The Density of Weakly Complete Problems under Adaptive Reductions*
The Density of Weakly Complete Problems under Adaptive Reductions Jack H. Lutz Yong Zhao Department of Computer Science LANshark Systems Iowa State University 784 Morrison Road Ames, IA 50011 U.S.A. Columbus, OH 43230 U.S.A. [email protected] [email protected]
Abstract
Given a real number < 1, every language that is weakly Pn =2 ;T -hard for E or weakly Pn ;T -hard for E2 is shown to be exponentially dense. This simultaneously strengthens results of Lutz and Mayordomo(1994) and Fu(1995).
1 Introduction In the mid-1970's, Meyer15] proved that every Pm-complete language for exponential time|in fact, every Pm-hard language for exponential time|is dense. That is, E 6 Pm(DENSEc )
(1)
where E = DTIME(2linear), DENSE is the class of all dense languages, DENSEc is the complement of DENSE, and Pm(DENSEc ) is the class of all languages that are Pm-reducible to non-dense languages. (A language A 2 f0 1g is dense if there is a real number > 0 such that jAn j > 2n for all suciently large n, where An = A \ f0 1gn .) Since that time, a major objective of computational complexity theory has been to extend Meyer's result from Pm-reductions to PT -reductions, i.e., to prove that every PT -hard language for E is dense. That is, the objective is to prove that E 6 PT (DENSEc )
(2)
where PT (DENSEc) is the class of all languages that are PT-reducible to non-dense languages. The importance of this objective derives largely from the fact (noted by Meyer15]) that the class PT (DENSEc) contains all languages that have subexponential circuit-size complexity. (A language A f0 1g has subexponential circuit-size complexity if, for every real number > 0, for every suciently large n, there is an n-input, 1-output Boolean This research was supported in part by National Science Foundation Grant CCR-9157382, with matching funds from Rockwell International, Microware Systems Corporation, and Amoco Foundation.
1
circuit that decides that the set A=n = A \ f0 1gn and has fewer than 2n gates. Otherwise, we say that A has exponential circuit-size complexity.) Thus a proof of (2) would tell us that E contains languages with exponential circuit-size complexity, thereby answering a major open question concerning the relationship between (uniform) time complexity and (nonuniform) circuit-size complexity. Of course (2) also implies the more modest, but more famous conjecture, that E 6 PT (SPARSE)
(3)
where SPARSE is the class of all sparse languages. (A language A f0 1g is sparse if there is a polynomial q(n) such that jAn j q(n) for all n 2 N .) As noted by Meyer15], the class PT (SPARSE) consists precisely of all languages that have polynomial circuit-size complexity, so (3) asserts that E contains languages that do not have polynomial circuit-size complexity. Knowing (1) and wanting to prove (2), the natural strategy has been to prove results of the form E 6 Pr (DENSEc ) for successively larger classes Pr (DENSEc) in the range Pm(DENSEc ) Pr (DENSEc) PT (DENSEc ): The rst major step beyond (1) in this program was the proof by Watanabe17] that E 6 PO(log n);tt (DENSEc )
(4)
i.e., that every language that is PO(log n);tt -hard for E is dense. The next big step was the proof by Lutz and Mayordomo10] that, for every real number < 1, E 6 Pn ;tt (DENSEc ):
(5)
This improved Watanabe's result from O(log n) truth-table (i.e., nonadaptive) queries to n such queries for arbitrarily close to 1 (e.g., to n0:99 truth-table queries). Moreover, Lutz and Mayordomo10] proved (5) by rst proving the stronger result that for all < 1,
p(Pn
tt (DENSE
c )) = 0
(6) which implies that every language that is weakly Pn;tt -hard for E or for E2 = DTIME(2poly ) is dense. (A language A is weakly Pr -hard for a complexity class C if (Pr (A) j C ) 6= 0, i.e., if Pr (A) \ C is a nonnegligible subset of C in the sense of the resource-bounded measure developed by Lutz9]. A language A is weakly Pr -complete for C if A 2 C and A is weakly P -hard for C . See 12] or 2] for a survey of resource-bounded measure and weak comr pleteness.) The set of weakly Pn ;tt -hard languages for E is now known to have p-measure 1 3], hence measure 1 in the class C of all languages, while the set of all Pn ;tt -hard languages for E has measure 0 unless E BPP 4, 1]. Thus, if E 6 BPP (which is generally conjectured to be true), almost every language is weakly Pn;tt -hard, but not Pn;tt -hard, for E, so the result of Lutz and Mayordomo 10] is much more general than the fact that every Pn ;tt -hard language for E is dense. ;
2
A word on the relationship between hardness notions for E and E2 is in order here. It is well known that a language is Pm-hard for E if and only if it is Pm-hard for E2 this is because E2 = Pm(E). The same equivalence holds for PT -hardness. It is also clear that every language that is Pn ;tt -hard for E2 is Pn;tt -hard for E. However, it is not generally the case that Pm(Pn ;tt (A)) = Pn ;tt (A), so it may well be the case that a language can be Pn;tt -hard for E, but not for E2 . These same remarks apply to Pn ;T -hardness. The relationship between weak hardness notions for E and E2 is somewhat dierent. Juedes and Lutz 8] have shown that weak Pm-hardness for E implies weak Pm-hardness for E2 , and their proof of this fact also works for weak PT -hardness. However, Juedes and Lutz 8] also showed that weak Pm-hardness for E2 does not generally imply weak Pm-hardness for E, and it is reasonable to conjecture (but has not been proven) that the same holds for weak PT -hardness. We further conjecture that the notions of weak Pn ;tt -hardness for E and weak Pn ;tt -hardness E2 are incomparable, and similarly for weak Pn ;T -hardness. In any case, (6) implies that, for every < 1, every language that is weakly Pn ;tt -hard for either E or E2 is dense. Shortly after, but independently of 10], Fu7] used very dierent techniques to prove that, for every < 1, E 6 Pn=2 ;T (DENSEc)
(7)
E2 6 Pn;T (DENSEc):
(8)
and That is, every language that is Pn=2 ;T -hard for E or Pn ;T -hard for E2 is dense. These results do not have the measure-theoretic strength of (6), but they are a major improvement over previous results on the densities of hard languages in that they hold for Turing reductions, which have adaptive queries. In the present paper, we prove results which simultaneously strengthen results of Lutz and Mayordomo10] and the results of Fu7]. Specically, we prove that, for every < 1,
p (Pn
=2 ;
c T (DENSE )) = 0
(9)
c )) = 0:
(10)
and
p2 (Pn
T (DENSE
;
These results imply that every language that is weakly Pn=2 ;T -hard for E or weakly P n=2 ;T -hard for E2 is dense. The proof of (9) and (10) is not a simple extension of the proof in 10] or the proof in 7], but rather combines ideas from both 10] and 7] with the martingale dilation technique introduced by Ambos-Spies, Terwijn, and Zheng 3]. Our results also show that the strong hypotheses p (NP) 6= 0 and p2 (NP) 6= 0 (surveyed in 12] and 2]) have consequences for the densities of adaptively hard languages for NP. Mahaney 13] proved that P 6= NP ) NP 6 Pm(SPARSE) 3
(11)
and Ogiwara and Watanabe 16] improved this to P 6= NP ) NP 6 Pbtt (SPARSE):
(12)
That is, if P 6= NP, then no sparse language can be Pbtt -hard for NP. Lutz and Mayordomo 10] used (6) to obtain a stronger conclusion from a stronger hypothesis, namely, for all < 1,
p(NP) 6= 0 ) NP 6 Pn
c ):
(13)
c T (DENSE )
(14)
c T (DENSE ):
(15)
tt (DENSE
;
By (9) and (10), we now have, for all < 1,
p(NP) 6= 0 ) NP 6 Pn
=2 ;
and
p2 (NP) 6= 0 ) NP 6 Pn
;
Thus, if p (NP) 6= 0, then every language that is Pn0:49 ;T -hard for NP is dense. If p2 (NP) 6= 0, then every language that is Pn0:99 ;T -hard for NP is dense.
2 Preliminaries The Boolean value of a condition, is ] =
1 if 0 if not :
The standard enumeration of f0 1g is s0 = s1 = 0 s2 = 1 s3 = 00 : : : This enumeration induces a total ordering of f0 1g which we denote by c. The success set of a martingale d is the set
S 1d] = fA 2 Cjd succeeds on Ag: The unitary success set of d is
S 1d] =
w2f01g d(w)1
Cw :
The following result was proven by Juedes and Lutz 8] and independently by Mayordomo 14].
Lemma 2.1 (Exact Computation Lemma) Let t : N ! N be nondecreasing with t(n) n2. Then, for every t(n)-martingale d, there is an exact n t(2n + 2)-martingale de such that S 1 d] S 1de]. A sequence
1 X
k=0
ajk
(j = 0 1 2 : : : )
of series of terms ajk 2 0 1) is uniformly p-convergent if there is a polynomial m : N 2 ! N 1 X ajk 2;r , where we write mj (r) = m(j r). The following such that, for all j r 2 N , k=mj (r)
sucient condition for uniform p-convergence is easily veried by routine calculus. 5
Lemma 2.2 Let ajk 2 0 1) for all j k 2 N . If there exist a real number > 0 and a polynomial g : N ! N such that ajk e;k for all j k 2 N with k g(j ), then the series
1 X
k=0
ajk (j = 0 1 2 : : : ) are uniformly p-convergent.
A uniform, resource-bounded generalization of the classical rst Borel-Cantelli lemma was proved by Lutz 9]. Here we use the following precise variant of this result.
Theorem 2.3 Let e 2 R with 1 e, and let d : N N f0 1g ! Q \ 0 1) be an exactly 2(log n) -time-computable function with the following two properties. (i) For each j k 2 N , the function djk dened by djk (w) = d(j k w) is a martingale. (ii) The series
1 X
k=0
djk (j = 0 1 2 : : : ) are uniformly p-convergent.
Then there is an exact 2(log n)e -martingale e such that 1 \ 1 1
j =0 t=0 k=t
S 1 djk ] S 1 de]:
Proof (sketch).0 Assume the hypothesis, and x 0 2 Q such that < 0 < e. Since0 n 2(log(2n+2)) = o(2(log n) e ), it suces by Lemma 2.1 to show that there is a 2(log n)
martingale d0 such that
1 \ 1 1
j =0 t=0 k=t
Fix a polynomial m :
N2
S 1djk ] S 1 d0 ]:
! N testifying that the series
uniformly p-convergent, and dene
d (w) = 0
1 X 1 1 X X
j =0 t=0 k=mj (2t)
(16) 1 X
k=0
2t;j djk (w)
for all w 2 f0 1g . Then, for each w 2 f0 1g , 1 X 1 1 X X
d0 (w)
j =0 t=0 k=mj (2t)
2jwj =
1 X
j =0 2jwj+2
2;j
6
1 X
t=0
2t;j +jwjdjk () 2t 2;2t
djk (j = 0 1 2 : : : ) are
so d0 : f0 1g ! 0 1). It is clear by linearity that d0 is a martingale. To see that (16) 1 \ 1 1 holds, assume that A 2 S 1 djk ], and let c 2 N be arbitrary. Then there exist j =0 t=0 k=t
j 2 N and k mj (2j + 2c) such that A 2 S 1 djk ]. Fix w v A such that djk (w) 1. Then d0 (w) 2c+j ;j djk (w) 2c . Since c is arbitrary here, it follows that A 2 S 1 d0 ], conrming
(16). 0 To see that d0 is 2(log n) -time-computable, dene dA dB dC : N f0 1g ! 0 1) as follows, using the abbreviation s = r + jwj + 2.
dA(r w) = dB (r w) = dC ( r w ) =
s X 1 1 X X
j =0 t=0 k=mj (2t) s X 2s X 1 X
2t;j djk (w) 2t;j djk (w)
j =0 t=0 k=mj (2t) 2 +4s+t) s X 2s mj (2sX X j =0 t=0 k=mj (2t)
2t;j djk (w)
(17)
For all r 2 N and w 2 f0 1g , it is clear that dC (r w) dB (r w) dA(r w) d0(w) and it is routine to verify the inequalities d0 (w) ; dA (r w) 2;(r+1) dA(r w) ; dB (r w) 2;(r+2) dB (r w) ; dC (r w) 2;(r+2) whence we have d0 (w) ; 2;r dC (r w) d0(w) (18) for all r 2 N and w 2 f0 1g . Using formula (17), the time required to compute dC (r w) exactly is no greater than O((s + 1)(2s + 1)m(s 2s2 + 4s + 2s)2(log n) ) = O(q(n) 2(log n) ) 0
where n = r + jwj and q is0 a polynomial. Since q(n) 2(log n) = o(2(log n) ),0 it follows that dC (r w) is exactly 2(log n) -time-computable. By (18), then, d0 is a 2(log n) -martingale. The proof of our main theorem uses the techniques of weak stochasticity and martingale dilation, which we briey review here. As usual, an advice function is a function h : N ! f0 1g . Given a function q : N ! N , we write ADV(q) for the set of all advice functions h such that jh(n)j q(n) for all n 2 N . Given a language B and an advice function h, we dene the language B=h = fx 2 f0 1g j< x h(jxj) >2 B g 7
where < > is a standard string-pairing function, e.g., < x y >= 0jxj1xy. Given functions t q : N ! N , we dene the advice class DTIME(t)=ADV(q) = fB=h j B 2 DTIME(t) and h 2 ADV(q)g:
Denition (Lutz and Mayordomo10], Lutz11]) For t q : N ! N , a language A is weakly (t q )-stochastic if, for all B C 2 DTIME(t)=ADV(q) such that jC=n j (n) for all suciently large n, lim j(A 4jCB ) \j C=nj = 12 : n!1 =n
We write WS(t q ) for the set of all weakly (t q )-stochastic languages. The following result resembles the weak stochasticity theorems proved by Lutz and Mayordomo 10] and Lutz 11], but gives a more careful upper bound on the time complexity of the martingale.
Theorem 2.4 (Weak Stochasticity Theorem) Assume that 2 R satisfy 1 1 > 0, and > . Then there is an exact 2(log n) -martingale d such that S 1d] WS(2n n 2n ) = C:
Proof. Assume the hypothesis, and assume without loss of generality that 0 2 Q . Fix 0 0 00 2 Q such that < 0 and 0 < 00 < 0 < . Let U 2 DTIME(2n ) be a language that is universal for DTIME(2n ) DTIME(2n ) in the following sense. For each i 2 N , let Ci = fx 2 f0 1g j < si 0x >2 U g Di = fx 2 f0 1g j < si 1x >2 U g: Then DTIME(2n ) DTIME(2n ) = f(Ci Di )ji 2 N g. Dene a function d0 : N 3 f0 1g ! Q \ 0 1) as follows. If k is not a power of 2, then 0 dijk (w) = 0. Otherwise, if k = 2n , where n 2 N , then
d0ijk (w) =
X
yz2f01g
n
Pr(Yijkyz jCw )
where the sets Yijkyz are dened as follows. If j(Ci =y)=n j < 2n , then Yijkyz = . If j(Ci =y)=nj 2n, then Yijkyz is the set of all A 2 C such that
j (A 4 (Di =z)) \ (Ci=y)=n j ; 1 1 : j(Ci =y)=nj 2 j + 1
The denition of conditional probability immediately implies that, for each i j k 2 N , the 0 0 n function dijk is a martingale. Since U 2 DTIME(2 ) and 0 < 00 , the time required 00 to compute each Pr(Yijkyz jCw ) using binomial coecients is at most O(2(log(i+j +k)) ) 00 steps, so the time required to compute d0ijk (w) is at most O((2n + 1)2 2(log(i+j +k)) ) = 0 0 O(2(log(i+j +k)) ) steps. Thus d0 is exactly 2(log n) -time-computable. 8
As in 10] and 11], the Cherno bound tells us that, for all i j n 2 N and y z 2 f0 1gn , writing k = 2n, Pr(Yijkyz ) 2e;k =2(j +1)2 whence
d0ijk ()
(2n + 1)2 2e;k =2(j +1)2
< e2n
+3;k =2(j +1)2 :
Let a = d 1 e, let = 4 , and x k0 2 N such that
k2 > k + 2(log k) + 3 for all k k0 . Dene g : N ! N by
g(j ) = 4a (j + 1)4a + k0 for all j 2 N . Then g is a polynomial and, for all i j n 2 N , writing k = 2n , 8 < k = k 2 k 2 k g (j ) ) : > 4a (j + 1)4a ]2(k + 2(log k) + 3)
2(j + 1)2 (k + 2n + 3)
) d0ijk () < e;k : 1 X 0
It follows by Lemma 2.2 that the series
k=0
dijk (), for i j 2 N , are uniformly p-convergent.
Since 1 < 0 < , it follows by Theorem 2.3 that there is an exact 2(log n) -martingale d such that 1 1 \ 1 1
i=0 j =0 t=0 k=t
S 1 d0ijk ] S 1 d]:
(19)
Now assume that A 62 WS(2n n 2n ). Then, by the denition of weak stochasticity, we can x i j 2 N , functions h1 h2 2 ADV(n ), and an innite set J N such that, for all n 2 J , A 2 Yijkh1(n)h2 (n) , where k = 2n . For each n 2 J , then, there is a prex w v A such that Cw Yijkh1(n)h2 (n), whence
d0ijk(w) Pr(Yijkh1(n)h2 (n) jCw ) = 1 i.e., A 2 S 1 d0ijk ]. This argument shows that 1 1 \ 1 1
i=0 j =0 t=0 k=t
S 1 d0ijk ] WS(2n n 2n ) = C:
It follows by (19) that
S 1d] WS(2n n 2n ) = C:
9
The technique of martingale dilation was introduced by Ambos-Spies, Terwijn, and Zheng 3]. It has also been used by Juedes and Lutz8] and generalized considerably by Breutzmann and Lutz 6]. We use the notation of 8] here. The restriction of a string w = b0 b1 bn;1 2 f0 1g to a language A f0 1g is the string wA obtained by concatenating the successive bits bi for which si 2 A. If f : f0 1g ! f0 1g is strictly increasing and d is a martingale, then the f -dilation of d is the function f^d : f0 1g ! 0 1) dened by
f^d(w) = d(wrange(f )) for all w 2 f0 1g .
Lemma 2.5 (Martingale Dilation Lemma - Ambos-Spies, Terwijn, and Zheng3]) If f : f0 1g ! f0 1g is strictly increasing and d is a martingale, then f^d is also a martingale. Moreover, for every language A 2 f0 1g , if d succeeds on f ;1(A), then f^d succeeds on A. Finally, we summarize the most basic ideas of resource-bounded measure in E and E2 . A p-martingale is a martingale that is, for some kk 2 N , an nk -martingale. A p2 -martingale is a martingale that is, for some k 2 N , a 2(log n) -martingale.
Denition (Lutz 9]) 1. A set X of languages has p-measure 0, and we write p (X ) = 0, if there is a pmartingale d such that X S 1d]. 2. A set X of languages has p2 -measure 0, and we write p2 (X ) = 0, if there is a p2 -martingale d such that X S 1 d]. 3. A set X of languages has measure 0 in E, and we write (X jE) = 0, if p (X \ E) = 0. 4. A set X of languages has measure 0 in E2 , and we write (X jE2 ) = 0, if p2 (X \E2 ) = 0. 5. A set X of languages has measure 1 in E, and we write (X jE) = 1, if (X c jE) = 0. In this case, we say that X contains almost every element of E. 6. A set X of languages has measure 1 in E2 , and we write (X jE2 ) = 1, if (X c jE2 ) = 0. In this case, we say that X contains almost every element of E2 . 7. The expression (X jE) 6= 0 means that X does not have measure 0 in E. Note that this does not assert that \(X jE)" has some nonzero value. Similarly, the expression (X jE2 ) 6= 0 means that X does not have measure 0 in E2 .
It is shown in 9] that these denitions endow E and E2 with internal measure structure. This structure justies the intuition that, if (X jE) = 0, then X \ E is a negligibly small subset of E (and similarly for E2 ).
10
3 Results The key to our main theorem is the following lemma, which says that languages that are P -reducible to non-dense languages cannot be very stochastic. n ;T
Lemma 3.1 (Main Lemma) For all real numbers < 1 and > 1 + , Pn ;T (DENSEc ) \ WS(2n n 2 2 ) = : Proof. Let < 1 and > 1 + , and assume without loss of generality that and are rational. Let A 2 Pn ;T (DENSEc ). It suces to show that A is not weakly (2n n 2 2 )n
n
stochastic. Since A 2 Pn ;T (DENSEc), there exist a non-dense language S , a polynomial q(n), and a q(n)-time-bounded oracle Turing machine M such that A = L(M S ) and, for every x 2 f0 1g and B f0 1g , M makes exactly bjxj cqueries (all distinct) on input x with oracle B . Call these queries QB (x 1) : : : QB (x bjxj c) in the order in which M makes them. For each B 2 f0 1g and n 2 N , dene an equivalence relation Bn on f0 1gq(n) by
u Bn v , (8w)u w v ) w 2 B ] = u 2 B ] ] and an equivalence relation Bn on f0 1gn by
x Bn y , (8i)1 i n ) QB (x i) Bn QB (y i)]: Note that Bn has at most 2jBq(n) j+1 equivalence classes, so Bn has at most (2jBq(n) j+ 1)n equivalence classes. Let = 1;2 , and let J be the set of all n 2 N for which the following three conditions hold. (i) 2jSq(n) j + 1 2n : (ii) n + n2 : (iii) n (2n + 1) n : Since + < 1 and > 1 + , conditions (ii) and (iii) hold for all suciently large n. Since > 0 and S is not dense, condition (i) holds for innitely many n. Thus the set J is innite. Dene an advice function h : N ! f0 1g as follows. If n 62 J , then h(n) = . If n 2 J , then let Dn be a maximum-cardinality equivalence class of the relation Sn. For each 1 i bn c, x strings yni zni 2 Dn such that, for all x 2 Dn ,
QS (yni i) QS (x i) QS (zni i): Let
h1 (n) h2 (n) h3 (n) h(n)
= = = =
yn1 ynbn c zn1 znbn c QS (yn1 1) 2 S ] QS (ynbn c bn c) 2 S ] h1 (n)h2 (n)h3 (n):
11
Note that jh(n)j = bn c(2n + 1) n for all n 2 J , so h 2 ADV(n ). For each n 2 N , let t = bn c, and let Cn be the set of all coded pairs
< x y1 yt z1 zt b1 bt > such that x y1 : : : yt z1 : : : zt 2 f0 1gn , b1 : : : bt 2 f0 1g, and, for each 1 i t, Qb1 bt (yi i) Qb1 bt (x i) Qb1 bt (zi i) where Qb1 bt (w i) denotes the ith query of M on input w when the successive oracle answers are b1 : : : bt . Let Bn be the set of all such coded pairs in Cn such that M accepts on input x when the successive oracle answers are b1 : : : bt . Finally, dene the languages
B = f< x v >j v = or < x v >2 Bjxjg C = f< x v >j v = or < x v >2 Cjxjg:
It is clear that B C 2 DTIME(2n ). Also, by our construction of these sets and the advice function h, for each n 2 N , we have
and
if n 2 J f0 1gn if n 62 J
(C=h)=n =
Dn
(B=h)=n =
A \ Dn if n 2 J : f0 1gn if n 62 J
For each n 2 J , if (n) is the number of equivalence classes of Sn, then
(n) (2jSq(n) j + 1)n so
n
(2n )n = 2n+
jDnj 2(n) 2n;n + 2 2 :
n
It follows that j(C=h)=n j 2 n2 for all n 2 N . Finally, for all n 2 J ,
(A 4 (B=h)) \ (C=h)=n = (A 4 (A \ Dn )) \ Dn = : Since J is innite, it follows that j(A 4 (B=h)) \ (C=h)=n j 6! 1 j(C=h)=n j 2 as n ! 1. Since B C 2 DTIME(2nn), h 2 ADV(n ), and jC=n j 2n2 for all n 2 N , this shows that A is not weakly (2n n 2 2 )-stochastic. We now prove our main result. 12
Theorem 3.2 (Main Theorem) For every real number < 1, p(Pn
c c T (DENSE )) = p2 (Pn ;T (DENSE )) = 0: = 3+2 , so that 1 + < < 2. By Theorem
=2 ;
Proof. Let 2 < 1, and let
exact
2(log n)
-martingale d such that
2.4, there is an
S 1 d] WS(2n n 2 2 ) = C: n
By Lemma 3.1, we then have Pn ;T (DENSEc ) S 1 d]: Since d is a p2 -martingale, this implies that p2 (Pn ;T (DENSEc)) = 0. Dene f : f0 1g ! f0 1g by
f (x) = 0jxj2;jxj;11x: Then f is strictly increasing, so f^d, the f -dilation of d, is a martingale. The time required to compute f^d(w) is O(jwj2 + 2(log jw0j)2 ) steps, where w0 = wrange(f ). (This allows O(jwj2 ) steps to compute w0 and then 0 j)2 (log j w O(2 ) steps to compute d(w0 ).) Now jw0 j is bounded above by the number of strings x such that jxj2 jsjwjj = blog(1 + jwj)c, so p jw0 j < 21+ log(1+jwj): Thus the time required to compute f^d(w) is
plog(1+jwj) )2
O(jwj2 + 2(1+
) = O(jwj2 )
steps, so f^d is an n2 -martingale. Now let A 2 Pn=2 ;T (DENSEc ). Then f ;1 (A) 2 Pn ;T (DENSEc) S 1 d], so A 2 1 S f^d] by Lemma 2.5. This shows that Pn=2 ;T(DENSEc) S 1 f^d]. Since f^d is an n2 -martingale, it follows that p(Pn=2 ;T (DENSEc)) = 0. We now develop a few consequences of the Main Theorem. The rst is immediate.
Corollary 3.3 For every real number < 1, (Pn 2 ;T (DENSEc) j E) = (Pn
c T (DENSE ) j E2 ) = 0:
;
=
The following result on the density of weakly complete (or weakly hard) languages now follows immediately from Corollary 3.3.
Corollary 3.4 For every real number < 1, every language that is weakly
for E or weakly Pn;T -hard for E2 is dense.
13
P n=2 ;T -hard
Our nal two corollaries concern consequences of the strong hypotheses p (NP) 6= 0 and p2 (NP) 6= 0. The relative strengths of these hypotheses are indicated by the known implications
(NP j E) 6= 0 ) (NP j E2 ) 6= 0 , p2 (NP) 6= 0 ) p(NP) 6= 0 ) P 6= NP: (The leftmost implication was proven by Juedes and Lutz8]. The remaining implications follow immediately from elementary properties of resource-bounded measure.)
Corollary 3.5 Let < 1. If p (NP) 6= 0, then every language that is Pn 2;T-hard for NP is dense. If p2 (NP) = 6 0, then every language that is Pn ;T-hard for NP is dense. =
We conclude by considering the densities of languages to which SAT can be adaptively reduced. Denition A function g : N ! N is subradical if log g(n) = o(log n). p It is easy to see that a function g is subradical if and only if, for all k > 0, g(n) = o( k n). (This is the reason for the name \subradical.") Subradical functions include very slowgrowing functions such as log n and (log n)5 , as well as more rapidly growing functions such 0:99 (log n ) as 2 .
Corollary 3.6 If p(NP) 6= 0, g : N ! N is subradical, and SAT dense.
P g(n);T
H , then H is
Proof. Assume the hypothesis. Let A 2 NP. Then there is a Pm-reduction f of A to SAT. Fix a polynomial q(n) such that, for all x 2 f0 1g , jf (x)j q(jxj). Composing f with the Pg(n);T -reduction of SAT to H that we have assumed to exist then gives a Pg(q(n));T reduction of A to H . Since g is subradical, log g(q(n)) = o(log q(n)) = o(log n), so for all log n suciently large n, g(q(n)) 2 4 = n 41 . Thus A P 41 H . n ;T The above argument shows that H is P 14 -hard for NP. Since we have assumed n ;T p (NP) 6= 0, it follows by Corollary 3.5 that H is dense.
To put the matter dierently, Corollary 3.6 tells us that if SAT is polynomial-time reducible to a non-dense language with at most 2(log n)0:99 adaptive queries, then NP has measure 0 in E and in E2 .
4 Questions As noted in the introduction, the relationships between weak hardness notions for E and E2 under reducibilities such as PT Pn ;T , and Pn ;tt remain to be resolved. Our main theorem also leaves open the question whether Pn;T -hard languages for E must be dense when 1 < 1. We are in the curious situation of knowing that the classes P 0:99 (DENSEc ) n ;tt 2 and Pn0:49 ;T (DENSEc) have p-measure 0, but not knowing whether Pn0:50 ;T (DENSEc ) has p-measure 0. Indeed, at this time we cannot even prove that E 6 Pn0:50 ;T (SPARSE). Further progress on this matter would be illuminating. 14
References 1] K. Ambos-Spies, Randomness, relativizations, and polynomial reducibilities. In Proceedings of the First Structure in Complexity Theory Conference, pp. 23-34. SpringerVerlag, 1986. 2] K. Ambos-Spies and E. Mayordomo, Resource-bounded measure and randomness, in A. Sorbi(ed.), Complexity, Logic, and Recursion Theory, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, to appear. 3] K. Ambos-Spies, S. A. Terwijn, and X. Zheng, Resource-bounded randomness and weakly complete problems. Theoretical Computer Science. To appear. 4] C. H. Bennett and J. Gill, Relative to a random oracle A, PA 6= NP 6= co ; NPA with probability 1, SIAM Journal on Computing 10 (1981), pp.96-113. 5] L. Berman and J. Hartmanis, On isomorphism and density of NP and other complete sets. SIAM Journal on Computing, 6:305-322, 1977. 6] J. M. Breutzmann and J. H. Lutz, Equivalence of measures of complexity classes, Proceedings of the Fourteenth Symposium on Theoretical Aspects of Computer Science, Springer-Verlag, 1997, to appear. 7] B. Fu, With quasi-linear queries, EXP is not polynomial time Turing reducible to sparse sets. SIAM Journal on Computing, 24:1082-1090, 1995. 8] D. W. Juedes and J. H. Lutz, Weak completeness in E and E2 . Theoretical Computer Science, 143:149-158, 1995. 9] J. H. Lutz, Almost everywhere high nonuniform complexity, Journal of Computer and System Sciences, 19:1100-1131, 1990. 10] J. H. Lutz and E. Mayordomo, Measure stochasticity, and the density of hard languages. SIAM Journal on Computing, 23:762-779, 1994. 11] J. H. Lutz, One-way functions and balanced NP. Theoretical Computer Science, to appear. 12] J. H. Lutz, The quantitative structure of exponential time, in L. A. Hemaspaandra and A. L. Selman (eds.), Complexity Theory Retrospective II, Springer-Verlag, 1996, to appear. A preliminary version appears in Proceedings of The Eighth Annual Structure in Complexity Theory Conference (San Diego, CA, May 18-21, 1993), IEEE Computer Society Press, 1993, pp. 158-175. 13] S. R. Mahaney, Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis. Journal of Computer and System Sciences, 25:130-143, 1982. 14] E. Mayordomo, Contributions to the Study of Resource-Bounded Measure. PhD thesis, Universitat Polit ecnica de Catalunya, Barcelona, Spain, 1994. 15
15] A. R. Meyer, 1977. Reported in 5]. 16] M. Ogiwara and O. Watanabe, On polynomial bounded truth-table reducibility of NP sets to sparse sets. SIAM Journal on Computing, 20(3):471-483, June 1991. 17] O. Watanabe, On the Structure of Intractable Complexity Classes. PhD thesis, Tokyo Institute of Technology, 1987.
16
Abstract
Given a real number < 1, every language that is weakly Pn =2 ;T -hard for E or weakly Pn ;T -hard for E2 is shown to be exponentially dense. This simultaneously strengthens results of Lutz and Mayordomo(1994) and Fu(1995).
1 Introduction In the mid-1970's, Meyer15] proved that every Pm-complete language for exponential time|in fact, every Pm-hard language for exponential time|is dense. That is, E 6 Pm(DENSEc )
(1)
where E = DTIME(2linear), DENSE is the class of all dense languages, DENSEc is the complement of DENSE, and Pm(DENSEc ) is the class of all languages that are Pm-reducible to non-dense languages. (A language A 2 f0 1g is dense if there is a real number > 0 such that jAn j > 2n for all suciently large n, where An = A \ f0 1gn .) Since that time, a major objective of computational complexity theory has been to extend Meyer's result from Pm-reductions to PT -reductions, i.e., to prove that every PT -hard language for E is dense. That is, the objective is to prove that E 6 PT (DENSEc )
(2)
where PT (DENSEc) is the class of all languages that are PT-reducible to non-dense languages. The importance of this objective derives largely from the fact (noted by Meyer15]) that the class PT (DENSEc) contains all languages that have subexponential circuit-size complexity. (A language A f0 1g has subexponential circuit-size complexity if, for every real number > 0, for every suciently large n, there is an n-input, 1-output Boolean This research was supported in part by National Science Foundation Grant CCR-9157382, with matching funds from Rockwell International, Microware Systems Corporation, and Amoco Foundation.
1
circuit that decides that the set A=n = A \ f0 1gn and has fewer than 2n gates. Otherwise, we say that A has exponential circuit-size complexity.) Thus a proof of (2) would tell us that E contains languages with exponential circuit-size complexity, thereby answering a major open question concerning the relationship between (uniform) time complexity and (nonuniform) circuit-size complexity. Of course (2) also implies the more modest, but more famous conjecture, that E 6 PT (SPARSE)
(3)
where SPARSE is the class of all sparse languages. (A language A f0 1g is sparse if there is a polynomial q(n) such that jAn j q(n) for all n 2 N .) As noted by Meyer15], the class PT (SPARSE) consists precisely of all languages that have polynomial circuit-size complexity, so (3) asserts that E contains languages that do not have polynomial circuit-size complexity. Knowing (1) and wanting to prove (2), the natural strategy has been to prove results of the form E 6 Pr (DENSEc ) for successively larger classes Pr (DENSEc) in the range Pm(DENSEc ) Pr (DENSEc) PT (DENSEc ): The rst major step beyond (1) in this program was the proof by Watanabe17] that E 6 PO(log n);tt (DENSEc )
(4)
i.e., that every language that is PO(log n);tt -hard for E is dense. The next big step was the proof by Lutz and Mayordomo10] that, for every real number < 1, E 6 Pn ;tt (DENSEc ):
(5)
This improved Watanabe's result from O(log n) truth-table (i.e., nonadaptive) queries to n such queries for arbitrarily close to 1 (e.g., to n0:99 truth-table queries). Moreover, Lutz and Mayordomo10] proved (5) by rst proving the stronger result that for all < 1,
p(Pn
tt (DENSE
c )) = 0
(6) which implies that every language that is weakly Pn;tt -hard for E or for E2 = DTIME(2poly ) is dense. (A language A is weakly Pr -hard for a complexity class C if (Pr (A) j C ) 6= 0, i.e., if Pr (A) \ C is a nonnegligible subset of C in the sense of the resource-bounded measure developed by Lutz9]. A language A is weakly Pr -complete for C if A 2 C and A is weakly P -hard for C . See 12] or 2] for a survey of resource-bounded measure and weak comr pleteness.) The set of weakly Pn ;tt -hard languages for E is now known to have p-measure 1 3], hence measure 1 in the class C of all languages, while the set of all Pn ;tt -hard languages for E has measure 0 unless E BPP 4, 1]. Thus, if E 6 BPP (which is generally conjectured to be true), almost every language is weakly Pn;tt -hard, but not Pn;tt -hard, for E, so the result of Lutz and Mayordomo 10] is much more general than the fact that every Pn ;tt -hard language for E is dense. ;
2
A word on the relationship between hardness notions for E and E2 is in order here. It is well known that a language is Pm-hard for E if and only if it is Pm-hard for E2 this is because E2 = Pm(E). The same equivalence holds for PT -hardness. It is also clear that every language that is Pn ;tt -hard for E2 is Pn;tt -hard for E. However, it is not generally the case that Pm(Pn ;tt (A)) = Pn ;tt (A), so it may well be the case that a language can be Pn;tt -hard for E, but not for E2 . These same remarks apply to Pn ;T -hardness. The relationship between weak hardness notions for E and E2 is somewhat dierent. Juedes and Lutz 8] have shown that weak Pm-hardness for E implies weak Pm-hardness for E2 , and their proof of this fact also works for weak PT -hardness. However, Juedes and Lutz 8] also showed that weak Pm-hardness for E2 does not generally imply weak Pm-hardness for E, and it is reasonable to conjecture (but has not been proven) that the same holds for weak PT -hardness. We further conjecture that the notions of weak Pn ;tt -hardness for E and weak Pn ;tt -hardness E2 are incomparable, and similarly for weak Pn ;T -hardness. In any case, (6) implies that, for every < 1, every language that is weakly Pn ;tt -hard for either E or E2 is dense. Shortly after, but independently of 10], Fu7] used very dierent techniques to prove that, for every < 1, E 6 Pn=2 ;T (DENSEc)
(7)
E2 6 Pn;T (DENSEc):
(8)
and That is, every language that is Pn=2 ;T -hard for E or Pn ;T -hard for E2 is dense. These results do not have the measure-theoretic strength of (6), but they are a major improvement over previous results on the densities of hard languages in that they hold for Turing reductions, which have adaptive queries. In the present paper, we prove results which simultaneously strengthen results of Lutz and Mayordomo10] and the results of Fu7]. Specically, we prove that, for every < 1,
p (Pn
=2 ;
c T (DENSE )) = 0
(9)
c )) = 0:
(10)
and
p2 (Pn
T (DENSE
;
These results imply that every language that is weakly Pn=2 ;T -hard for E or weakly P n=2 ;T -hard for E2 is dense. The proof of (9) and (10) is not a simple extension of the proof in 10] or the proof in 7], but rather combines ideas from both 10] and 7] with the martingale dilation technique introduced by Ambos-Spies, Terwijn, and Zheng 3]. Our results also show that the strong hypotheses p (NP) 6= 0 and p2 (NP) 6= 0 (surveyed in 12] and 2]) have consequences for the densities of adaptively hard languages for NP. Mahaney 13] proved that P 6= NP ) NP 6 Pm(SPARSE) 3
(11)
and Ogiwara and Watanabe 16] improved this to P 6= NP ) NP 6 Pbtt (SPARSE):
(12)
That is, if P 6= NP, then no sparse language can be Pbtt -hard for NP. Lutz and Mayordomo 10] used (6) to obtain a stronger conclusion from a stronger hypothesis, namely, for all < 1,
p(NP) 6= 0 ) NP 6 Pn
c ):
(13)
c T (DENSE )
(14)
c T (DENSE ):
(15)
tt (DENSE
;
By (9) and (10), we now have, for all < 1,
p(NP) 6= 0 ) NP 6 Pn
=2 ;
and
p2 (NP) 6= 0 ) NP 6 Pn
;
Thus, if p (NP) 6= 0, then every language that is Pn0:49 ;T -hard for NP is dense. If p2 (NP) 6= 0, then every language that is Pn0:99 ;T -hard for NP is dense.
2 Preliminaries The Boolean value of a condition, is ] =
1 if 0 if not :
The standard enumeration of f0 1g is s0 = s1 = 0 s2 = 1 s3 = 00 : : : This enumeration induces a total ordering of f0 1g which we denote by c. The success set of a martingale d is the set
S 1d] = fA 2 Cjd succeeds on Ag: The unitary success set of d is
S 1d] =
w2f01g d(w)1
Cw :
The following result was proven by Juedes and Lutz 8] and independently by Mayordomo 14].
Lemma 2.1 (Exact Computation Lemma) Let t : N ! N be nondecreasing with t(n) n2. Then, for every t(n)-martingale d, there is an exact n t(2n + 2)-martingale de such that S 1 d] S 1de]. A sequence
1 X
k=0
ajk
(j = 0 1 2 : : : )
of series of terms ajk 2 0 1) is uniformly p-convergent if there is a polynomial m : N 2 ! N 1 X ajk 2;r , where we write mj (r) = m(j r). The following such that, for all j r 2 N , k=mj (r)
sucient condition for uniform p-convergence is easily veried by routine calculus. 5
Lemma 2.2 Let ajk 2 0 1) for all j k 2 N . If there exist a real number > 0 and a polynomial g : N ! N such that ajk e;k for all j k 2 N with k g(j ), then the series
1 X
k=0
ajk (j = 0 1 2 : : : ) are uniformly p-convergent.
A uniform, resource-bounded generalization of the classical rst Borel-Cantelli lemma was proved by Lutz 9]. Here we use the following precise variant of this result.
Theorem 2.3 Let e 2 R with 1 e, and let d : N N f0 1g ! Q \ 0 1) be an exactly 2(log n) -time-computable function with the following two properties. (i) For each j k 2 N , the function djk dened by djk (w) = d(j k w) is a martingale. (ii) The series
1 X
k=0
djk (j = 0 1 2 : : : ) are uniformly p-convergent.
Then there is an exact 2(log n)e -martingale e such that 1 \ 1 1
j =0 t=0 k=t
S 1 djk ] S 1 de]:
Proof (sketch).0 Assume the hypothesis, and x 0 2 Q such that < 0 < e. Since0 n 2(log(2n+2)) = o(2(log n) e ), it suces by Lemma 2.1 to show that there is a 2(log n)
martingale d0 such that
1 \ 1 1
j =0 t=0 k=t
Fix a polynomial m :
N2
S 1djk ] S 1 d0 ]:
! N testifying that the series
uniformly p-convergent, and dene
d (w) = 0
1 X 1 1 X X
j =0 t=0 k=mj (2t)
(16) 1 X
k=0
2t;j djk (w)
for all w 2 f0 1g . Then, for each w 2 f0 1g , 1 X 1 1 X X
d0 (w)
j =0 t=0 k=mj (2t)
2jwj =
1 X
j =0 2jwj+2
2;j
6
1 X
t=0
2t;j +jwjdjk () 2t 2;2t
djk (j = 0 1 2 : : : ) are
so d0 : f0 1g ! 0 1). It is clear by linearity that d0 is a martingale. To see that (16) 1 \ 1 1 holds, assume that A 2 S 1 djk ], and let c 2 N be arbitrary. Then there exist j =0 t=0 k=t
j 2 N and k mj (2j + 2c) such that A 2 S 1 djk ]. Fix w v A such that djk (w) 1. Then d0 (w) 2c+j ;j djk (w) 2c . Since c is arbitrary here, it follows that A 2 S 1 d0 ], conrming
(16). 0 To see that d0 is 2(log n) -time-computable, dene dA dB dC : N f0 1g ! 0 1) as follows, using the abbreviation s = r + jwj + 2.
dA(r w) = dB (r w) = dC ( r w ) =
s X 1 1 X X
j =0 t=0 k=mj (2t) s X 2s X 1 X
2t;j djk (w) 2t;j djk (w)
j =0 t=0 k=mj (2t) 2 +4s+t) s X 2s mj (2sX X j =0 t=0 k=mj (2t)
2t;j djk (w)
(17)
For all r 2 N and w 2 f0 1g , it is clear that dC (r w) dB (r w) dA(r w) d0(w) and it is routine to verify the inequalities d0 (w) ; dA (r w) 2;(r+1) dA(r w) ; dB (r w) 2;(r+2) dB (r w) ; dC (r w) 2;(r+2) whence we have d0 (w) ; 2;r dC (r w) d0(w) (18) for all r 2 N and w 2 f0 1g . Using formula (17), the time required to compute dC (r w) exactly is no greater than O((s + 1)(2s + 1)m(s 2s2 + 4s + 2s)2(log n) ) = O(q(n) 2(log n) ) 0
where n = r + jwj and q is0 a polynomial. Since q(n) 2(log n) = o(2(log n) ),0 it follows that dC (r w) is exactly 2(log n) -time-computable. By (18), then, d0 is a 2(log n) -martingale. The proof of our main theorem uses the techniques of weak stochasticity and martingale dilation, which we briey review here. As usual, an advice function is a function h : N ! f0 1g . Given a function q : N ! N , we write ADV(q) for the set of all advice functions h such that jh(n)j q(n) for all n 2 N . Given a language B and an advice function h, we dene the language B=h = fx 2 f0 1g j< x h(jxj) >2 B g 7
where < > is a standard string-pairing function, e.g., < x y >= 0jxj1xy. Given functions t q : N ! N , we dene the advice class DTIME(t)=ADV(q) = fB=h j B 2 DTIME(t) and h 2 ADV(q)g:
Denition (Lutz and Mayordomo10], Lutz11]) For t q : N ! N , a language A is weakly (t q )-stochastic if, for all B C 2 DTIME(t)=ADV(q) such that jC=n j (n) for all suciently large n, lim j(A 4jCB ) \j C=nj = 12 : n!1 =n
We write WS(t q ) for the set of all weakly (t q )-stochastic languages. The following result resembles the weak stochasticity theorems proved by Lutz and Mayordomo 10] and Lutz 11], but gives a more careful upper bound on the time complexity of the martingale.
Theorem 2.4 (Weak Stochasticity Theorem) Assume that 2 R satisfy 1 1 > 0, and > . Then there is an exact 2(log n) -martingale d such that S 1d] WS(2n n 2n ) = C:
Proof. Assume the hypothesis, and assume without loss of generality that 0 2 Q . Fix 0 0 00 2 Q such that < 0 and 0 < 00 < 0 < . Let U 2 DTIME(2n ) be a language that is universal for DTIME(2n ) DTIME(2n ) in the following sense. For each i 2 N , let Ci = fx 2 f0 1g j < si 0x >2 U g Di = fx 2 f0 1g j < si 1x >2 U g: Then DTIME(2n ) DTIME(2n ) = f(Ci Di )ji 2 N g. Dene a function d0 : N 3 f0 1g ! Q \ 0 1) as follows. If k is not a power of 2, then 0 dijk (w) = 0. Otherwise, if k = 2n , where n 2 N , then
d0ijk (w) =
X
yz2f01g
n
Pr(Yijkyz jCw )
where the sets Yijkyz are dened as follows. If j(Ci =y)=n j < 2n , then Yijkyz = . If j(Ci =y)=nj 2n, then Yijkyz is the set of all A 2 C such that
j (A 4 (Di =z)) \ (Ci=y)=n j ; 1 1 : j(Ci =y)=nj 2 j + 1
The denition of conditional probability immediately implies that, for each i j k 2 N , the 0 0 n function dijk is a martingale. Since U 2 DTIME(2 ) and 0 < 00 , the time required 00 to compute each Pr(Yijkyz jCw ) using binomial coecients is at most O(2(log(i+j +k)) ) 00 steps, so the time required to compute d0ijk (w) is at most O((2n + 1)2 2(log(i+j +k)) ) = 0 0 O(2(log(i+j +k)) ) steps. Thus d0 is exactly 2(log n) -time-computable. 8
As in 10] and 11], the Cherno bound tells us that, for all i j n 2 N and y z 2 f0 1gn , writing k = 2n, Pr(Yijkyz ) 2e;k =2(j +1)2 whence
d0ijk ()
(2n + 1)2 2e;k =2(j +1)2
< e2n
+3;k =2(j +1)2 :
Let a = d 1 e, let = 4 , and x k0 2 N such that
k2 > k + 2(log k) + 3 for all k k0 . Dene g : N ! N by
g(j ) = 4a (j + 1)4a + k0 for all j 2 N . Then g is a polynomial and, for all i j n 2 N , writing k = 2n , 8 < k = k 2 k 2 k g (j ) ) : > 4a (j + 1)4a ]2(k + 2(log k) + 3)
2(j + 1)2 (k + 2n + 3)
) d0ijk () < e;k : 1 X 0
It follows by Lemma 2.2 that the series
k=0
dijk (), for i j 2 N , are uniformly p-convergent.
Since 1 < 0 < , it follows by Theorem 2.3 that there is an exact 2(log n) -martingale d such that 1 1 \ 1 1
i=0 j =0 t=0 k=t
S 1 d0ijk ] S 1 d]:
(19)
Now assume that A 62 WS(2n n 2n ). Then, by the denition of weak stochasticity, we can x i j 2 N , functions h1 h2 2 ADV(n ), and an innite set J N such that, for all n 2 J , A 2 Yijkh1(n)h2 (n) , where k = 2n . For each n 2 J , then, there is a prex w v A such that Cw Yijkh1(n)h2 (n), whence
d0ijk(w) Pr(Yijkh1(n)h2 (n) jCw ) = 1 i.e., A 2 S 1 d0ijk ]. This argument shows that 1 1 \ 1 1
i=0 j =0 t=0 k=t
S 1 d0ijk ] WS(2n n 2n ) = C:
It follows by (19) that
S 1d] WS(2n n 2n ) = C:
9
The technique of martingale dilation was introduced by Ambos-Spies, Terwijn, and Zheng 3]. It has also been used by Juedes and Lutz8] and generalized considerably by Breutzmann and Lutz 6]. We use the notation of 8] here. The restriction of a string w = b0 b1 bn;1 2 f0 1g to a language A f0 1g is the string wA obtained by concatenating the successive bits bi for which si 2 A. If f : f0 1g ! f0 1g is strictly increasing and d is a martingale, then the f -dilation of d is the function f^d : f0 1g ! 0 1) dened by
f^d(w) = d(wrange(f )) for all w 2 f0 1g .
Lemma 2.5 (Martingale Dilation Lemma - Ambos-Spies, Terwijn, and Zheng3]) If f : f0 1g ! f0 1g is strictly increasing and d is a martingale, then f^d is also a martingale. Moreover, for every language A 2 f0 1g , if d succeeds on f ;1(A), then f^d succeeds on A. Finally, we summarize the most basic ideas of resource-bounded measure in E and E2 . A p-martingale is a martingale that is, for some kk 2 N , an nk -martingale. A p2 -martingale is a martingale that is, for some k 2 N , a 2(log n) -martingale.
Denition (Lutz 9]) 1. A set X of languages has p-measure 0, and we write p (X ) = 0, if there is a pmartingale d such that X S 1d]. 2. A set X of languages has p2 -measure 0, and we write p2 (X ) = 0, if there is a p2 -martingale d such that X S 1 d]. 3. A set X of languages has measure 0 in E, and we write (X jE) = 0, if p (X \ E) = 0. 4. A set X of languages has measure 0 in E2 , and we write (X jE2 ) = 0, if p2 (X \E2 ) = 0. 5. A set X of languages has measure 1 in E, and we write (X jE) = 1, if (X c jE) = 0. In this case, we say that X contains almost every element of E. 6. A set X of languages has measure 1 in E2 , and we write (X jE2 ) = 1, if (X c jE2 ) = 0. In this case, we say that X contains almost every element of E2 . 7. The expression (X jE) 6= 0 means that X does not have measure 0 in E. Note that this does not assert that \(X jE)" has some nonzero value. Similarly, the expression (X jE2 ) 6= 0 means that X does not have measure 0 in E2 .
It is shown in 9] that these denitions endow E and E2 with internal measure structure. This structure justies the intuition that, if (X jE) = 0, then X \ E is a negligibly small subset of E (and similarly for E2 ).
10
3 Results The key to our main theorem is the following lemma, which says that languages that are P -reducible to non-dense languages cannot be very stochastic. n ;T
Lemma 3.1 (Main Lemma) For all real numbers < 1 and > 1 + , Pn ;T (DENSEc ) \ WS(2n n 2 2 ) = : Proof. Let < 1 and > 1 + , and assume without loss of generality that and are rational. Let A 2 Pn ;T (DENSEc ). It suces to show that A is not weakly (2n n 2 2 )n
n
stochastic. Since A 2 Pn ;T (DENSEc), there exist a non-dense language S , a polynomial q(n), and a q(n)-time-bounded oracle Turing machine M such that A = L(M S ) and, for every x 2 f0 1g and B f0 1g , M makes exactly bjxj cqueries (all distinct) on input x with oracle B . Call these queries QB (x 1) : : : QB (x bjxj c) in the order in which M makes them. For each B 2 f0 1g and n 2 N , dene an equivalence relation Bn on f0 1gq(n) by
u Bn v , (8w)u w v ) w 2 B ] = u 2 B ] ] and an equivalence relation Bn on f0 1gn by
x Bn y , (8i)1 i n ) QB (x i) Bn QB (y i)]: Note that Bn has at most 2jBq(n) j+1 equivalence classes, so Bn has at most (2jBq(n) j+ 1)n equivalence classes. Let = 1;2 , and let J be the set of all n 2 N for which the following three conditions hold. (i) 2jSq(n) j + 1 2n : (ii) n + n2 : (iii) n (2n + 1) n : Since + < 1 and > 1 + , conditions (ii) and (iii) hold for all suciently large n. Since > 0 and S is not dense, condition (i) holds for innitely many n. Thus the set J is innite. Dene an advice function h : N ! f0 1g as follows. If n 62 J , then h(n) = . If n 2 J , then let Dn be a maximum-cardinality equivalence class of the relation Sn. For each 1 i bn c, x strings yni zni 2 Dn such that, for all x 2 Dn ,
QS (yni i) QS (x i) QS (zni i): Let
h1 (n) h2 (n) h3 (n) h(n)
= = = =
yn1 ynbn c zn1 znbn c QS (yn1 1) 2 S ] QS (ynbn c bn c) 2 S ] h1 (n)h2 (n)h3 (n):
11
Note that jh(n)j = bn c(2n + 1) n for all n 2 J , so h 2 ADV(n ). For each n 2 N , let t = bn c, and let Cn be the set of all coded pairs
< x y1 yt z1 zt b1 bt > such that x y1 : : : yt z1 : : : zt 2 f0 1gn , b1 : : : bt 2 f0 1g, and, for each 1 i t, Qb1 bt (yi i) Qb1 bt (x i) Qb1 bt (zi i) where Qb1 bt (w i) denotes the ith query of M on input w when the successive oracle answers are b1 : : : bt . Let Bn be the set of all such coded pairs in Cn such that M accepts on input x when the successive oracle answers are b1 : : : bt . Finally, dene the languages
B = f< x v >j v = or < x v >2 Bjxjg C = f< x v >j v = or < x v >2 Cjxjg:
It is clear that B C 2 DTIME(2n ). Also, by our construction of these sets and the advice function h, for each n 2 N , we have
and
if n 2 J f0 1gn if n 62 J
(C=h)=n =
Dn
(B=h)=n =
A \ Dn if n 2 J : f0 1gn if n 62 J
For each n 2 J , if (n) is the number of equivalence classes of Sn, then
(n) (2jSq(n) j + 1)n so
n
(2n )n = 2n+
jDnj 2(n) 2n;n + 2 2 :
n
It follows that j(C=h)=n j 2 n2 for all n 2 N . Finally, for all n 2 J ,
(A 4 (B=h)) \ (C=h)=n = (A 4 (A \ Dn )) \ Dn = : Since J is innite, it follows that j(A 4 (B=h)) \ (C=h)=n j 6! 1 j(C=h)=n j 2 as n ! 1. Since B C 2 DTIME(2nn), h 2 ADV(n ), and jC=n j 2n2 for all n 2 N , this shows that A is not weakly (2n n 2 2 )-stochastic. We now prove our main result. 12
Theorem 3.2 (Main Theorem) For every real number < 1, p(Pn
c c T (DENSE )) = p2 (Pn ;T (DENSE )) = 0: = 3+2 , so that 1 + < < 2. By Theorem
=2 ;
Proof. Let 2 < 1, and let
exact
2(log n)
-martingale d such that
2.4, there is an
S 1 d] WS(2n n 2 2 ) = C: n
By Lemma 3.1, we then have Pn ;T (DENSEc ) S 1 d]: Since d is a p2 -martingale, this implies that p2 (Pn ;T (DENSEc)) = 0. Dene f : f0 1g ! f0 1g by
f (x) = 0jxj2;jxj;11x: Then f is strictly increasing, so f^d, the f -dilation of d, is a martingale. The time required to compute f^d(w) is O(jwj2 + 2(log jw0j)2 ) steps, where w0 = wrange(f ). (This allows O(jwj2 ) steps to compute w0 and then 0 j)2 (log j w O(2 ) steps to compute d(w0 ).) Now jw0 j is bounded above by the number of strings x such that jxj2 jsjwjj = blog(1 + jwj)c, so p jw0 j < 21+ log(1+jwj): Thus the time required to compute f^d(w) is
plog(1+jwj) )2
O(jwj2 + 2(1+
) = O(jwj2 )
steps, so f^d is an n2 -martingale. Now let A 2 Pn=2 ;T (DENSEc ). Then f ;1 (A) 2 Pn ;T (DENSEc) S 1 d], so A 2 1 S f^d] by Lemma 2.5. This shows that Pn=2 ;T(DENSEc) S 1 f^d]. Since f^d is an n2 -martingale, it follows that p(Pn=2 ;T (DENSEc)) = 0. We now develop a few consequences of the Main Theorem. The rst is immediate.
Corollary 3.3 For every real number < 1, (Pn 2 ;T (DENSEc) j E) = (Pn
c T (DENSE ) j E2 ) = 0:
;
=
The following result on the density of weakly complete (or weakly hard) languages now follows immediately from Corollary 3.3.
Corollary 3.4 For every real number < 1, every language that is weakly
for E or weakly Pn;T -hard for E2 is dense.
13
P n=2 ;T -hard
Our nal two corollaries concern consequences of the strong hypotheses p (NP) 6= 0 and p2 (NP) 6= 0. The relative strengths of these hypotheses are indicated by the known implications
(NP j E) 6= 0 ) (NP j E2 ) 6= 0 , p2 (NP) 6= 0 ) p(NP) 6= 0 ) P 6= NP: (The leftmost implication was proven by Juedes and Lutz8]. The remaining implications follow immediately from elementary properties of resource-bounded measure.)
Corollary 3.5 Let < 1. If p (NP) 6= 0, then every language that is Pn 2;T-hard for NP is dense. If p2 (NP) = 6 0, then every language that is Pn ;T-hard for NP is dense. =
We conclude by considering the densities of languages to which SAT can be adaptively reduced. Denition A function g : N ! N is subradical if log g(n) = o(log n). p It is easy to see that a function g is subradical if and only if, for all k > 0, g(n) = o( k n). (This is the reason for the name \subradical.") Subradical functions include very slowgrowing functions such as log n and (log n)5 , as well as more rapidly growing functions such 0:99 (log n ) as 2 .
Corollary 3.6 If p(NP) 6= 0, g : N ! N is subradical, and SAT dense.
P g(n);T
H , then H is
Proof. Assume the hypothesis. Let A 2 NP. Then there is a Pm-reduction f of A to SAT. Fix a polynomial q(n) such that, for all x 2 f0 1g , jf (x)j q(jxj). Composing f with the Pg(n);T -reduction of SAT to H that we have assumed to exist then gives a Pg(q(n));T reduction of A to H . Since g is subradical, log g(q(n)) = o(log q(n)) = o(log n), so for all log n suciently large n, g(q(n)) 2 4 = n 41 . Thus A P 41 H . n ;T The above argument shows that H is P 14 -hard for NP. Since we have assumed n ;T p (NP) 6= 0, it follows by Corollary 3.5 that H is dense.
To put the matter dierently, Corollary 3.6 tells us that if SAT is polynomial-time reducible to a non-dense language with at most 2(log n)0:99 adaptive queries, then NP has measure 0 in E and in E2 .
4 Questions As noted in the introduction, the relationships between weak hardness notions for E and E2 under reducibilities such as PT Pn ;T , and Pn ;tt remain to be resolved. Our main theorem also leaves open the question whether Pn;T -hard languages for E must be dense when 1 < 1. We are in the curious situation of knowing that the classes P 0:99 (DENSEc ) n ;tt 2 and Pn0:49 ;T (DENSEc) have p-measure 0, but not knowing whether Pn0:50 ;T (DENSEc ) has p-measure 0. Indeed, at this time we cannot even prove that E 6 Pn0:50 ;T (SPARSE). Further progress on this matter would be illuminating. 14
References 1] K. Ambos-Spies, Randomness, relativizations, and polynomial reducibilities. In Proceedings of the First Structure in Complexity Theory Conference, pp. 23-34. SpringerVerlag, 1986. 2] K. Ambos-Spies and E. Mayordomo, Resource-bounded measure and randomness, in A. Sorbi(ed.), Complexity, Logic, and Recursion Theory, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, to appear. 3] K. Ambos-Spies, S. A. Terwijn, and X. Zheng, Resource-bounded randomness and weakly complete problems. Theoretical Computer Science. To appear. 4] C. H. Bennett and J. Gill, Relative to a random oracle A, PA 6= NP 6= co ; NPA with probability 1, SIAM Journal on Computing 10 (1981), pp.96-113. 5] L. Berman and J. Hartmanis, On isomorphism and density of NP and other complete sets. SIAM Journal on Computing, 6:305-322, 1977. 6] J. M. Breutzmann and J. H. Lutz, Equivalence of measures of complexity classes, Proceedings of the Fourteenth Symposium on Theoretical Aspects of Computer Science, Springer-Verlag, 1997, to appear. 7] B. Fu, With quasi-linear queries, EXP is not polynomial time Turing reducible to sparse sets. SIAM Journal on Computing, 24:1082-1090, 1995. 8] D. W. Juedes and J. H. Lutz, Weak completeness in E and E2 . Theoretical Computer Science, 143:149-158, 1995. 9] J. H. Lutz, Almost everywhere high nonuniform complexity, Journal of Computer and System Sciences, 19:1100-1131, 1990. 10] J. H. Lutz and E. Mayordomo, Measure stochasticity, and the density of hard languages. SIAM Journal on Computing, 23:762-779, 1994. 11] J. H. Lutz, One-way functions and balanced NP. Theoretical Computer Science, to appear. 12] J. H. Lutz, The quantitative structure of exponential time, in L. A. Hemaspaandra and A. L. Selman (eds.), Complexity Theory Retrospective II, Springer-Verlag, 1996, to appear. A preliminary version appears in Proceedings of The Eighth Annual Structure in Complexity Theory Conference (San Diego, CA, May 18-21, 1993), IEEE Computer Society Press, 1993, pp. 158-175. 13] S. R. Mahaney, Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis. Journal of Computer and System Sciences, 25:130-143, 1982. 14] E. Mayordomo, Contributions to the Study of Resource-Bounded Measure. PhD thesis, Universitat Polit ecnica de Catalunya, Barcelona, Spain, 1994. 15
15] A. R. Meyer, 1977. Reported in 5]. 16] M. Ogiwara and O. Watanabe, On polynomial bounded truth-table reducibility of NP sets to sparse sets. SIAM Journal on Computing, 20(3):471-483, June 1991. 17] O. Watanabe, On the Structure of Intractable Complexity Classes. PhD thesis, Tokyo Institute of Technology, 1987.
16
