Hardness of Sparse Sets and Minimal Circuit Size Problem
aa r X i v : . [ c s . CC ] J u l Hardness of Sparse Sets and Minimal Circuit SizeProblem
Bin FuDepartment of Computer Science,University of Texas Rio Grande Valley, Edinburg, TX 78539, [email protected]
Abstract
We study the magnification of hardness of sparse sets in nondetermin-istic time complexity classes on a randomized streaming model. One ofour results shows that if there exists a 2 n o (1) -sparse set in NTIME(2 n o (1) )that does not have any randomized streaming algorithm with n o (1) up-dating time, and n o (1) space, then NEXP = BPP, where a f ( n )-sparseset is a language that has at most f ( n ) strings of length n . We also showthat if MCSP is ZPP-hard under polynomial time truth-table reductions,then EXP = ZPP.
1. Introduction
Hardness magnification has been intensively studied in the recent years [13, 4,10, 12]. A small lower bound such as Ω( n ǫ ) for one problem may bring a largelower bound such as super-polynomial lower bound for another problem. Thisresearch is closely related to Minimum Circuit Size Problem (MCSP) that is todetermine if a given string of length n = 2 m with integer m can be generatedby a circuit of size k . For a function s ( n ) : N → N , MCSP[ s ( n )] is that givena string x of length n = 2 m , determine if there is a circuit of size at most s ( n ) to generate x . This problem has received much attention in the recentyears [3, 2, 13, 8, 7, 6, 5, 4, 12, 11, 10].Hardness magnification results are shown in a series of recent papers aboutMCSP [13, 4, 10, 12]. Oliveira and Santhanam [13] show that n ǫ -size lowerbounds for approximating MKtP[ n β ] with an additive error O (log n ) impliesEXP P / poly. Oliveira, Pich and Santhanam [12] show that for all small β > n ǫ -size lower bounds for approximating MCSP[ n βm ] with factor O ( m )error implies NP P / poly. McKay, Murray, and Williams [10] show that anΩ( n poly(log n )) lower bound on poly(log n ) space deterministic streaming modelfor MCSP[poly(log n )] implies separation of P from NP.The hardness magnification of non-uniform complexity for sparse sets isrecently developed by Chen, Jin and Williams [4]. Since MCSP[ s ( n )] are of1ub-exponential density for s ( n ) = n o (1) , the hardness magnification for sub-exponential density sets is more general than the hardness magnification forMCSP. They show that if there is an ǫ > { L b } (indexed over b ∈ (0 , L b is a 2 n b -sparse language in NP, and L b Circuit[ n ǫ ], then NP Circuit[ n k ] for all k , where Circuit[ f ( n )] is theclass of languages with nonuniform circuits of size bounded by function f ( n ).Their result also holds for all complexity classes C with ∃ C = C .On the other hand, it is unknown if MCSP is NP-hard. Murray and Williams[11] show that NP-completeness of MCSP implies the separation of EXP fromZPP, a long standing unsolved problem in computational complexity theory.Hitchcock and Pavan [8, 11] if MCSP is NP-hard under polynomial time truth-table reductions, then EXP NP ∩ P / poly.Separating NEXP from BPP, and EXP from ZPP are two of major openproblems in the computational complexity theory. We are motivated by furtherrelationship about sparse sets and MCSP, and the two separations NEXP =BPP and EXP = ZPP. We develop a polynomial method on finite fields tomagnify the hardness of sparse sets in nondeterministic time complexity classesover a randomized streaming model. One of our results show that if there existsa 2 n o (1) -sparse set in NTIME(2 n o (1) ) that does not have a randomized streamingalgorithm with n o (1) updating time, and n o (1) space, then NEXP = BPP, wherea f ( n )-sparse set is a language that has at most f ( n ) strings of length n . Ourmagnification result has a flexible trade off between the spareness and timecomplexity.We use two functions d ( n ) and g ( n ) to control the sparseness of a tally set T . Function d ( n ) gives an upper bound for the number of elements of in T and g ( n ) is the gap lower bound between a string 1 n and the next string 1 m in T ,which satisfy g ( n ) < m . The class TALLY( d ( n ) , g ( n )) defines the class of allthose tally sets. By choosing d ( n ) = log log n , and g ( n ) = 2 n , we prove thatif MCSP is ZPP ∩ TALLY( d ( n ) , g ( n ))-hard under polynomial time truth-tablereductions, then EXP = ZPP. Comparing with some existing results about sparse sets hardness magnificationin this line [4], there are some new advancements in this paper.1. Our magnification of sparse set is based on a uniform streaming model. Aclass of results in [4] are based on nonuniform models. In [10], they showthat if there is A ∈ PH, and a function s ( n ) ≥ log n , search-MCSP A [ s ( n )]does not have s ( n ) c updating time in deterministic streaming model for allpositive, then P = NP. MCSP[ s ( n )] is a s ( n ) O( s ( n )) -sparse set.2. Our method is conceptually simple, and easy to understand. It is a polyno-mial algebraic approach on finite fields.3. A flexible trade off between sparseness and time complexity is given in ourpaper. 2roving NP-hardness for MCSP implies EXP = ZPP [8, 11]. We considerthe implication of ZPP-hardness for MCSP, and show that if MCSP is ZPP ∩ TALLY( d ( n ) , g ( n ))-hard for a function pair such as d ( n ) = log log n and g ( n ) =2 n , then EXP = ZPP. It seems that proving MCSP is ZPP-hard is much easierthan proving MCSP is NP-hard since ZPP ⊆ (NP ∩ co-NP) ⊆ NP. According tothe low-high hierarchy theory developed by Sch¨oning [14], the class NP ∩ co-NPis the low class L . Although MCSP may not be in the class ZPP, it is possibleto be ZPP-hard.
2. Notations
Minimum Circuit Size Problem (MCSP) is that given an integer k , and a binarystring T of length n = 2 m for some integer m ≥
0, determine if T can begenerated by a circuit of size k . Let N = { , , · · ·} be the set of all naturalnumbers. For a language L , L n is the set of strings in L of length n , and L ≤ n isthe set of strings in L of length at most n . For a finite set A , denote | A | to be thenumber of elements in A . For a string s , denote | s | to be its length. If x, y, z arenot empty strings, we have a coding method that converts a x, y into a string h x, y i with | x | + | y | ≤ |h x, y i| ≤ | x | + | y | ) and converts x, y, z into h x, y, z i with | x | + | y | + | z | ≤ |h x, y, z i| ≤ | x | + | y | + | z | ). For example, for x = x · · · x n , y = y · · · y n , z = z · · · z n , let h x, y, z i = 1 x · · · x n y · · · y n z · · · z n .Let DTIME( t ( n )) be the class of languages accepted by deterministic Tur-ing machines in time O( t ( n )). Let NTIME( t ( n )) be the class of languages ac-cepted by nondeterministic Turing machines in time O( t ( n )). Define EXP = ∪ ∞ c =1 DTIME(2 n c ) and NEXP = ∪ ∞ c =1 NTIME(2 n c ). P/poly, which is also calledPSIZE, is the class of languages that have polynomial-size circuits.We use a polynomial method on a finite field F . It is classical theory thateach finite field is of size p k for some prime number p and integer k ≥ F , we denote R ( F ) = ( p, t F ( u )) to represent F , where t F ( u ) isa irreducible polynomial over field GF( p ) for the prime number p and its degreeis deg( t F ( . )) = k . The polynomial t F ( u ) is equal to u if F is of size p , whichis a prime number. Each element of F with R ( F ) = ( p, t F ( u )) is a polynomial q ( u ) with degree less than the degree of t F ( u ). For two elements q ( u ) and q ( u ) in F , their addition is defined by ( q ( u ) + q ( u ))(mod t F ( u )), and theirmultiplication is defined by ( q ( u ) · q ( u ))(mod t F ( u )) (see [9]). Each elementin GF(2 k ) is a polynomial P k − i =0 b i u i ( b i ∈ { , } ), which is represented by abinary string b k − · · · b of length k .We use GF(2 k ) field in our randomized streaming algorithm for hardnessmagnification . Let F be a GF(2 k ) field (a field of size q = 2 k ) and has its R ( F ) = (2 , t F ( u )). Let s = a · · · a m − be a binary string of length m with m ≤ k , and u be a variable. Define w ( s, u ) to be the element P m − i =0 a i u i in GF(2 k ). Let x be a string in { , } ∗ and k be an integer at least 1. Let x = s r − s t − · · · s s such that each s i is a substring of x of length k for i =1 , , · · · , r −
1, and the substring s has its length | s | ≤ k . Each s i is calleda k -segment of x for i = 0 , , · · · , r −
1. Define the polynomial d x ( z ) = z r +3 r − i =0 w ( s i , u ) z i , which converts a binary string into a polynomial in GF(2 k ).We develop a streaming algorithm that converts an input string into anelement in a finite field. We give the definition to characterize the properties ofthe streaming algorithm developed in this paper. Our streaming algorithm isto convert an input stream x into an element d x ( a ) ∈ F = GF(2 k ) by selectinga random element a from F . Definition 1.
Let r ( n ) , r ( n ) , r ( n ) , s ( n ) , u ( n ) be nondecreasing functionsfrom N to N . Define Streaming( r ( n ) , r ( n ) , s ( n ) , u ( n ) , r ( n )) to be the classof languages L that have one-pass streaming algorithms that has input ( n, x )with n = | x | ( x is a string and read by streaming), it satisfiesi. It takes r ( n ) time to generate a field F = GF(2 k ), which is representedby (2 , t F ( . )) with a irreducible polynomial t f ( . ) over GF(2) of degree k .ii. It takes O ( r ( n )) random steps before reading the first bit from the inputstream x .iii. It uses O ( s ( n )) space that includes the space to hold the field represen-tation generated by the algorithm. The space for a field representation isΩ((deg( t F ( . )) + 1)) and O((deg( t F ( . )) + 1)) for the irreducible polynomial t F ( . ) over GF(2).iv. It takes O ( u ( n )) field conversions to elements in F and O ( u ( n )) field op-erations in F after reading each bit.v. It runs O ( r ( n )) randomized steps after reading the entire input.
3. Overview of Our Methods
In this section, we give a brief description about our methods used in thispaper. Our first result is based on a polynomial method on a finite field whosesize affects the hardness of magnification. The second result is a translationalmethod for zero-error probabilistic complexity classes.
We have a polynomial method over finite fields. Let L be f ( n )-sparse languagein NTIME( t ( n )). In order to handle an input string of size n , a finite field F = GF( q ) with q = 2 k for some integer k is selected, and is represented by R ( F ) = (2 , t F ( z )), where t F ( z ) is a irreducible polynomial over GF(2). Aninput y = a a · · · a n is partitioned into k -segments s r − · · · s s such that each s i is converted into an element w ( s i , u ) in F , and y is transformed into anpolynomial d y ( z ) = z r + P r − i =0 w ( s i , u ) z i . A random element a ∈ F is chosen inthe beginning of streaming algorithm before processing the input stream. Thevalue d y ( a ) is evaluated with the procession of input stream. The finite F islarge enough such that for different y and y of the same length, d y ( . ) and4 y ( . ) are different polynomials due to their different coefficients derived from y and y , respectively. Let H ( y ) be the set of all h n, a, d y ( a ) i with a ∈ F and n = | y | . Set A ( n ) is the union of all H ( y ) with y ∈ L n . The set of A is ∪ ∞ i =1 A ( n ). A small lower bound for the language A is magnified to large lowerbound for L .The size of field F depends on the density of set L and is O( f ( n ) n ). Bythe construction of A , if y ∈ L , there are q tuples h n, a, d y ( a ) i in A that aregenerated by y via all a in F . For two different y and y of length n , theintersection H ( y ) ∩ H ( y ) is bounded by the degree of d y ( . ). If y L , thenumber of items h n, a, d y ( a ) i generated by y is at most q in A . If y ∈ L , thenumber of items h n, a, d y ( a ) i generated by y is q in A . This enables us to converta string x of length n in L into some strings in A of length much smaller than n , make the hardness magnification possible. ZPP -hardness of
MCSP
Our another result shows that ZPP-hardness for MCSP implies EXP = ZPP.We identify a class of functions that are padding stable, which has the propertyif T ∈ TALLY( d ( n ) , g ( n )), then { n +2 n : 1 n ∈ T } ∈ TALLY( d ( n ) , g ( n )). Thefunction pair d ( n ) = log log n and g ( n ) = 2 n has this property. We constructa very sparse tally set L ∈ EXP ∩ TALLY( d ( n ) , g ( n )) that separates ZPEXPfrom ZPP, where ZPEXP is the zero error exponential time probabilistic class.It is based on a diagonal method that is combined with a padding design. Atally language L has a zero-error 2 n -time probabilistic algorithm implies L ′ = { n +2 n : 1 n ∈ L } has a zero-error 2 n -time probabilistic algorithm. Adapting tothe method of [11], we prove that if MCSP is ZPP ∩ TALLY( d ( n ) , g ( n ))-hardunder polynomial time truth-table reductions, then EXP = ZPP.
4. Hardness Magnification via Streaming
In this section, we show a hardness magnification of sparse sets via a streamingalgorithm. A classical algorithm to find irreducible polynomial [15] is used toconstruct a field that is large enough for our algorithm.
Theorem 2. [15] There is a deterministic algorithm that constructs a irre-ducible polynomial of degree n in O( p (log p ) n ǫ + (log p ) n ǫ ) operations in F , where F is a finite field GF( p ) with prime number p . Definition 3.
Let f ( n ) be a function from N to N . For a language A ⊆ { , } ∗ ,we say A is f ( n )-sparse if | A n | ≤ f ( n ) for all large integer n . The algorithm Streaming(.) is based on a language L that is f ( n )-sparse. Itgenerates a field F = GF(2 k ) and evaluates d x ( a ) with a random element a in5 . A polynomial z r + P r − i =0 b i z i = z r + b r − z r − + b r − z r − + · · · + b canbe evaluated by ( · · · (( z + b r − ) z + b r − ) z + ... ) z + b according to the classicalHorner’s algorithm. For example, z + z + 1 = ( z + 1) z + 1. Algorithm
Streaming( n, x )Input: an integer n , and string x = a · · · a n of the length n ;Steps:1. Select a field size q = 2 k such that 8 f ( n ) n < q ≤ f ( n ) n .2. Generate an irreducible polynomial t F ( u ) of degree k over GF(2) such that(2 , t F ( u )) represents finite F = GF( q ) (by Theorem 2 with p = 2);3. Let a be a random element in F ;4. Let r = (cid:6) nk (cid:7) ; (Note that r is the number of k -segments of x . See Section 2)5. Let j = r − v = 1;7. Repeat8. {
9. Receive the next k -segment s j from the input stream x ;10. Convert s j into an element b j = w ( s j , u ) in GF( q );11. Let v = v · a + b j ;12. Let j = j − }
13. Until j < h n, a, v i ; End of Algorithm
Now we have our magnification algorithm. Let M ( . ) be a randomized Turingmachine to accept a language A that contains all h| x | , a, d x ( a ) i with a ∈ F and x ∈ L . We have the following randomized streaming algorithm to accept L viathe randomized algorithm M ( . ) for A . Algorithm
Magnification( n, x )Input integer n and x = a · · · a n as a stream;Steps: Let y =Streaming( n, x ); Accept if M ( y ) accepts; End of Algorithm .2. Hardness Magnification In this section, we derive some results about hardness magnification via sparseset. Our results show a trade off between the hardness magnification and sparse-ness via the streaming model.
Definition 4.
For a nondecreasing function t ( . ) : N → N , define BTIME( t ( n ))the class of languages L that have two-side bounded error probabilistic algo-rithms with time complexity O ( t ( n )). Define BPP = ∪ ∞ c =1 BTIME( n c ). Theorem 5.
Assume that u ( m ) be nondecreasing function for the time to gen-erate an irreducible polynomial of degree m in GF(2) , and u ( m ) be the nonde-creasing function of a time upper bound for the operations ( + , . ) in GF(2 m ) . Let f ( . ) , t ( . ) , t ( . ) , t ( n ) be nondecreasing functions N → N with f ( n ) ≤ n , v ( n ) =(log n + log f ( n )) , and v ( n ) + t ( n ) + u (10 v ( n )) + n · u (10 v ( n )) ≤ t ( v ( n )) for all large n . If there is a f ( n ) -sparse set L with L ∈ NTIME( t ( n )) and L Streaming( u (10 v ( n ))) , v ( n ) , v ( n ) , , t (10 v ( n ))) , then there is a language A such that A ∈ NTIME( t ( n )) and A BTIME( t ( n )) . Proof:
Select a finite field GF( q ) with q = 2 k for an integer k by line 1 of thealgorithm streaming(.). For each x ∈ L n , let x be partitioned into k -segments: s r − s r − · · · s . Let w ( s i , u ) convert s i into an element of GF( q ) (See Section 2).Define polynomial d x ( z ) = z r + P r − i =0 w ( q, s i ) z i . For each x , let H ( x ) be theset {h n, a, d x ( a ) i| a ∈ GF( q ) } , where n = | x | . Define set A ( n ) = ∪ y ∈ L n H ( y ) for n = 1 , · · · , and language A = ∪ + ∞ n =1 A ( n ). Claim 1.
For any x L n with n = | x | , we have | H ( x ) ∩ A ( n ) | < q . Proof:
Assume that for some x L n with n = | x | , | H ( x ) ∩ A ( n ) | ≥ q . It iseasy to see that r ≤ n and k ≤ n for all large n by the algorithm Streaming(.)and the condition of f ( . ) in the theorem. Assume that | H ( x ) ∩ H ( y ) | < r + 1for every y ∈ L n . Since A ( n ) is the union H ( y ) with y ∈ L n and | L n | ≤ f ( n ),there are at most rf ( n ) ≤ nf ( n ) < q elements in H ( x ) ∩ A ( n ) by line 1 of thealgorithm Streaming(.). Thus, | H ( x ) ∩ A ( n ) | < q . This brings a contradiction.Therefore, there is a y ∈ L n to have | H ( x ) ∩ H ( y ) | ≥ r +1. Since the polynomials d x ( . ) and d y ( . ) are of degrees at most r , we have d x ( z ) = d y ( z ) (two polynomialsare equal). Thus, x = y . This brings a contradiction because x L n and y ∈ L n . Claim 2. If x ∈ L , then Streaming( | x | , x ) ∈ A . Otherwise, with probabilityat most , Streaming( | x | , x ) ∈ A . Proof:
For each x , it generates h n, a, d x ( a ) i for a random a ∈ GF( q ). Each a ∈ GF( q ) determines a random path. We have that if x ∈ L , then h n, a, d x ( a ) i ∈ A , and if x L , then h n, a, d x ( a ) i ∈ A with probability at most by Claim 1. Claim 3. A ∈ NTIME( t ( m )). Proof:
Let z = 10 v ( n ) = 10(log n + log f ( n )). Each element in field F =GF(2 k ) is of length k . For each u = h n, a, b i ( a, b ∈ F ), we need to guess a string7 ∈ L n such that b = d x ( a ). It is easy to see that v ( n ) ≤ |h n, a, b i| ≤ v ( n )for all large n if h n, a, b i ∈ A (See Section 2 about coding). Let m = |h n, a, b i| .It takes at most u ( z ) steps to generate a irreducible polynomial t F ( . ) for thefield F by our assumption.Since L ∈ NTIME( t ( n )), checking if u ∈ A takes nondeterministic t ( n )steps to guess a string x ∈ L n , u ( z ) deterministic steps to generate t F ( u ) forthe field F , O ( z ) nondeterministic steps to generate a random element a ∈ F ,and additional O( n · u ( z )) steps to evaluate d x ( a ) in by following algorithmStreaming(.) and check b = d x ( a ). The polynomial t F ( u ) in the GF(2) hasdegree at most z . Each polynomial operation (+ or . ) in F takes at most u ( z )steps. Since z + t ( n ) + u ( z ) + n · u ( z ) ≤ t ( m ) time by the condition of thistheorem, we have A ∈ NTIME( t ( m )). Claim 4. If A ∈ BTIME( t ( m )), then L ∈ Streaming( u (10 v ( n )) , v ( n ) , v ( n ) , , t (10 v ( n ))). Proof:
The field generated at line 2 in algorithm Streaming(.) takes u (10(log n + log f ( n ))) time. Let x = a · · · a n be the input string. The string x partitioned into k -segments s r − · · · s . Transform each s i into an element b i = w ( s i , u ) in GF( q ) in the streaming algorithm. We generate a polynomial d x ( z ) = z r + P r − i =0 b i z i = z r + b r − z r − + b r − z r − + · · · + b . Given a randomelement a ∈ GF( q ), we evaluate d x ( a ) = ( · · · (( a + b r − ) a + b r − ) a + ... ) a + b ac-cording to the classical algorithm. Therefore, d x ( a ) is evaluated in Streaming(.)with input ( | x | , x ).If A ∈ BTIME( t ( m )), then L has a randomized streaming algorithm thathas at most t (10 v ( n )) random steps after reading the input, and at mostO( v ( n )) space. After reading one substring s i from x , it takes one conver-sion from a substring of the input to an element of field F by line 10, and atmost two field operations by line 11 in the algorithm Streaming(.).Claim 4 brings a contradiction to our assumption about the complexity of L in the theorem. This proves the theorem. Proposition 6.
Let f ( n ) : N → N be a nondecreasing function. If for each fixed ǫ ∈ (0 , , f ( n ) ≤ n ǫ for all large n , then there is a nondecreasing unboundedfunction g ( n ) : N → N with f ( n ) ≤ n g ( n ) . Proof:
Let n = 1. For each k ≥
1, let n k be the least integer such that n k ≥ n k − and f ( n ) ≤ n k for all n ≥ n k . Clearly, we have the infinite list n ≤ n · · · ≤ n k ≤ · · · such that lim k → + ∞ n k = + ∞ . Define function g ( k ) : N → N such that g ( n ) = k for all n ∈ [ n k − , n k ). For each n ≥ n k , we have f ( n ) ≤ n k .Our Definition 7 is based Proposition 6. It can simplify the proof when wehandle a function that is n o (1) . 8 efinition 7. A function f ( n ) : N → N is n o (1) if there is a nondecreasingfunction g ( n ) : N → N such that lim n → + ∞ g ( n ) = + ∞ and f ( n ) ≤ n g ( n ) for alllarge n . A function f ( n ) : N → N is 2 n o (1) if there is a nondecreasing function g ( n ) : N → N such that lim n → + ∞ g ( n ) = + ∞ and f ( n ) ≤ n g ( n ) for all large n . Corollary 8.
If there exists a n o (1) -sparse language L in NTIME(2 n o (1) ) suchthat L does not have any randomized streaming algorithm with n o (1) updatingtime, and n o (1) space, then NEXP = BPP . Proof:
Let g ( n ) : N → N be an arbitrary unbounded nondecreasing functionthat satisfies lim n → + ∞ g ( n ) = + ∞ and g ( n ) ≤ log log n . Let t ( n ) = f ( n ) =2 n g ( n ) and Let t ( n ) = 2 n , t ( n ) = n √ g ( n ) , and v ( n ) = (log n + log f ( n )).It is easy to see that v ( n ) = n o(1) , and both u ( n ) and u ( n ) are n O(1) (seeTheorem 2). For any fixed c >
0, we have t ( v ( n )) > t (log f ( n )) ≥ t ( n g ( n ) ) >t ( n ) + n c for all large n . For all large n , we have t (10 v ( n )) ≤ t (20 log f ( n )) = t (20 n g ( n ) ) (1) ≤ (20 n g ( n ) ) q g (20 n g ( n ) ) ≤ ( n g ( n ) ) √ g ( n ) = n o (1) . (2)Clearly, these functions satisfy the inequality of the precondition in Theorem 5.Assume L ∈ Streaming(poly( v ( n )) , v ( n ) , v ( n ) , , t (10 v ( n ))). With O ( v ( n )) = n o (1) space, we have a field representation (2 , t F ( . )) with deg( t F ( . )) = n o (1) .Thus, each field operation takes n o (1) time by the brute force method for poly-nomial addition and multiplication. We have t (10 v ( n )) = n o (1) by inequality(2). Thus, the streaming algorithm updating time is n o (1) . Therefore, we havethat L has a randomized streaming algorithm with n o (1) updating time, and n o (1) space. This gives a contradiction. So, L Streaming(poly( v ( n )) , v ( n ) , v ( n ) , , t (10 v ( n ))). By Theorem 5, there is A ∈ NTIME( t ( n )) such that A BTIME( t ( n )). Therefore, A BPP. Thus,NEXP = BPP.
5. Implication of
ZPP -Hardness of
MCSP
In this section, we show that if MCSP is ZPP ∩ TALLY-hard, then EXP = ZPP.The conclusion still holds if TALLY is replaced by a very sparse subclass ofTALLY languages. Definition 9.
For a nondecreasing function t ( . ) : N → N , define ZTIME( t ( n ))the class of languages L that have zero-error probabilistic algorithms with timecomplexity O ( t ( n )). Define ZPP = ∪ ∞ c =1 ZTIME( n c ), andZPEXP = ∪ ∞ c =1 ZTIME(2 n c ). 9 efinition 10. For an nondecreasing function f ( n ) : N → N , defineTALLY[ f ( k )] to be the class of tally set A ⊆ { } ∗ such that for each 1 m ∈ A ,there is an integer i ∈ N with m = f ( i ). For a tally language T ⊆ { } ∗ , definePad( T ) = { n + n | n ∈ T } . Definition 11.
For two languages A and B , a polynomial time truth-table re-duction from A to B is a polynomial time computable function f ( . ) such thatfor each instance x for A , f ( x ) = ( y , · · · , y m , C ( . )) to satisfy x ∈ A if and onlyif C ( B ( y ) , · · · , B ( y m )) = 1, where C ( . ) is circuit of m input bits and B ( . ) isthe characteristic function of B .Let ≤ Pr be a type of polynomial time reductions ( ≤ Ptt represents polynomialtime truth-table reductions), and C be a class of languages. A language A is C -hard under ≤ Pr reductions if for each B ∈ C , B ≤ Pr A . Definition 12.
Let k be an integer. Define two classes of functions withrecursions: (1) log (1) ( n ) = log n , and log ( k +1) ( n ) = log (log ( k ) ( n )). (2)exp (1) ( n ) = 2 n , and exp ( k +1) ( n ) = 2 exp ( k ) ( n ) . Definition 13.
For two nondecreasing functions d ( n ) , g ( n ) : N → N , the pair( d ( n ) , g ( n )) is time constructible if ( d ( n ) , g ( n )) can be computed in time d ( n ) + g ( n ) steps. Definition 14.
Define TALLY( d ( n ) , g ( n )) to be the class of tally sets T suchthat | T ≤ n | ≤ d ( n ) and for any two strings 1 n , m ∈ T with n < m , they satisfy g ( n ) < m . We call d ( n ) to be the density function and g ( n ) to be the gapfunction . A gap function g ( n ) is padding stable if g (2 n + n ) < g ( n ) + g ( n ) forall n > Lemma 15. i. Assume the gap function g ( n ) is padding stable. If T ∈ TALLY( d ( n ) , g ( n )) ,then Pad( T ) ∈ TALLY( d ( n ) , g ( n )) .ii. For each integer k > , g ( n ) = exp ( k ) (2 n ) is padding stable. Proof:
Part i. Let 1 n be a string in T . The next shortest string 1 m ∈ T with n < m satisfies g ( n ) < m . We have 1 n + n and 1 m + m are two consecutiveneighbor strings in Pad( T ) such that there is no other string 1 k ∈ Pad( T ) with2 n + n < k < m + m . We have g (2 n + n ) < g ( n ) + g ( n ) < m + m . Sincethe strings in Pad( T ) ≤ n are one-one mapped from the strings in T with lengthless than n , | Pad( T ) ≤ n | ≤ | T ≤ n | ≤ d ( n ), we have Pad( T ) ∈ TALLY( d ( n ) , g ( n )).This proves Part (i).Part ii. We have inequality g (2 n + n ) = exp ( k ) (2(2 n + n )) < exp ( k ) (4 · n ) =exp ( k ) (2 n +2 ) ≤ exp ( k ) (2 n ) = 2 g ( n ) < g ( n ) + g ( n ). Therefore, gap function g ( n )is padding stable. This proves Part ii. 10 emma 16. Let d ( n ) and g ( n ) be nondecreasing unbounded functions from N to N , and ( d ( n ) , g ( n )) is time constructible. Then there exists a time con-structible increasing unbounded function f ( n ) : N → N such that TALLY[ f ( n )] ⊆ TALLY( d ( n ) , g ( n )) . Proof:
Compute the least integer n with d ( n ) >
0. Let s be the numberof steps for the computation. Define f (1) = max( s , n ). Assume that f ( k − f ( k ) below.For an integer k >
0, compute g ( f ( k − k numbers n < n < · · · < n k such that 0 < d ( n ) < d ( n ) < · · · < d ( n k ). Assume the computationabove takes s steps. Define f ( k ) to be the max(2 s, n k , g ( f ( k − T ∈ TALLY[ f ( n )], there are at most k strings in T with lengthat most f ( k ). On the other hand, d ( n k ) ≥ k by the increasing list 0 < d ( n ) Let d ( n ) and g ( n ) be nondecreasing unbounded functions. Iffunction g ( n )) is padding stable, then there is a language A such that A ∈ ZTIME(2 O ( n ) ) ∩ TALLY( d ( n ) , g ( n )) and A ZPP . Proof: It is based on the classical translational method. AssumeZTIME(2 O ( n ) ) ∩ TALLY( d ( n ) , g ( n )) ⊆ ZPP. Let f ( . ) be a time constructibleincreasing unbounded function via Lemma 16 such thatTALLY[ f ( n )] ⊆ TALLY( d ( n ) , g ( n )). Let t ( n ) = 2 n and t ( n ) = 2 n − .Let L be a tally language in DTIME( t ( n )) ∩ TALLY[ f ( n )], but it is not inDTIME( t ( n )). Such a language L can be constructed via a standard diagonalmethod. Let M , · · · , M be the list of Turing machines such that each M i hastime upper bound by function t ( n ). Define language L ∈ TALLY[ f ( n )] suchthat for each k , 1 f ( k ) ∈ L if and only if M k (1 f ( k ) ) rejects in t ( f ( k )) steps. Wehave L ∈ TALLY( d ( n ) , g ( n )) by Lemma 16.Let L = Pad( L ). We have L ∈ TALLY( d ( n ) , g ( n )) by Lemma 15.We have L ∈ DTIME(2 O ( n ) ) ⊆ ZTIME(2 O ( n ) ). Thus, L ∈ ZPP. So, L ∈ ZTIME(2 O ( n ) ). Therefore, L ∈ ZTIME(2 O ( n ) ) ∩ TALLY( d ( n ) , g ( n )). Wehave L ∈ ZPP. Thus, L ∈ DTIME(2 n O(1) ) ⊆ DTIME(2 n − ). This brings acontradiction. Theorem 18. Let d ( n ) and g ( n ) be nondecreasing unbounded functions from N to N . Assume that g ( n ) is padding stable. If MCSP is ZPP ∩ TALLY( d ( n ) , g ( n )) -hard under polynomial time truth-table reductions, then EXP = ZPP . roof: Assume that MCSP is (ZPP ∩ TALLY( d ( n ) , g ( n ))-hard under poly-nomial time truth-table reductions, and EXP = ZPP.Let L be a language in ZTIME(2 O ( n ) ) ∩ TALLY[ d ( . ) , g ( . )], but L ZPP byLemma 17. Let L ′ = Pad( L ). Clearly, every string 1 y in L ′ has the propertythat y = 2 n + n for some integer n . This property is easy to check and wereject all strings without this property in linear time. We have L ′ ∈ ZPP.Therefore, there is a polynomial time truth-table reduction from L ′ to MCSP viaa polynomial time truth-table reduction M ( . ). Let polynomial p ( n ) = n c be therunning time for M ( . ) for a fixed c and n ≥ R = { (1 n , i, j ) , the i -th bit of j -th query of M (1 n +2 n )is equal to 1 , and i, j ≤ p ( n + 2 n ) } . We can easily prove that R is in EXP.Therefore, R ∈ ZPP ⊆ P / poly (See [1]).Therefore, there is a class of polynomial size circuits { C n } ∞ n =1 to recognize R such that C n ( . ) recognize all (1 n , i, j ) with i, j ≤ p ( n +2 n ) in R . Assume that thesize of C n is of size at most q ( n ) = n t + t for a fixed t . For an instance x = 1 n for L , consider the instance y = 1 n +2 n for L ′ . We can compute all non-adaptivequeries h T, s ( n ) i to MCSP in 2 n O (1) time via M ( y ). If s ( n ) ≥ q ( n ), the answerfrom MCSP for the query h T, s ( n ) i is yes since h T, s ( n ) i can be generated asone of the instances via the circuit C n ( . ). If s ( n ) < q ( n ), we can use a bruteforce method to check if there exists a circuit of size at most q ( n ) to generate T . It takes 2 n O (1) time. Therefore, L ∈ EXP. Thus, L ∈ ZPP. This bring acontradiction as we already assume L ZPP. Corollary 19. For any integer k , if MCSP is ZPP ∩ TALLY(log ( k ) ( n ) , exp ( k ) (2 n )) -hard under polynomial time truth-table reductions, then EXP = ZPP . Proof: It follows from Theorem 18 and Lemma 15. Corollary 20. For any integer k , if MCSP is ZPP ∩ TALLY -hard under poly-nomial time truth-table reductions, then EXP = ZPP . Corollary 21. If MCSP is ZPP -hard under polynomial time truth-table reduc-tions, then EXP = ZPP . 6. Conclusions In this paper, we develop an algebraic method to magnify the hardness of sparsesets in nondeterministic classes via a randomized streaming model. It has a flexi-ble trade off between the sparseness and time complexity. This shows connectionto the major problems to prove NEXP = BPP. We also prove that if MCSP isZPP-hard, then EXP = ZPP. Acknowledgements: This research was supported in part by NationalScience Foundation Early Career Award 0845376, and Bensten Fellowship ofthe University of Texas Rio Grande Valley. 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