not apply to NEXP; however, he was able to show that every m-complete sets for NEXP has an infinite E-subset. Homer pointed out that this infinite set can be made dense and in UP [Hom90]. In this paper, we show that every m-complete sets for NEXP does have an infinite P-subset, and furthermore its complement also has an infinite P-subset. In
Theorem 1 (Seiferas et al.) If t(n) and T(n) are time-constructible functions such that t(n + 1) E o(T(n)), then there is a tally set T E NTIME(T(n)) - NTIME(t(n)).
As a corollary, there exists a tally set Tk NTIME(2n4+1 ) - NTIME(2nk) for each Ic 2 1.
3
Main
E
Result
Theorem 2 Every m-complete complements contaan an infinite
set for NEXP P-subset.
and its
Let A be an m-complete set for NEXP. By definition A E NTIME(2”k) for some k > 1. By Theorem 1, there exists a tally set T in NTIME(2nkt1) - NTIME(2”k). Define C = {Oci,“> : ]fi(O)] > ~~~~:;‘io’ ,or 0” E T}. Clearly C E,NTIME(~~‘+,‘) ) can be computed m time exponential in 10 1, and hence C is m-reducible to A via a polynomial-time computable function fj. But {O : ]fj(O)] > /O]/j’} must be infinite, or else for large enough x, O E C
Proof:
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Proceedings of the 10th Annual Structure in Complexity Theory Conference (SCT'95) 1063-6870/95 $10.00 © 1995 IEEE
iff 0” E T; this means that for large enough 2, 0” E T iff j’i (O
