On P-Immunity of Nondeterministic Complete Sets - Semantic Scholar
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<j+> : ]fj(O<j,z>)] > /O<J+>]/j’} must be infinite, or else for large enough x, O E C
Proof:
262
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 (Ol/j:! = j24j" = 2 = IO”], T can be accepted /u nondeterministically in time 2nk, a contradiction. Define D = {y : 32 = O<jazc>, Iz] < j2]y] & fj(z) y}. D is clearly in P, and from the paragraph above, is infinite. Furthermore, y E D + 30[~Ol j21yl & f’(O’jJ>) = y] 3 O<j,z> E C =+ y E Hence D is an infinite P-subset of A. Now define F = {O : If;(O)l ]~]/i’ (112d 0,” E T}. Again, F and hence iv is n-reducible to A time reduction j’i. Also, {O E IO l/1”} must be infinite, or else
= D < A.
K. GANESAN AND S. HOMER, Complete problems and strong polynomial reducibilities, SIAM J. Comput., 21 (1992), pp. 7333 742.
[HornSO]
S. HOMER, Structural properties of nondeterministic complete sets, in Proc. 5th Structure in Complexity Conference, IEEE, 1990, pp. 3310.
[KLD87]
K. Ko, T. LONG, AND D. Du, A note on one-way functions and polynomial-time isomorphisms, Theoretical Comput. Sci., 47 (1987), pp. 263-276.
[SFM78]
J. SEIFERAS, M. FISCHER,AND A. MEYER, Separating nondeternainzstic time complexity classes, J. Assoc. Comput. Mach., 25 (1978), pp. 1466167.
[Z&k831
S. ZAK, A Turing machine hierarchy, Theoretical Comput. Sci., 26 (1983), pp. 3277333.
5
E NTIME(2”k+1), via a polynomialF : lf~(O)j > T can be accepted
nondeterministically in time 2nk, a contradiction. Define G = {y : 32 = O, ]z] < 12]y] & Ii(z) = y}. G is clearly in P, and from the paragraph above, G is infinite. Furthermore, y E G + 30[]011~z>] < l”lyl & f~(O’:‘~z>) := y] =+ O $ F + y $ A. Hence G is an infinite P-subset of the complement of A. n Combining the theorem above Buhrman, ;Spaan, and Torenvliet trut,h-table-‘complete set for NEXP [BSTSl] yields
[GH92]
with the result by that every oneis also m-complete
1 Every one-truth-table-complete Corollary NEXP and its complements have an infinite
set for P-subset.
In contrast, one can easily construct via diagonalization a two-truth-table-complete set for NEXP that does not have an infinite subset in P. Theorem 2 can be generalized to hold for any reasonable class that 1) h as complete problems, 2) has a universal function for the polynomial-time computable functions, and 3) for every complete set C, there is a set in t#he class that cannot be compressed in length by a. reduction to C infinitely often.
References [Ber76]
L. BERMAN, On the structure of complete .ret,s: almost everywhere complexity and in,finitely often speedup, in Proc. 17th FOCS, IE:EE, 1976, pp. 76-83.
[BI177]
L
BERMAN
AND J. HARTMANIS,
morphisms and density of NP complete sets, SIAM J. Comput., pp. 3055322. [BSTSl]
H. BUHRMAN, E. SPAAN, AND L. TORENin Proc. 8th VIAET, Bounded reductions, STACS, vol. 480 Computer Science, pp. 41Om-421.
[FKR89]
On isoand other 6 (1977),
of Lecture Notes in Berlin, 1991, Springer,
S
FENNER, S. KURTZ, AND J. ROYER, Every polynomial-time l-degree collapses ifl P = PSPACE, in Proc. 30th FOCS, IEEE, 1989, pp. 624-629.
263
Proceedings of the 10th Annual Structure in Complexity Theory Conference (SCT'95) 1063-6870/95 $10.00 © 1995 IEEE
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<j+> : ]fj(O<j,z>)] > /O<J+>]/j’} must be infinite, or else for large enough x, O E C
Proof:
262
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 (Ol/j:! = j24j" = 2 = IO”], T can be accepted /u nondeterministically in time 2nk, a contradiction. Define D = {y : 32 = O<jazc>, Iz] < j2]y] & fj(z) y}. D is clearly in P, and from the paragraph above, is infinite. Furthermore, y E D + 30[~Ol j21yl & f’(O’jJ>) = y] 3 O<j,z> E C =+ y E Hence D is an infinite P-subset of A. Now define F = {O : If;(O)l ]~]/i’ (112d 0,” E T}. Again, F and hence iv is n-reducible to A time reduction j’i. Also, {O E IO l/1”} must be infinite, or else
= D < A.
K. GANESAN AND S. HOMER, Complete problems and strong polynomial reducibilities, SIAM J. Comput., 21 (1992), pp. 7333 742.
[HornSO]
S. HOMER, Structural properties of nondeterministic complete sets, in Proc. 5th Structure in Complexity Conference, IEEE, 1990, pp. 3310.
[KLD87]
K. Ko, T. LONG, AND D. Du, A note on one-way functions and polynomial-time isomorphisms, Theoretical Comput. Sci., 47 (1987), pp. 263-276.
[SFM78]
J. SEIFERAS, M. FISCHER,AND A. MEYER, Separating nondeternainzstic time complexity classes, J. Assoc. Comput. Mach., 25 (1978), pp. 1466167.
[Z&k831
S. ZAK, A Turing machine hierarchy, Theoretical Comput. Sci., 26 (1983), pp. 3277333.
5
E NTIME(2”k+1), via a polynomialF : lf~(O)j > T can be accepted
nondeterministically in time 2nk, a contradiction. Define G = {y : 32 = O, ]z] < 12]y] & Ii(z) = y}. G is clearly in P, and from the paragraph above, G is infinite. Furthermore, y E G + 30[]011~z>] < l”lyl & f~(O’:‘~z>) := y] =+ O $ F + y $ A. Hence G is an infinite P-subset of the complement of A. n Combining the theorem above Buhrman, ;Spaan, and Torenvliet trut,h-table-‘complete set for NEXP [BSTSl] yields
[GH92]
with the result by that every oneis also m-complete
1 Every one-truth-table-complete Corollary NEXP and its complements have an infinite
set for P-subset.
In contrast, one can easily construct via diagonalization a two-truth-table-complete set for NEXP that does not have an infinite subset in P. Theorem 2 can be generalized to hold for any reasonable class that 1) h as complete problems, 2) has a universal function for the polynomial-time computable functions, and 3) for every complete set C, there is a set in t#he class that cannot be compressed in length by a. reduction to C infinitely often.
References [Ber76]
L. BERMAN, On the structure of complete .ret,s: almost everywhere complexity and in,finitely often speedup, in Proc. 17th FOCS, IE:EE, 1976, pp. 76-83.
[BI177]
L
BERMAN
AND J. HARTMANIS,
morphisms and density of NP complete sets, SIAM J. Comput., pp. 3055322. [BSTSl]
H. BUHRMAN, E. SPAAN, AND L. TORENin Proc. 8th VIAET, Bounded reductions, STACS, vol. 480 Computer Science, pp. 41Om-421.
[FKR89]
On isoand other 6 (1977),
of Lecture Notes in Berlin, 1991, Springer,
S
FENNER, S. KURTZ, AND J. ROYER, Every polynomial-time l-degree collapses ifl P = PSPACE, in Proc. 30th FOCS, IEEE, 1989, pp. 624-629.
263
Proceedings of the 10th Annual Structure in Complexity Theory Conference (SCT'95) 1063-6870/95 $10.00 © 1995 IEEE
