On P-selective Sets and EXP Hard Sets - Semantic Scholar

On P-selective Sets and EXP Hard Sets

Bin Fu

Department of Computer Science, Yale University, New Haven, CT 06520 and UMIACS, University of Maryland at College Park, MD 20742 May 1997



Email: [email protected]

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Abbreviated form of the title:

P-selective Sets and EXP Hard Sets Mailing Address: Bin Fu 5119 Hollywood Road College Park, MD 20740.

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Abstract

Let PSel be the class of all P-selective sets. We show that DTIME(2nk a ) 6 Pnk ?T (PSel) for all k; a > 0. It implies EXP 6 Pnc ?T (PSel) for all c > 0. This greatly improves Toda's result that EXP 6 Ptt(PSel) since Ptt (PSel) is equal to PO(log n)?T (PSel). We construct an oracle A such that DTIME(2nk )A  PAnk a -T (PSel). We also show that the symmetric dierence of a E-Pm -hard set and a P-selective set is a E-P2?T -hard. This generalizes a result by Rao [18] who showed the symmetric dierence of a E-Pm -hard set and a P-selective set is exponentially dense. +

+1+

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Symbols used in the paper: a; b; c; d; e; f; g; h; i; j; k; l; m;n; o; p; q; r; s;t; u; v; w; x;y; z A; B; C; D; E; F; G; H; I; J; K; L; M; N; O; P; Q; R; S; T; U; V; W; X; Y; Z [; \; ; ?; ; ; ;  ; ; 9; 8; ; ; 4; k; [; ](; ); f:g; (); =); ?!; 6=; =; 2; ; ;; ; 6; 8; ^

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1. Introduction The study of reducibilities of complexity classes to the low sets is an important topics in complexity theory. It has long and rich history. One of the most notable low notions is P-selectivity. It was introduced by Selman [20, 22, 21] as a complexity-theoretic analog to the semi-recursive sets [12] from recursive theory. He used it to distinguish polynomial time m-reducibility from Turing-reducibility in NP under the assumption E6=NE, and proved that if every set in NP is positively truth-table reducible to Pselective sets then P=NP. Ko [14] extended the notion to weakly P-selectivity and showed that no NP-hard set is polynomial time disjunctively reducible to weakly Pselective sets unless P=NP. The class of languages that are polynomial time Turing reducible to P-selective sets are the same as P/poly [20]. Separating complexity class from P/poly is one of the most fundamental problems in complexity theory. It is still open if NEXP 6P/poly. The line of research about the consequence of NP sets reducing to a P-selective set is very active in the recent years. It has attracted a great deal of interest and have been many advances in understanding P-selectivity. Buhrman, Torenvliet and Emde Bosa [6] generalized Selman's result that if a NP-hard set under Turing positive reductions is reducible to a P-selective set then P=NP. Beigel [2] and Toda [26] showed that if a NP-Ptt-hard set is P-selective then P=FewP and NP=RP. Thierauf, Toda and Watanabe [24] showed that if a NP-Pbtt-hard set is P-selective then NP p DTIME(2nO = n ). Agrawal and Arvind [1], Beigel, Kummer and Stephan [3], and Ogiwara [17] independently proved that if a NP-Pbtt-hard set is P-selective, then P=NP. Cai, Naik and Selmank showed that if a NP-Ptt-hard set is P-selective then then NP DTIME(2nO = n ) for all k > 0. Toda [26] showed that EXP is not polynomial time truth-table reducible to Pselective sets. Recently Buhrman and Longre [5], and Wang [27] independently proved that the class of languages that are polynomial time truth-table reducible to P-selective sets has measure in E. Ptt(PSel) is the same as PO(log n)-T (PSel). We show that EXP 6 Pnc ?T (PSel) for all c > 0. This improves the Toda's result that EXP 6 Ptt(PSel). We construct an oracle A such that DTIME(2nk )A  PAnk +1+a-T (PSel). Since PT (PSel) = P/poly, it is impossible to prove the no P-Selective set is E- PT -hard unless the most fundamental problem in complexity theory can be solved. This indicates our result is almost optimal by the relativizable methods. We also study the symmetric dierence between hard set with P-selective sets. Yesha [29] rst studied the symmetric dierence between NP-Ptt-hard sets and the sets in P. Schoning showed that no the symmetric dierence between a E-PT -hard and a set in P is of exponential density. The symmetric dierence of NP-hard sets and P-selective was investigated by Fu and Li [9]. Rao showed that the symmetric dierence of a EXP- Ptt-hard set and a P-selective set is of exponential density. We study the symmetric dierence between a EXP- Pm-hard set and a P-selective set from the stability point of view in [23] We generalize Rao's result by proving that the symmetric dierence of a E-Pm-hard set and a P-selective set is a E-P2?T -hard. (1

log

)

(1 (log

) )

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2. Preliminaries

We use  = f0; 1g as our alphabet. By \string" we mean an element of , jxj denotes the length of x. We use lexicographic order on . For any strings x and y, x is smaller than y (write x < y) if either jxj < jyj, or jxj = jyj and there exists some k, 1  k  jxj, such that (8i : 1  i < k[xi = yi] and [xk = 0 and yk = 1]), where xi is the ith symbol of the string x. For two strings x; y, x  y if x is an initial segement of y. For S  , the cardinality of S is denoted by kS k. Set S =n (S n ) consists of all words of length = n( n) in S . In particular, let n = fx : x 2  and jxj = ng and n = fx : x 2  and jxj  ng. For a language A, A(x) is the characteristic function of A. N = f0; 1; 2;


g. For real number x, bxc is the largest integer  x.

De nition 1. [20] A set A is P-selective if there is a total polynomial time function g so that, for every x; y 2 ,  g(x; y) 2 fx; yg, and  if x 2 A or y 2 A, then g(x; y) 2 A. We call g as the selector function for A. The above g induces a linear order g that x g y if there exists z1;


; zt such that g(x; z1) = x; g(zi; zi+1) = zi; and g(zt; y) = zt. The P-selective set A is an initial segment of the linear order g . De nition 2.  E = S1k=1 DTIME(2kn+k ).  EXP = S1k=1 DTIME(2nk +k ). De nition 3. A PT -reduction of A to B is a polynomial-time oracle Tur ing machine P  M such that for each x 2 , x 2 A () M B accepts x. For a function g : N ?! N , a Pg(n)?T -reduction of A to B is a polynomial-time oracle Turing machine M such that A PT B is witnessed by M and M will not query the oracle more that g(n) times for each input with length n.

De nition 4. Let H  , C be a class of languages and Pr be a type of reductions. Pr(H ) is the class of all languages which are Pr-reducible to H . Pr(C) is the class of languages which are Pr-reducible to some languages in C. If C Pr(H ), then we say H is C-Pr-hard. De nition 5. For a function d from N to N , DENSITY(d(n)) is the class of all languages that has density bounded by d(n). If there exists a polynomial p(n) such that k An k< p(n) for all n, then we say A is sparse. \SPARSE" represents the class of all sparse languages.

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Hartmanis [11] introduced a \generalized Kolmogorov complexity measure". We employ this tool in the proof of our theorems. We consider standard deterministic time- bounded Turing machines that act as transducers. We assume a standard enumeration of such transducers, say N1; N2;


. For each i, let fi be the function computed by Ni, and let Ti be the running time of the transducer Ni . We assume that the enumeration has the property that there exists a universal Turing transducer Nu and a description function d with the following property: for every i > 0 there is a constant ci such that for all x 2 , (a)d(i) is not a pre x of d(j ) if i 6= j , (b) fu (d(i)x) = fi(x), and (c)Tu(d(i)x)  ci
Ti(x)
log Ti(x) + ci. We then de ne the following classes of strings.

De nition 6.  Ki[g; t] = fy : 9x[jxj  g(jyj); fi(x) = y and Ti(x)  t(jyj)]g:  K[g; t] = Ku[g; t], where u denotes the index of the universal Turing machine. Let fu be the transducer for the universal Turing machine.  For two integers m; n, de ne K[m; n] = fy : 9x[jxj  m; fu(x) = y, and Ti(x)  n]g.

Lemma 7 ([11]). Let i be any index and let g(n) and t(n) be time-constructible functions. Then there exists c > 0 such that Ki [g (n); t(n)]  Ku [g (n)+c; c
t(n) log t(n)+c].

3. E-PT-hard sets

The function cod :  ?!  is de ned by cod(x) = 1a11a2


1ar 02, where x = a1a2


ar 2 . The function bin : N ?!  is de ned so that for each i 2 N , bin(i) is the binary expression of i (for example, bin(5) = 101 and bin(8) = 1000). For tuple hx1;


; xni, we use cod(bin(jx1j))x1


cod(bin(jxn j))xn to encode it into a string. It is very easy to see that hx1;


; xti can be encoded and decoded in polynomial time. The length of cod(bin(jx1j))x1


cod(bin(jxn j))xn is bounded by

Xn jx j + 4n(1 + log(Xn jx j)): i=1

i

i=1

i

The following proposition is easy to verify (because uvwvx can be compressed into cod(bin(juj))cod(bin(jvj))cod(bin(jwj))uvwz). Proposition 8 ([8]). There exists a polynomial p(n) such that for all large n, if there exist strings  2 n and u; v; w; x to satisfy  = uvwvx and jvj > 7 log n, then  2 K[n ? 1; p(n)].

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Theorem 9. Let f (n) be a nO(1) time computable nondecreasing function from N to N . if f (n) = 6 O(log n) and f (n)  nc for some constant c0 > 0, then DTIME(2(3+a)f (n)) \ DENSITY(f (n)) 6 Pf (n)?T (PSel) 0

for all a > 0.

Proof: Suppose

DTIME(2(3+a)f (n)) \ DENSITY(f (n)) 6 Pf (n)?T (PSel) for some a > 0. We will derive a contradiction. In this chapter we give informal description about the idea of proof. The idea is to use the resource bounded Kolmogorov complexity point of view. A selective set is a initial segment of a linear order g . For a P-selective set S , if A PT S via oracle Turing machine M that makes small number f (n) of queries. Let's consider all of strings queried by M with inputs of length n. Among them there is a string y that is at the boundary of S . In other words, for every string u queried by M, u g y =) u 2 S and y 0. 6

Corollary 15. EXP 6 Pnc ?T (PSel) for every c > 0. Corollary 16. DTIME(2nk a ) 6 Pnk ?T (PSel) for every a > 0. +

Corollary 17. SPARSE 6 Pnc ?T (PSel) for every c > 0. Theorem 18. For any c > 0, Pnc ?T (PSel) \ EXP has measure 0 in EXP. Theorem 19. For any c > 0, Pcn?T (PSel) \ E has measure 0 in E.

4. Oracle In this section we constrcut an oracle to indicate the limit of relativizable method.

Theorem k20. For every k > 1 and a > 0, there exists an oracle A such that DTIME(2n )A  PAnk a ?T (PSel). Proof: For anyk oracle B , let K B = fx10i : MBi accepts x in 2jxjk steps g. K B is +1+

computable in 2n time relative oracle Bk . There is a oracle Turing machine M such that M B accepts K B and M runs in 2n time. De ne Query(M; B; x) be the set of all strings that are queried by M B with input x. Construction of set A: Stage 0: Set s0 = ;, A0 = ; and P0 = ;. Stage n: Let [ Query(M; A ; x) Qn = n?1 x2n Pn?1 [ Qn:

Pn = Since M runs in 2nk time, k Query(M; An?1; x) k 2nk for any x 2 n . Therefore,

k Qn k  2n+1
2nk = 2nk +n+1 n X k Pn k  k Qi k i=1

j k ak

= n
2nk +n+1 :

There exists yn 2  n such that hyn; xi 62 Pn for any x 2 n. Let sn = sn?1yn. Let Gn be the set of all hyn ; xi such that x 2 K An? . Set An = An?1 [ Gn . End of Stage n. De ne A = S1 n=1 Gn . End of the consruction of set A. +2

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The following claim is easy to verify by the construction of A. Claim M An? (x) = M A(x) for all x 2 n . From the above the construction we get in nite strings 1

s1  s2 : : :  sn  : : :: Let s1 be the string of length 1 such that every si is an initial segement of it. Let S = fx : x  s1g. Thus, s1 is the boundary of the P-selective set S under the lexicographic order. Since s1 is the boundary of S , 1  s1 i 1 2 S . Assume zn 2 n is an initial segement of s1 . One of zn0 and zn1 is an initial segement of s1. Furthermore, zn1 2 S ) zn1  s1 and zn1 62 S ) zn0  s1 . Therefore, the rst n bits of s1 can be found in polynomial j k+ a k time by making n queries to set S . Since jynj = n , 2

jsnj = 

Xn jy j i=1 n

i

X jik + a k i=1 j k+ a k 2

 nn  nk+1+ a : 2

2

So, we can nd yn by making no more than nk+1+ a queries to S . For each x 2 n , hyn ; xi 2 A () x 2 K A . So, K A 2 PAnk a ?T (S ). For every language L accepted by MAi in time 2nk , we have x 2 L () x10i 2 K A. So, L 2 PAnk a ?T (S ). Form the Theorem 20, we know the Corollary 16 is almost the optimal result that we can get by the relativizable method. 2

+1+

2

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5. Symmetric Dierences

Let g(n) be a time constructible super-polynomial function from N to N . A gm;Preduction from A to B is a function f such that for some polynomial p(n), f is computable in g(n)-time , jf (x)j  p(jxj) and x 2 A () f (x) 2 B for each input x.

Theorem 21. Let g(n) be a time constructible super polynomial function from N to N . Let C be a class of languages which has Pm-complete set and is closed under gm;Preductions, H be Pm-hard for C , A be a P-selective set , then A 4 H is P2?T -hard for C .

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Proof: Let f1; f2;


, be an eective enumeration of all Pm-reductions and K is a Pm-complete set for C . Let f be the selector functor function for A. Construction of L Input hi; xi Within g(jhi; xij)=2 steps, compute fi(hi; x; 0i) = y0; fi(hi; xi) = y1 and fi(hi; x; 2i) = y2 and let yi f yi f yi , 0

1

2

where i0; i1; i2 is a permutation of 0; 1; 2. If the above computation can be nished in the given time, then Accept hi; x; i1i and ( for w in fhi; x; i0i; hi; x; i2ig accepts w i x 2 K ).

End of the construction. Since C is closed under gm;P-reductions, it is easy to see that L 2 C . Since H is C - Pm-hard, so L Pm H via fj for some j . Lemma 22. For almost all x, let fj (hj; x; 0i) = y0 ,fj (hj; x; 1i) = y1, fj (hj; x; 2i) = y2 and yi f yi f yi , if yi 62 A 4 H , then x 2 K () yi 2 A 4 H , and; if yi 2 A 4 H , then x 2 K () yi 62 A 4 H . Proof: Since hj; x; i1i 2 L, we have yi 2 H . If yi 2 A 4 H ,then yi 62 A. It implies yi 62 A since yi f yi . Hence, x 2 K () yi 2 A 4 H . If yi 62 A 4 H , then yi 2 A. It implies yi 2 A since yi f yi . So, x 2 K () yi 62 A 4 H . From the above discussion, we have K P2?T A. So, A is C - P2?T -hard. 0

1

2

1

2

1

0

2

1

2

2

1

1

1

1

0

1

0

1

0

Corollary 23. The symmetric dierence of a DTIME(nO(log n) )- Pm-hard set and a P-selective set is DTIME(nO(log n) )- P2?T -hard. Corollary 24. The symmetric dierence of a E-Pm-hard set and a P-selective set is E- P2?T -hard. Corollary 25. The symmetric dierence of a NE- Pm-hard set and a P-selective set is NE- P2?T -hard. We know that every E- P2?T -hard is of exponential density [25]. The Corollary 24 generalizes a result by Rao [18] who showed the symmetric dierence of a E-Pm-hard set a P-selective set is of exponential density. We note that the symmetric dierence of a NP-Pm-hard set and a P-selective set is NP- PT-hard. 9

References [1] M.Arvind and V.Arvind: Polynomial-time Truth-table Reductions to P-selective Sets. Proceedings of the 9th Structure in Complexity Theory Conference. IEEE Computer Society Press, June/July 1994. [2] R.Beigel: NP-hard sets are p-superterse unless R=NP, TR 88-04, Dept. of Computer Science, John Hopkins University, 1988. [3] R.Beigel, M.Kummer, and F.Stephan: Approximable Sets. In Proceedings of the 9th Structure in Complexity Theory Conference. IEEE Computer Society Press, June/July 1994. [4] L.Berman and J.Hartmanis: On Isomorphisms and Density of NP and other Complete Sets. SIAM Journal on Computing 1(1977), 305-322. [5] Compressibility and Resource Bounded Measure, STACS 1996. [6] H.Buhrman, L. Torenvliet, and P. van Emde Boas: Twenty Questions to a P-selector. Manuscript, 1993. [7] J. Cai, A. Naik, and A. Selman: On P-selective Sets and Adaptive versus Non-adaptive Queries to NP. TR University of New York at Bualo, Feb., 1994. [8] B. Fu: With Quasi-linear Queries EXP is Not Polynomial-time Turning Reducible to Sparse Sets. SIAM Journal on Computing, October, 1995, pp.1082-1090. [9] B.Fu and H.Li: On Symmetric Dierence of NP-hard Sets with Weakly-P-Selective Sets. Theoretical Computer Science, 120(1993), pp.279-291. [10] R.Gavalda and O.Watanabe: On the Computational Complexity of Small Descriptions. Proc.of 6th IEEE Conference on Structure in Complexity Theory 1991, 89-101.

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[11] J.Hartmanis: Generalized Kolmogorov Complexity and the Structure of Feasible Computations. Proc.24th Annual IEEE Symposium on Foundations of Computer Science 1983,439-445. [12] C.Jockusch: Semirecursive Sets and Positive Reducibility, Transaction of the AMS, 131(1968), 420-436. [13] R.Karp and R.Lipton: Some Connections between Nonuniform and Uniform Complexity Classes. Proc. of the 12th ACM Symposium on Theory of Computing 1980, 302-309. [14] K.Ko: On Self-reducibility and Weak P-Selectivity, Journal of Computer and System Sciences, 26(1983),209-221. [15] J.Lutz: Category and Measure in Complexity Classes, SIAM Journal on Computing, 19(1990), 1100-1131. [16] J.Lutz and E.Mayordomo: Measure, Stochasticity, and the Density of Hard Languages. SIAM Journal on Computing, 23(1994), pp 762-779. [17] M.Ogihara: Polynomial-time Membership Comparable Sets. SIAM Journal on Computing. October 1995, [18] R.Rao: On P-Selectivity and Closeness, Information Processing Letters, 54(1995), 179-185. [19] U.Schoning: Complete Sets and Closeness to Complexity Classes, Mathematical System Theory, 19(1986), 29-42. [20] A.Selman: P-selectivity Set, Tally Languages, and the behavior of Polynomial Time Reducibilities on NP, Mathematical System Theory, 13(1979), 55-65. [21] A.Selman: Analogies of Semirecursive Sets and Eective Reducibilities to the Study of NP Complexity. Information and Control, 52(1982), 36-51. [22] A.Selman: Reduction on NP and P-selective Sets. Theoretical Computer Science, 19(1982), 287-304. 11

[23] S.Tang, B.Fu and T.Liu: Exponential Time and Sub-exponential Time Sets. Theoretical Computer Science, 115(1993), pp.371-381. [24] T. Thierauf, S. Toda, and O. Watanabe: On Sets Bounded Truthtable Reducible to P-selective Sets, STACS 1994. [25] O.Watanabe: Polynomial Time Reducibility to a Set of Small Density. Proc.of 2th IEEE Conference on Structure in Complexity Theory 1987, 138-146. [26] S.Toda, On Polynomial-time Truth-table Reducibilities of Intractable Sets to P-selective Sets, Mathematical System Theory, 24(1991), 69-82. [27] Y. Wang: NP-hard Sets Are Superterse unless NP Is Small. TR96-020, ECCC, 1996. [28] C.B.Wilson: Relativized Circuit Complexity. Journal of Computer and System Science 31(1985) 169-181. [29] Y.Yesha: On Certain Polynomial-time Truth-table Reducibilities of Complete Sets to Sparse Sets. SIAM Journal on Computing, 12(1983), 411-425.

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