On Reductions of P Sets to Sparse Sets. Dieter van Melkebeek. The University of Chicago. June 18, Abstract

Transcription

1 On Reductions of P Sets to Sparse Sets Dieter van Melkebeek The University of Chicago June 18, 1995 Abstract We prove unlikely consequences of the existence of sparse hard sets for P under deterministic as well as one-sided error randomized truth-table reductions. Our main results are as follows. We establish that the existence of a polynomially dense hard set for P under (randomized) logspace bounded truth-table reductions implies that P (R)L, and that the collapse goes down to P (R)NC 1 in case of reductions computable in (R)NC 1. We also prove that the existence of a quasipolynomially dense hard set for P under (randomized) polylog-space truth-table reductions using polylogarithmically many queries implies that P (R)SPACE[polylog n]. The randomized space complexity classes we consider are based on the multiple access randomness concept. 1 Introduction A lot of research eort in complexity theory has been spent on the sparse hard set problem for NP, i.e., the question whether there are sparse hard sets for NP under various polynomial-time reducibilities. Two major motivations for the study of this problem are the following: One concerns many-one reductions and is the Berman-Hartmanis isomorphism conjecture that all NP-complete sets are polynomial-time isomorphic [BH77]. In support of it, Berman and Hartmanis also conjectured that there are no sparse NP-complete sets. Although the isomorphism conjecture itself is considered doubtful nowadays, the latter conjecture was proved by Mahaney [Mah82]: There is a sparse hard set for NP under many-one reductions i P = NP. Another motivation deals with Turing reductions, and is based on the fact that the languages polynomial-time Turing reducible to sparse sets are precisely those that have polynomial-size (non-uniform) circuits. As evidence that NP is not so easy in that respect, Karp and Lipton [KL82] showed that if NP has polynomial-size circuits, then the polynomial-time hierarchy collapses to the second level. So, there are no sparse hard sets for NP under Turing reductions, unless p 2 = p 2. In bridging the gap between many-one reductions and Turing reductions, a breakthrough result is the extension by Ogiwara and Watanabe of Mahaney's theorem to bounded truth-table reductions, i.e., there is a sparse hard set for NP under bounded truth-table reductions i P = NP [OW91]. Research Assistant of the Belgian National Fund for Scientic Research. address: dieter@cs.uchicago.edu 1

2 For some unbounded truth-table reductions, the proof shows other improbable inclusions of NP to follow from the existence of sparse hard sets for NP, as analyzed in [HL94]. This paper deals with the sparse hard set problem for P, i.e., whether there are sparse hard sets for P under logspace reducibilities. The research on this problem was initiated by Hartmanis. While studying logspace isomorphisms between P-complete languages, he conjectured that there are no sparse P-complete sets [Har78]. Following signicant progress made by Ogihara [Ogi95], the conjecture was recently settled by Cai and Sivakumar [CS95]: There is a sparse hard set for P under many-one reductions i P = L. Cai, Naik and Sivakumar next considered truth-table reductions. They showed that the existence of a sparse hard set for P under bounded truth-table reductions implies P = NC 2, but left the implication P = L open [CNS95]. We establish here that this ultimate collapse does follow from that hypothesis, and that the collapse goes down to P = NC 1 if the reductions are computable in NC 1. We obtain this result as an instance of a generic theorem analogous to the one for NP in [HL94], which allows to vary the sparseness condition of the hypothesis, and the resource bounds as well as the bound on the number of queries of the truth-table reduction. The construction in the proof is less complicated than in [CNS95]. As a result, by another instantiation of our theorem, we are able to prove that there is a quasipolynomially dense hard set for P under polylog-space truth-table reductions using polylogarithmically many queries i P is in polylog-space, resolving another open problem mentioned in [CNS95]. Cai, Naik and Sivakumar also studied randomized many-one reductions with one-sided error, and showed that there is a sparse hard set for P under such reductions computable in randomized logspace and with condence at least inversely polynomial i P RL, where the randomized space complexity class involved is dened using Turing machines endowed with a two-way read access random tape. We strengthen our generic theorem to also hold for randomized truth-table reductions with one-sided error, which allows us to extend the result of [CNS95] to truth-table reductions. The strengthening also yields that there is a quasipolynomially dense hard set for P under one-sided error randomized polylog-space truth-table reductions with condence at least inversely quasipolynomial and using polylogarithmically many queries, i P is in randomized polylog-space. The organization of this paper is as follows. After dening our notation in section 2, we start section 3 with a more detailed account of the recent history of the sparse hard set problem for P, pointing out the ideas of previous papers we use, and give an outline of the proof of our generic theorem. Next, we formally establish the generic theorem for deterministic reductions, and make several instantiations yielding results on the sparse hard set problem for P, and also some consequences for other complexity classes than P. Section 5 deals with one-sided error randomized reductions and consists of the extension of the generic theorem for these reductions, followed by instantiations of it and other corollaries. Finally, we mention possible directions for further research in section 6. 2 Preliminaries and Notation All of our notation and denitions are standard, except for the randomized space complexity classes, and the classes NC k and RNC k. denotes the binary alphabet f0; 1g. For n 2 N, 6n : = [ n k=1 k [fg, and : = [ 1 n=1 n [fg, where represents the empty string. We consider the lexicographic ordering 6` on. An interval of is a subset of the form [w 1 ; w 2 ] = : fw 2 j w 1 6` w 6` w 2 g for some w 1 ; w 2 2. A language 2

3 L is a subset of. The characteristic function L :! of L is such that L (x) = 1 if x 2 L, and L (x) = 0 otherwise. A function f : N! N is said to be polynomial if there is c 2 N such that f 2 O(n c ), polylogarithmic if there is a c 2 N such that f 2 O(log c n), and quasipolynomial if there is c 2 N such that f 2 2 O(logc n). The class of all polynomial functions is denoted by poly, and the one containing all polylogarithmic functions by polylog. For any set S, c S denotes the census function of S, i.e., c S : N! N : n! js \ 6n j. S is polynomially dense if c S is polynomial, and quasipolynomially dense if c S is quasipolynomial. A sparse set is a polynomially dense set. Let f :!, s : N! N, and : N! [0; 1). We say that f is computable in RSPACE[s] with condence if there is a Turing machine M with 2 input tapes that halts and uses no more than O(s(jxj)) space for all inputs (x; ) 2 1 such that for any x 2 : Pr [M(x; ) = f(x)] > (jxj); where the probability is taken over the uniform distribution of 2 1. The second input tape is the source of randomness and models a random bit tape, to which M has full (i.e., 2-way) read access, but which is not taken into account for the space complexity of M. In case M(x; ) is independent of, f is said to be computable in DSPACE[s]. A language L is in RSPACE[s] if, for any constant 2 (0; 1), its characteristic function is computable in RSPACE[s] with condence by a machine M that never errs on inputs x 62 L. L is in NSPACE[s] if the above holds for 0; if M(x; ) is independent of, L is said to be in DSPACE[s]. In particular, RL is short for RSPACE[log], NL for NSPACE[log], and L for DSPACE[log]. Analogously, using time resource bounds instead of space resource bounds, for any t : N! N, we dene computability in RTIME[t] and DTIME[t], as well as the language classes RTIME[t], NTIME[t], and DTIME[t]. In particular, RP stands for RTIME[poly], NP for NTIME[poly], and P for DTIME[poly]. A randomized oracle-augmented bounded fan-in Boolean circuit C with n inputs, r random bits, p oracle-gates, and m gates is a labeled acyclic directed graph with p + m internal nodes (i.e., nodes with positive indegree), such that the labels of the leaves (i.e., nodes with indegree 0) are in f1; : : :; n + rg, p of the internal nodes are labeled with a positive integer and have their incoming arcs ordered, and the remaining m internal nodes have either label \:" and indegree 1, or else label \_" or \^" and indegree 2. The size of the circuit is the number of gates m. We will count the oracle-gates separately. The depth of the circuit is the maximum length of the directed paths in the graph. Given an oracle f :! 1, with an oracle-gate of indegree d labeled i, we associate the i-th component of the restriction of f to d, and with the gates the Boolean function indicated by their labels. We dene the function g : n+r! computed by a node of the circuit as follows: For inputs and random bits, g is the corresponding projection function. For gates and oracle-gates, g is the composition of the associated function with the functions computed by the begin nodes of the incoming arcs. Here, in case of oracle-gates, the arguments are taken in the order corresponding to 3

4 the ordering of the incoming arcs. The circuit itself is said to compute all the functions computed by its nodes. In case r = 0, we will drop the term \randomized", if p = 0, the term \oracle-augmented", and \circuit" is short for \bounded fan-in Boolean circuit". A family of circuits is a sequence (C n ) 1, where C n=1 n is a circuit with n inputs. The family is called logspace-uniform if there is a transducer that on input n outputs a description of C n and uses space at most logarithmic in the size of C n. Let f :! be a function that maps strings of length n to strings of length s(n), k 2 N, : N! [0; 1), and g :! 1. The function f is computable in RNC k with condence using oracle g if there is a logspace-uniform family (C n ) 1 n=1 of randomized oracle-augmented circuits of depth O(log k n) such that for each n 2 N with n > 1 the following holds: C n, using oracle g, computes functions g 1 ; : : :; g s(n) such that 8 x 2 n : Pr [ 8i 2 f1; : : :; s(n)g : g i (x; ) = f i (x)] > (n); where f i (x) is the i-th bit of f(x), and the probability is over the uniform distribution of 2 r(n), r(n) being the number of random bits of C n. Note that we do not put an explicit polynomial bound on the size of the circuits. In case r 0, we say that f is computable in NC k (using oracle g); in case no circuit of the family has oracle-gates, we omit the oracle. A language L is in RNC k, if, for any constant 2 (0; 1), its characteristic function is computable in RNC k with condence by circuits that never err on inputs x 62 L. If these circuits have no random bits, L is said to be in NC k. Let k : N! N and : N! [0; 1]. A one-sided error randomized k-truth-table reduction with condence of a set A to a set B is a couple (f; g), where f and g are functions with domains 1, f maps inputs in n 1 to k(n)-tuples of strings (called queries), and g maps inputs in n 1 to Boolean functions in k(n) variables, such that for any x 2 : x 2 A ) Pr[g(x; )( B (f(x; ))) = 1] > (jxj) x 62 A ) : g(x; )( B (f(x; ))) = 0: : Here, B (f(x; )) = ( B (f i (x; ))) k(jxj) i=1 and f i (x; ) is the i-th component of f(x; ) for i 2 f1; : : :; k(jxj)g, and the probability is taken over the uniform distribution of 2 1. In case 1, the reduction becomes deterministic, and f and g can be dened as functions with domains. In case k 1 and g id, id representing the identity function on, f is called a (one-sided error randomized) many-one reduction of A to B. A bounded truth-table reduction is a k-truth-table reduction for some constant k. Let s; t : N! N. The reduction (f; g) is computable in RSPACE[s] respectively RTIME[t], if there are transducers M f and M g with 2 input tapes that, on input (x; ) 2 1, compute f(x; ) respectively g(x; ) in space O(s(jxj)) respectively time O(t(jxj)). If the reduction is deterministic, we say that (f; g) is computed in DSPACE[s] respectively DTIME[t]. In this context, we also use the shorthands RL, RP, L, and RL as above. Let ` 2 N. The reduction (f; g) is computable in RNC` if there is a logspace-uniform family (C n ) 1 of randomized circuits of depth O(log` n=1 n), such that C n computes each of the components of f(x; ) as well as the complete truth-table g(x; ) for inputs (x; ) 2 n 1, where the initial segment of is fed to the random bit nodes of the circuit. In case the reduction is deterministic, we call (f; g) computable in NC`. 4

5 An instance of the circuit value problem, CVP for short, consists of the description of a circuit C with one marked node and an input x, where C has jxj inputs. The instance <C; x> is positive if the marked node of C outputs 1 on input x. CVP is complete for P under logspace many-one reductions. Hence, it suces to prove that CVP is in (R)L to establish that P (R)L. It also suces to prove that CVP is in (R)NC k to establish that P (R)NC k (k 2 N, k > 1) [Coo85]. 3 Deterministic Reductions 3.1 Informal Description of the Generic Theorem Our goal is to give evidence that there are no sparse hard sets for P under logspace reductions, by proving that the hypothesis that there are, implies some unlikely inclusion of P in another complexity class, ideally P L. In the analogous problem for NP, where we wish to obtain the implication NP P from the existence of sparse hard sets under polynomial-time reductions, this is usually done by constructing a set W of possibly invalid membership witnesses, each of which can be checked on validity in P, that contains at least one valid witness for a positive instance of the problem. In the case of P, such a witness system in L does not exist by denition, but the complete problem CVP has a natural one: gate assignments can be checked on validity in L. So, the task is to construct such a witness set W within the given resource bounds, by making use of reductions of some language(s) L in the class we want to collapse, to a sparse set S. Much of the progress on this kind of problems relies on an appropriate choice of L. In [Ogi95], Ogihara considers the language L 1 = f<c; x; y; b> j C is the description of a circuit with ` inputs and m gates; x 2 `; y 2 m ; b 2 and m (y j ^ g j ) = bg; (1) where g j denotes the value of the j-th gate of C on input x for j 2 f1; : : :; mg, which clearly is in P and has the following trivial, but interesting properties: Let f be a many-one reduction of L 1 to a set S. Property 1 If f(<c; x; y; b>) 2 S, then denotes exclusive or. mm (y j ^ g j ) = b: (2) Property 2 For any syntactically correct <C; x; y >, one of f(<c; x; y; 0>) and f(<c; x; y; 1>) is in S. If S is sparse, for any xed instance < C; x > of CVP, Property 2 guarantees that many (y; b) 2 m are mapped by f to the same string w = f(< C; x; y ; b >) 2 S, so that according to Property 1 mm (y j ^ g j ) = mm (y j ^ g j) b b: (3) This equation actually also holds if w 62 S. In any case, if y 6= y, it allows to express one of the gate values as the parity of some other gate values. 5

6 Based on this observation, Ogihara manages in DSPACE[log 2 n] to construct an O(log 2 n) sized subset G of the gates of C and transform each of the gates not in G into parity gates. Since CVP for parity circuits is in DSPACE[log 2 n], cycling through all possible assignments to the O(log 2 n) gates in G yields a DSPACE[log 2 n] enumerable witness set W for CVP, whence a DSPACE[log 2 n] algorithm for CVP. Cai and Sivakumar [CS95] also use the language L 1, but view (2) as the linear equation mx y j g j = b (4) over Z 2 in the gate values g 1 ; : : :; g m, and try to set up a system of such equations of rank m? O(log m), which allows them to obtain a witness set W for CVP in NC 2 based on the following facts: Fact 1 The rank of a matrix can be determined in arithmetic NC 2. Fact 2 A square system of linear equations of full rank can be solved in arithmetic NC 2. Using Fact 1, given a system of equations of rank m? O(log m) over Z 2 in the gate values, they construct in NC 2 a subset of these equations and an O(log m) sized subset G of the gates, such that the resulting system in the other gate values is square and of full rank. The application of Fact 2 to these systems for all possible assignments to the O(log m) gates in G in parallel, results in an NC 2 witness set W for CVP. They show that the system consisting of all equations over Z 2 corresponding to (3) for (y; b); (y ; b ) 2 Y m for which 1 f(< C; x; y; b>) = f(< C; x; y ; b >), has rank m? O(log m) with high probability, if Y is a polynomial sized uniform sample from m. This yields a RNC 2 algorithm for CVP. Next they prove that the rank of this system is always m? O(log m), if Y = D with D the following \small-bias sample space" [NN90, AGsP90]: D = f(u i ; v ) m?1 i=0 j u; v 2 Fg; (5) where F denotes r seen as the extension eld GF(2 r ) constructed using the irreducible polynomial z r + z r= Z 2 [z], i.e., F = Z 2 [z]=(z r + z r=2 + 1); (6) for some r 2 f2:3 t jt 2 Ng [vl91] with r 2 O(log m) and ; : F 2! Z 2 : (u; v)! is the standard inner product of the vector space F over the eld Z 2. This way, they eectively derandomize their RNC 2 algorithm to obtain a NC 2 procedure for CVP. Finally, probably inspired by the construction of the sample space D used in the derandomization, Cai and Sivakumar consider the language L 2 = f<c; x; 1 r ; u; v > j C is the description of a circuit with ` inputs and m gates; rx u j v j x 2 `; r 2 f2:3 t jt 2 Ng; u; v 2 r and mx u j?1 g j = vg; (7) 1 The proposition actually also holds under the additional restriction that f(< C; x; y ; b >) 2 S, but the algorithm, of course, cannot check membership to S. 6

7 where g j denotes the value of the j-th gate of C on input x for j 2 f1; : : :; mg, and r is seen as the eld F dened by (6). This language also is in P and has the following similar properties to L 1 : Let f be a many-one reduction of L 2 to a set S. Property 3 If f(<c; x; 1 r ; u; v >) 2 S, then mx u j?1 g j = v (8) Property 4 For any syntactically correct < C; x; 1 r ; u >, there is (exactly) one v 2 r such that f(<c; x; 1 r ; u; v >) 2 S. In this context, we will use the following terminology. Denition 1 Let C be a circuit with ` inputs and m gates, x 2 `, and let g 1 ; : : :; g m denote the values of the gates of C on input x. Let F be the eld dened by (6) for some r 2 f2:3 t jt 2 Ng, and u; v 2 F. Then (u; v) is called the generator of the equation P m u j?1 z j = v in the variables z 1 ; : : :; z m, and (u; v) is the generator of a correct equation if v = P m u j?1 g j. The equations that result from Property 3 form a Vandermonde system, which is particularly interesting because Facts 1 and 2 can be strengthened as follows for such systems: Fact 3 Determining the rank of an m n Vandermonde matrix over the eld F only requires determining the cardinality of a multiset of m elements of F, and comparing it to n. Fact 4 An m m Vandermonde system of full rank over the eld F can be solved in NC 1 (m + log jfj) [CS95]. If r = log jfj is suciently large, Property 4 guarantees that for at least one pair (u ; v ) 2 F the set E(u ; v ) = f(u; v) 2 F 2 j f(<c; x; 1 r ; u; v >) = f(<c; x; 1 r ; u ; v >)g contains at least m generators, and f(< C; x; 1 r ; u ; v >) 2 S. Hence, it contains only generators of correct equations, and the equations that result from Property 3 form a Vandermonde system of full column rank m. If S is sparse, r 2 (log m) suces. So, applying Fact 4 to an m m subsystem generated by E(u ; v ) for all possible pairs (u ; v ) 2 F 2 yields a witness set W for CVP with complexity NC 1 modulo the complexity of the reduction, and complexity L if the logspace complexity of the reduction is taken into account. The corresponding circuit for CVP is sketched in Figure 1, where E j = E(u j ; v j ) and (u j ; v j ) ranges over F 2. This way, Cai and Sivakumar prove that the existence of a sparse hard set for P under logspace many-one reductions, implies that CVP 2 L. We will use the same scheme to prove that proposition for logspace bounded truth-table reductions, that is: Idea 1 Construct an oracle-augmented circuit for CVP with the structure outlined in Figure 1 using a logspace truth-table reduction of L 2 to a sparse set S as oracle. 7

8 ... reduction construct E and select n elements j n x n Vandermonde system solver solution checker... V CVP(<C,x>) Figure 1: Structure of a circuit for solving the circuit value problem. 8

9 When assuming the existence of a k-truth-table reduction (f; g) of L 1 or L 2 to a set S, there is no analogon of Facts 2 and 4, i.e., for no i 2 f1; : : :; kg, we can guarantee that many of the i-th queries belong to S, and hence, if S is sparse, many map to the same query (in S). However, for the case of a reduction from L 1, Cai, Naik and Sivakumar [CNS95] make the following observation: Let Y be any subset of m. Either there is an i 2 f1; : : :; kg and a string w i such that for many (y; b) 2 Y, f i (<C; x; y; b> ) = w i. In that case, we can reduce the problem to one involving a (k?1)-truth-table reduction by restricting the i-th query to w i and assuming both w i 2 S and w i 62 S in parallel. Since many couples (y; b) in Y map to w i, this restriction does not decrease the number of couples (y; b) by too much, and in one of the parallel executions, the assumption we make is correct. Either for all i 2 f1; : : :; kg and for any string w i, at most say q of the i-th queries for couples (y; b) 2 Y equal w i. If s is a bound on the length of the queries, this implies that there are at most k c S (s) q pairs (y; b) in Y for which not all queries are outside of S. Hence, if we partition Y into k c S (s) q + 1 classes of equal size ( 1), then for at least one class the assumption that all queries made are outside of S is correct, and that class contains at least jy j b kc c couples. All classes can be processed in parallel, and provided k c S (s)q+1 S(s) q is not too large compared to jy j, once again, we end up with large subclasses Y 0 of Y for at least one of which we make correct assumptions. Cai, Naik and Sivakumar then show that for a suciently large uniform sample Y of m, with high probability, for every large subset Y 0 of Y the system of equations mx y j g j = b + g(<c; x; y; b>)(( S (f i (<C; x; y; b>))) k i=1) + 1 (9) where (y; b) ranges over Y 0, has high rank. A class of large subsets Y 0 with correct assumptions about all the membership queries for at least one Y 0, can be constructed in NC 1 by the above procedure. Since Cai, Naik and Sivakumar prove that a polynomial sized sample Y suces to obtain rank m? O(log m) with high probability, the strategy of [CS95] discussed above allows them to solve CVP in RNC 2. If we want to apply the same idea using a truth-table reduction from L 2, there is an additional diculty related to the fact that, whereas in the case of L 1 for every (y; b) for which the memberships to S of the truth-table queries are known, we can construct a correct equation (9), even if g evaluates to 0, for L 2 this is only the case for generators (u; v) 2 F 2 of correct equations, i.e., when g evaluates to 1, which only happens for a small fraction of the generators. So, we have to restrict our attention to the set of generators (u; v) 2 F 2 for which g evaluates to 1. If there is an i 2 f1; : : :; kg and a string w i such that for many generators (u; v) of correct equations, f i (< C; x; 1 r ; u; v >) = w i, we can reduce the problem as above. However, there is a complication in the other case. In considering only the set G of generators (u; v) for which g(<c; x; 1 r ; u; v >)(0; : : :; 0) = 1, we exclude at most kc S (s)q of the generators of correct equations, so that should not be a problem. But there can be many more than k c S (s) q generators in G for which not all queries are outside of S, because the bound q only holds for generators of correct equations. Therefore, we cannot use the idea of partitioning G into subsets of equal size, at least one of which only contains generators all of whose queries are outside of S, since the resulting 9

10 subsets would have to be too small. Although there are other ways 2 to remedy this problem for small k, we solve the problem by using variable sized subsets of a particular kind: Idea 2 In case there is no popular i-th query among the generators of correct equations for any i 2 f1; : : :; kg, consider all subsets consisting of the generators in G whose i-th query is in I i for i 2 f1; : : :; kg, where I i is an interval of the set F i of all i-th queries made by the generators in G, for all combinations of intervals I i. Since S divides each set F i in at most c S (s) + 1 intervals disjoint from S, the generators in G all of whose queries are outside of S, form the union of (c S (s) + 1) k or less of such variable sized subsets. Hence, provided (c S (s) + 1) k and k c S (s) q are not too big compared to jfj, at least one of these subsets is large, and, as it only contains generators of correct equations, yields a Vandermonde system from which the gate values can be determined in NC 1 (m + log jfj). As before, all of these subsets can be processed in parallel. Before embarking on the formal proof that these ideas work, for the sake of completeness of the overview of recent progress on the sparse hard set problem for P, we mention the result of Cai, Naik and Sivakumar in [CNS95] that the existence of a sparse hard set for P under logspace bounded truth-table reductions implies that P NC 2. They use a reduction from L 1 for all elements of the set D dened by (5) and certain error-correcting capabilities of this small-bias sample space, which allows them to distill out of the equations (4) for all (y; b) 2 D for which g(< C; x; y; b >)(0; : : :; 0) = 1, a correct full rank Vandermonde system over F in the gate values, provided the assumption that all queries made by elements of D are outside S does not introduce too many false equations of the form (4). They show that otherwise, the NC 2 approach of [CS95] works. In no case 3 they make use of Fact 4, which is essential for our construction. 3.2 Formal Proof of the Generic Theorem First we will show in Lemma 1 that Idea 2 does indeed oer a way to construct a valid witness for positive instances of CVP. The proof of our main theorem then boils down to applying Idea 1 in order to eciently check the existence of a valid witness. Lemma 1 Let (f; g) be a truth-table reduction of the language L 2 dened by (7) to a set S, r 2 f2:3 t jt 2 Ng, and < C; x > an instance of size n of the circuit value problem. Suppose that (f; g) queries k strings of length s on inputs <C; x; 1 r ; u; v > with u; v 2 r. If 2 r > e n (c S (s) + 1) k k!; (10) then there are a subset I of f1; : : :; kg, strings w i for i 2 I, and intervals I i of disjoint from S for i 2 I c such that the set E = f(u; v) 2 r r j 8 >< >: 8 i 2 I : f i (<C; x; 1 r ; u; v >) = w i 8 i 2 I c : f i (<C; x; 1 r ; u; v >) 2 I i g(<c; x; 1 r ; u; v >)(b I;(wi ) i2i ) = 1 g; (11) 2 Remedies include limiting the number of generators that make queries of a particular kind, and case distinctions on g. 3 The Vandermonde system in [CNS95] has a prexed coecient matrix, whence solving it does not recur to Fact 4. 10

11 where has cardinality at least n. b I;(wi ) i2i = (b j) k with b j = ( S (w j ) if j 2 I 0 o.w.; (12) Note that E only contains generators of correct equations in the values of the gates of C on input x. Consequently, the rst components of the generators in E are all distinct. Proof Let the values of the m gates of C on input x be g 1 ; : : :; g m, let F denote r considered as the eld (6), and v : F! F : u! P m u j?1 g j. To simplify notation, we will write f i (u; v) instead of f i (<C; x; 1 r ; u; v >) and g(u; v) instead of g(<c; x; 1 r ; u; v >) for u; v 2 F and i 2 f1; : : :; kg. Let q : N! N be a function that will be specied later, once we know all the properties we need of it. It is obvious that there exist a subset I of f1; : : :; kg and strings w i for i 2 I such that 4 : jfu 2 Fj 8 i 2 I : f i (u; v(u)) = w i gj > q(ji c j) (13) 8 j 2 I c ; 8 w j 2 : jfu 2 Fj 8 i 2 I [ fjg : f i (u; v(u)) = w i gj 6 q(ji c j? 1); (14) provided that q(k) 6 jfj = 2 r. Note that I can be the empty set or the entire set f1; : : :; kg. Consider the following set G of generators of equations: where b I;(wj ) j2i G = f(u; v) 2 F 2 j 8 i 2 I : f i (u; v) = w i and g(u; v)(b I;(wj ) j2i ) = 1g; is dened by (12), the subset G 0 = f(u; v(u)) 2 F 2 j (u; v(u)) 2 G and 8 i 2 I c : f i (u; v(u)) 62 Sg consisting of generators of equations guaranteed correct, and the query projections F i = ff i (u; v) j (u; v) 2 Gg of G for i 2 f1; : : :; kg. Claim 1 There are intervals I i of for i 2 I c such that 8 i 2 I c : I i \ S =? (15) jf(u; v) 2 G 0 j 8 i 2 I c jg : f i (u; v) 2 I i gj > j (c S (s) + 1) ji c j : (16) This is because for each i 2 I c the set F i? S is the union of at most c S (s) + 1 intersections of F i with an interval, and every element of G 0 has its i-th query in F i? S for every i 2 I c. Condition (15) will guarantee that E only contains generators of correct equations. Condition (16) yields a lower bound on jej, when combined with the next claim. Claim 2 jg 0 j > q(ji c j)? ji c j c S (s) q(ji c j? 1), where q(?1) = 0. 4 The 6-sign in (14) can actually be replaced with a <-sign, but we will not need that strengthening. 11

12 Indeed, the set G 0 contains all couples (u; v(u)) 2 F 2 for which u is in the set of the left-hand side of (13), except those for which for at least one j 2 I c, f j (u; v(u)) is in S. Because of (14) the number of exceptions is bounded by ji c j c S (s) q(ji c j? 1). The combination of Claims 1 and 2 yields that jej > q(ji c j)? ji c j c S (s) q(ji c j? 1) ; (c S (s) + 1) ji c j which is at least n, provided that for i 2 f0; : : :; kg It is straightforward to check that the function q(i)? i c S (s) q(i? 1) (c S (s) + 1) i > n: q(i) = n (c S (s) + 1) i i! satises these conditions. The requirement q(k) 6 jfj = 2 r is also met if (10) holds, since P k j=0 e. The existence of such a function q concludes the proof of the lemma. 2 Following Idea 1, in order to prove the main theorem, we basically have to show how the set E from Lemma 1 can be constructed eciently in parallel. Theorem 2 (Generic Theorem) Let k; s : N! N and r : N! f2:3 t jt 2 Ng be space constructible functions, and suppose that the language L 2 dened by (7) is reducible to a set S by a truth-table reduction that queries k(n) strings of length s(n) on inputs < C; x; 1 r(n) ; u; v > with u; v 2 r(n) and <C; x> any instance of the circuit value problem, where n = j<c; x>j. If 2 r(n) > e n (c S (s(n)) + 1) k(n) (k(n))!; (17) then the circuit value problem can be computed by a logspace-uniform family of oracle-augmented bounded fan-in Boolean circuits of size 2 O(k(n)r(n)) poly(n; s(n)) and depth O(k(n) r(n) + log n + log s(n)) with the reduction as oracle, to which there are 2 2r(n) parallel calls on inputs of size O(n + r(n)). Proof Fix an instance <C; x> of size n of CVP. We will use the same notation as in the proof of Lemma 1, including the set E dened by (11) of generators of linear equations in the gate values g 1 ; : : :; g m of C on input x, which exists since the conditions of the Lemma are satised because of (17). As a technical aside, note that we can view these equations as equations in n variables, by adding the variables g m+1 ; : : :; g n. Since the rst components of generators in E are all distinct and jej > n, the extended Vandermonde system generated by E in the variables g 1 ; : : :; g n has rank n, and its unique solution forces all extra variables to zero. The outline is as follows: In parallel, we construct sets of generators E 1 ; E 2 ; : : :; E h, at least one of which is guaranteed to equal E. For each E j, we select n of its elements, (try to) solve the corresponding square Vandermonde system, and check whether the solution corresponds to a valid gate assignment with value 1. It is clear that <C; x> is a positive instance of CVP i for at least one E j the nal check is passed. This results in a circuit as depicted in Figure 1 with 5 levels, numbered from top to bottom: 12 ix j=0 1 j! 1 6 j!

13 Level 1 contains all the calls to the reduction oracle, namely one for each (u; v) 2 F 2. All these calls can be made in parallel. Level 2 consists of h similar parallel subcircuits, each corresponding to a set of generators E j. These subcircuits will be rened below. Level 2 is fully connected with level 1, and the subcircuit corresponding to E j outputs n generators in E j (if je j j > n). Level 3 are h identical n n Vandermonde system solvers, which output for each of the n components of the solution to the system, the least signicant bit of that component. The rst m of them are supposedly the values of the corresponding gate of C on input x. We use the logspace uniform family of bounded fan-in Boolean circuits of size O(poly(m; jfj)) and depth O(log m + log jfj) constructed in [CS95] to implement these modules. Each of the h solution checkers of level 4 is a simple circuit of size poly(n) and depth O(log n). Level 5 is an OR-tree of size O(h) and depth O(log h). To obtain a family E 1 ; : : ::E h of generator sets, at least one of which equals E, we can make use of the fact that if (u ; v ) 2 E, then where f(u; v) 2 F 2 j 8 >< >: 8 i 2 I : f i (u; v) = f i (u ; v ) 8 i 2 I c : f i (u; v) 2 I i g(u; v)(b) = 1; I; I i for i 2 I c ; and b = b I;(fi (u ;v )) i2i (18) are as in Lemma 1, equals E. We do not know the parameters (18), but we can check all possibilities in parallel. For I and b, it is obvious how to do this: there are 3 k possible pairs (I; b), since for each i 2 f1; : : :; kg it can be the case that i 62 I or (i 2 I and f i (u ; v ) 2 S) or (i 2 I and f i (u ; v ) 62 S). For the intervals I i (i 2 I c ), note that in the proof of Lemma 1, only the intersection of I i with the query projection g F i (u ; v ; I; b) = ff i (u; v)j ( 8 i 2 I : fi (u; v) = f i (u ; v ) g(u; v)(b) = 1 g is relevant, implying that there are 1 2 jf i(u ; v ; I; b)j(jf i (u ; v ; I; b)j + 1) 2 O(jFj 2 ) possibilities, each of which can be written as [f i (u i;1 ; v i;1 ); f i (u i;2 ; v i;2 )] for some (u i;1 ; v i;1 ; u i;2 ; v i;2 ) 2 F 4. Hence, if we consider all the generator sets E j of the form E(u ; v ; I; b; (u i;1 ; v i;1 ; u i;2 ; v i;2 ) i2i c) = f(u; v) 2 F 2 j 8 >< 8 i 2 I : f i (u; v) = f i (u ; v ) 8 i 2 I >: c : f i (u i;1 ; v i;1 ) 6 f i (u; v) 6 f i (u i;2 ; v i;2 ) g(u; v)(b) = 1 g (19) where u ; v ; u i;1 ; v i;1 ; u i;2 ; v i;2 range over F, and (I; b) ranges over the 3 k possibilities described above, it is clear that at least one of them equals E. The number h of such generator sets is in O(3 k :jfj 4k+2 ). The subcircuit of level 2 in Figure 1 corresponding to the generator set E j dened by (19) can be implemented in 2 sublevels as follows: 13

14 The rst sublevel determines for each (u; v) 2 F 2 whether (u; v) 2 E j. By a straightforward application of the denition of (19) of E j, each of these bits can be computed by a circuit of size poly(k; s) and depth O(log k + log s). The second sublevel determines for each i 2 f1; : : :; ng the i-th element of E j according to the lexicographic ordering, which can be performed by a circuit of size O(n:poly(jFj)) and depth O(log jfj). So, each of the h modules of level 2 has size poly(n; k; s; jfj) and depth O(log k + log s + log jfj). Finally, the resulting family of circuits for CVP has size O(3 k :jfj 4k+O(1) ):poly(n; k; s) 2 O(kr) :poly(n; s), and depth O(k: log jfj + log n + log s) = O(kr + log n + log s). It is straightforward to check that the family is logspace uniform. 2 We observe that the right-hand side of condition (17) is remarkably similar to the bound on the number of witness intervals that have to kept track of in the proof in [HL94] that the existence of a sparse hard set for NP under polynomial-time bounded truth-table reductions implies that NP P. Using our notation, that bound reads n 4 k(n) (c S (s(n)) + 1) k(n) (k(n) + 1)!. Theorem 2 will be applied in the following way: Corollary 3 Let b; c; k; s : N! N be such that b, k and s are space constructible, b(o(n)) O(b) and log s(o(n)) O(log s), and let S be a set for which c S (n) 6 c(n). If S is hard for P under DSPACE[b(n)] computable truth-table reductions that make k(n) queries of length s(n) on inputs of length n, then P DSPACE[(poly n)], where (n) = k(n) r(n) + log s(n) + b(n) (20) r(n) = log n + k(n) (log c(s(n)) + log k(n)); provided r 2 o(n), r(o(n)) O(r), and r is space constructible. Moreover, if (n) 2 O(log n) and the reductions are computable in NC 1, then P = NC 1. Proof (Sketch) It is argued in [CS95] that L 2 2 P, and it is straightforward to check that under the given conditions, Theorem 2 can be used (by slightly modifying r, k, and s) to show that CVP 2 DSPACE[(n)]. As CVP is hard for P under logspace many-one reductions and (n) 2 (log n), it follows that P DSPACE[(poly n)]. In case (n) 2 O(log n) and the reductions are computable in NC 1, Theorem 2 yields that CVP 2 NC 1, whence P NC Instantiations of the Generic Theorem We now make instantiations of Corollary 3 to Theorem 2 for various choices of the relevant parameters: the sparseness of S: polynomial or quasipolynomial density, the resource bounds of the reductions: logspace, NC 1 or polylog-space, and the bound on the number of truth-table queries: constant or polylogarithmic. For polynomially dense hard sets we obtain the following theorem: 14

15 Theorem 4 Let e > 1 and f > 0. If there is a sparse hard set for P under truth-table reductions computable in DSPACE[log e n] that make at most O(log f n) queries, then P DSPACE[log e+2f n]. Putting e = 1 and f = 0 in Theorem 4, we are able to solve an open problem in [CNS95]. Corollary 5 There is a sparse hard set for P under logspace bounded truth-table reductions i P = L. Using the full power of Corollary 3, we can also state: Corollary 6 There is a sparse hard set for P under bounded truth-table reductions computable in NC 1 i P = NC 1. Corollaries 5 and 6 lead to the following observations for higher complexity classes than P, when combined with the results of [OW91, OL91]. Corollary 7 Let C be NP, PP, C = P or Mod k P for some k > 2. There exists a sparse hard set for C under logspace bounded truth-table reductions i C = L. Corollary 8 Let C be NP, PP, C = P or Mod k P for some k > 2. There exists a sparse hard set for C under bounded truth-table reductions computable in NC 1 i C = NC 1. Note that because of the term k(n)log n in (20), our Corollary 3 only allows to obtain the conclusion P = L for a constant bound on the number of queries, no matter how sparse the set S is. This contrasts with the analogous theorem for the sparse hard set problem for NP in [HL94], which also gives the conclusion P = NP assuming the existence of a polylogarithmically dense hard set for NP under polynomial-time truth-table reductions with some non-constant bound on the number of queries (namely O( log n )). log log n The case e = f = 1 in Theorem 4 settles another open problem in [CNS95]. Corollary 9 If there is a sparse hard set for P under logspace O(log n)-truth-table reductions, then P DSPACE[log 3 n]. Essentially the same conclusion can be drawn, if we relax the sparseness condition to quasipolynomially dense, and allow the truth-table reductions to use polylogarithmic space as well as a polylogarithmic number of queries. Theorem 10 Let d > 0, e > 1 and f > 0. If there is a set with density bounded by 2 O(logd n) that is hard for P under DSPACE[log e n] truth-table reductions using at most O(log f n) queries, then P DSPACE[log a n], where a = max(de + 2f; e; f + 1). In particular, the following in some sense stronger resolution of the second open problem in [CNS95] mentioned above, holds: Corollary 11 There is a quasipolynomially dense hard set for P under polylog-space truth-table reductions using no more than polylogarithmically many queries i P DSPACE[polylog n]. Finally, we note that we can obtain collapses of lower complexity classes than P, by applying the idea of Theorem 2 to easier languages than CVP. 15

16 Theorem 12 There is a sparse hard set for L under bounded truth-table reductions computable in NC 1 i L = NC 1. Proof (Sketch) Consider the language L 3 = f<g; s; t; 1 r ; u; v > j G is the description of a forest with m edges; s; t 2 V (G); r 2 f2:3 t jt 2 Ng; u; v 2 r and mx u j?1 e j = vg; where e j = 1 if s and t are in the same component tree of G and the j-th edge of G is on the unique path connecting s and t, and e j = 0 otherwise, and r is the eld F dened by (6). Note that all arithmetic involved can be carried out in NC 1, which allows to show that L 3 2 L. Based on the strategy of the proof of Theorem 2, using a reduction of L 3 instead of L 2 to S, we can construct an NC 1 algorithm for the language f<g; s; t> j G is the description of a forest; s; t 2 V (G) and s and t are connected in Gg; which suces to show that L NC 1 [CM87]. 2 In the last theorem of this section, we apply the proof scheme of Theorem 2 to each individual language of a class instead of to a single complete one. Theorem 13 Let 1 6 ` 6 k. There is a sparse hard set for NC k under bounded truth-table reductions computable in NC` i NC k = NC`. Proof (Sketch) Fix a language L 2 NC k, and consider L 0 = f<x; 1 r ; u; v > j x 2 L; r 2 f2:3 t jt 2 Ng; u; v 2 r and mx u j?1 g j = vg; where g 1 ; : : :; g m are the values on input x of the m gates of the NC k circuit that decides membership to L for strings of length jxj, and r is the eld F dened by (6). Noting that all arithmetic involved can be carried out in NC 1, it is clear that L 0 2 NC k. Using an NC` computable reduction of L 0 instead of L 2 to S in the proof of Theorem 2, allows to construct an NC` family of circuits for L. 2 Theorems 12 and 13 extend analogous theorems proven in [CNS95] for many-one reductions. 4 Randomized Reductions 4.1 Generic Theorem We now show how we can extend our generic theorem to randomized truth-table reductions with one-sided error. Since the generalization is fairly straightforward, in the proofs we will only indicate how the corresponding ones from section 3.2 for deterministic truth-table reductions can be adapted. 16

17 Lemma 14 Let (f; g) be a one-sided error randomized truth-table reduction of the language L 2 dened by (7) to a set S, r 2 f2:3 t jt 2 Ng, 2 [0; 1], and < C; x > an instance of size n of the circuit value problem. Suppose that (f; g) queries k strings of length s and has condence at least on inputs <C; x; 1 r ; u; v > with u; v 2 r. If 2 r > e n (c S (s) + 1) k k! > e (c S (s) + 1) k k!; then there are a subset I of f1; : : :; kg, strings w i for i 2 I, and intervals I i of disjoint from S for i 2 I c such that the set E = f(u; v) 2 r r j Pr [ 8 >< >: 8 i 2 I : f i (<C; x; 1 r ; u; v >; ) = w i 8 i 2 I c : f i (<C; x; 1 r ; u; v >; ) 2 I i g(<c; x; 1 r ; u; v >; )(b I;(wi ) i2i ) = 1 ] > g; (21) where b I;(wi ) i2i is dened by (12), has cardinality at least n. As before, E only contains generators of correct equations in the values of the gates of C on input x, whence the rst components of the generators in E are all distinct. Proof (Sketch) We use analogous notation as in the proof of Lemma 1, and follow the same outline. We also consider functions ; : N! [0; 1] to be specied later. Provided that q(k) 6 jfj and (k) 6, there exist a subset I of f1; : : :; kg and strings w i for i 2 I such that: jfu 2 Fj Pr [ 8 i 2 I : f i (u; v(u); ) = w i and g(u; v(u); )(b u; ) = 1] > (ji c j)gj > q(ji c j) 8 j 2 I c ; 8 w j 2 : jfu 2 Fj Pr [ 8 i 2 I [ fjg : f i (u; v(u); ) = w i where b u; = ( S (f i (u; v(u); ))) k i=1. Considering the set of generators where b I;(wj ) j2i G 0 = f(u; v(u)) 2 F 2 j Pr and g(u; v(u); )(b u; ) = 1] > (ji c j? 1)gj 6 q(ji c j? 1); [ 8 >< >: 8 i 2 I : f i (u; v(u); ) = w i 8 i 2 I c : f i (u; v(u); ) 62 S g(u; v(u); )(b I;(wi ) i2i ) = 1 ] > (ji c j)g; is dened by (12), the following claims can be made: Claim 1 There are intervals I i of for i 2 I c such that 8 i 2 I c : I i \ S =? jf(u; v(u)) 2 F 2 j Pr > [ 8 >< >: jg 0 j (c S (s) + 1) ji c j : 8 i 2 I : f i (u; v(u); ) = w i 8 i 2 I c : f i (u; v(u); ) 2 I i g(u; v(u); )(b I;(wi ) i2i ) = 1 ] > (ji c j) (c S (s) + 1) ji c j gj 17

18 Claim 2 jg 0 j > q(ji c j)?ji c jc S (s)q(ji c j?1), provided that (ji c j) 6 (ji c j)?ji c jc S (s)(ji c j?1), where q(?1) = (?1) = 0. Hence, combining Claims 1 and 2, and letting (i) = (i)? i c S (s) (i? 1) for i 2 f0; : : :; kg, we obtain that jej > n, provided that for i 2 f0; : : :; kg: The functions q(i)? i c S (s) q(i? 1) (c S (s) + 1) i > n (i)? i c S (s) (i? 1) (c S (s) + 1) i > : q(i) = n (c S (s) + 1) i i! (i) = (c S (s) + 1) i i! satisfy these conditions, as well as the requirements q(k) 6 jfj = 2 r and (k) 6. 2 This leads to the following generic theorem: Theorem 15 (Generic Theorem) Let k; s; t : N! N and r : N! f2:3 t jt 2 Ng be space constructible functions, : N! [0; 1), and suppose that the language L 2 dened by (7) is reducible to a set S by a one-sided error randomized truth-table reduction that queries k(n) strings of length s(n) and has condence at least (n) on inputs < C; x; 1 r(n) ; u; v > with u; v 2 r(n) and < C; x > any instance of the circuit value problem, where n = j<c; x>j. If 2 r(n) > e n (c S (s(n)) + 1) k(n) (k(n))! (22) t(n) > e ln( n 1?(n) (c S (s(n)) + 1) k(n) (k(n))!; (n) (23) then the circuit value problem can be computed by a logspace-uniform family of randomized oracleaugmented bounded fan-in Boolean circuits with one-sided error and condence at least (n), of size 2 O(k(n)(r(n)+t(n))) poly(n; s(n)) and depth O(k(n) (r(n) + t(n)) + log n + log s(n)) with the randomized reduction as oracle. The oracle is called t(n) times for each of 2 2r(n) inputs of size O(n + r(n)), and is the only randomized component of the circuits. Proof (Sketch) Fix an instance <C; x> of size n of CVP. We will use analogous notation as in the proof of Theorem 2, with the set E now being dened by (21), in which we choose =. Note that E e(c S (s)+1) k k! exists and has size at least n, since the conditions of Lemma 14 are satised. The outline is the same as in Theorem 2, except that we now construct in parallel sets of generators E 1 ; E 2 ; : : :; E h such that at least one of them only contains generators of correct equations and, ix j=0 ix j=0 1 j! 1 j! 18

19 with probability at least, contains at least n (xed) elements of E. We consider all sets E j of the form E(u ; v ; ; I; b; (u i;1 ; v i;1 ; i;1 ; u i;2 ; v i;2 ; i;2 ) i2i c) = f(u; v) 2 F 2 j 9 2 f1; : : :; tg : 8 >< 8 i 2 I : f i (u; v; ) = f i (u ; v ; ) 8 i 2 I >: c : f i (u i;1 ; v i;1 ; i;1 ) 6 f i (u; v; ) 6 f i (u i;2 ; v i;2 ; i;2 ) g g(u; v; )(b) = 1 where u ; v ; u i;1 ; v i;1 ; u i;2 ; v i;2 range over F, (I; b) over the 3 k possibilities described in the proof of Theorem 2, and ; i;1 ; i;2 over f1; : : :; tg. Here f i (u; v; ) stands for f i (<C; x; 1 r ; u; v >; (u; v)) and g(u; v; ) for g(< C; x; 1 r ; u; v >; (u; v)), where (u; v) is the random string used by the reduction oracle during its -th call on input < C; x; 1 r ; u; v >. The number h of such generator sets is in O(3 k :jfj 4k+2 :t 2k+1 ). Note that at least one E j equals E = f(u; v) 2 F 2 j 9 2 f1; : : :; tg : 8 >< >: 8 i 2 I : f i (u; v; ) = w i 8 i 2 I c : f i (u; v; ) 2 I i g(u; v; )(b I;(wi ) i2i ) = 1 g: This set only contains generators of correct equations. For any xed (u 0 ; v 0 ) 2 E, the probability that it does not belong to E is at most (1? ) t. Hence, the probability that E does not contain all of n xed generators in E is bounded by n (1? ) t n 6 exp(ln n? t) 6 exp(ln n? ln( )) = 1? : 1? The structure of the circuits for CVP is again the one of Figure 1. The only dierences with the circuits constructed in Theorem 2 are the following: Level 1 now contains t calls to the reduction oracle for each (u; v) 2 F 2. The rst sublevel of the level 2 subcircuit corresponding to E j, which determines for each (u; v) 2 F 2 whether (u; v) 2 E j, can be implemented by a circuit of size poly(k; s; t) and depth O(log k + log s + log t) for each (u; v) 2 F 2. It follows that the resulting family of circuits for CVP has size O(3 k :jfj 4k+O(1) :t 2k+1 ):poly(n; k; s; t) 2 O(k(r+t)) :poly(n; s), and depth O(k: log jfj + k: log t + log n + log s) = O(k(r + t) + log n + log s). It is again straightforward to check that the family is logspace uniform. 2 Analogously to the deterministic case, we will use Theorem 15 as follows: Corollary 16 Let b; c; k; s : N! N be such that b, k and s are space constructible, b(o(n)) O(b) and log s(o(n)) O(log s), let : N! (0; 1] be such that ((O(n))?1 O(?1 ), and let S be a set for which c S (n) 6 c(n). If S is hard for P under one-sided error randomized truth-table reductions computable in RSPACE[b(n)] that make k(n) queries of length s(n) and have condence at least (n) on inputs of length n, then P RSPACE[(poly n)], where (n) = k(n) (n) + log s(n) + b(n) (n) = log n + log( log n ) + k(n) (log c(s(n)) + log k(n)); (n) provided 2 o(n), (O(n)) O(), and is space constructible. Moreover, if (n) 2 O(log n) and the reductions are computable in RNC 1, then P RNC 1. 19

20 Proof (Sketch) Analogous to the proof of Corollary 3. For the application of Theorem 15, We take (n) to be any constant in (0; 1). 2 Note that it is crucial for Corollary 16 that the randomized space complexity classes are dened using the multiple access concept, i.e., the standard Turing machine endowed with a random bit tape, which is considered as an additional input tape in the sense that the Turing machine has full read access to it and is not charged for the space of the tape segment accessed. This allows the Turing machine to simulate the circuits of Theorem 15 within the given space bounds, because it can force the same random bit sequence to be used in several calls to the reduction oracle on the same input, guaranteeing we obtain the same answer, without having to store these bit sequences on its work tape. Randomized circuits are implicitly granted the ability to regenerate random bit sequences, and the circuits of Theorem 15 make intensive use of it. In general, for randomized space bounded computation, Nisan gives strong evidence that this multiple access concept is more powerful than the natural read-once randomness concept, in which the Turing machine only has one-way read access to its random bit tape, or equivalently, is endowed with the ability to toss an unbiased coin instead of using an extra input tape with random bits [Nis93]. 4.2 Instantiations of the Generic Theorem We obtain the following results for polynomially dense hard sets, similar to those in section 3.3. They extend theorems of [CNS95] for randomized many-one reductions with one-sided error. Theorem 17 Let e > 1, f > 0, and g > 0. If there is a sparse hard set for P under onesided error randomized truth-table reductions computable in RSPACE[log e n] and with condence at least 2?O(logg n) that make at most O(log f n) queries, then P RSPACE[log a n], where a = max(2e + f; e + g). Corollary 18 There is a sparse hard set for P under one-sided error randomized bounded truthtable reductions computable in RL and with condence at least inversely polynomial i P RL. Corollary 19 There is a sparse hard set for P under one-sided error randomized bounded truthtable reductions computable in RNC 1 and with condence at least inversely polynomial i P RNC 1. Corollary 20 If there is a sparse hard set for P under one-sided error randomized O(log n)-truthtable reductions computable in RL and with condence at least 2?O(log2 n), then P RSPACE[log 3 n]. The results for quasipolynomially dense hard sets from section 3.3 can also be translated readily. Theorem 21 Let d > 0, e > 1, f > 0, and g > 0. If there is a set with density bounded by 2 O(logd n) that is hard for P under one-sided error randomized truth-table reductions computable in RSPACE[log e n] and with condence at least 2?O(logg n) that use at most O(log f n) queries, then P RSPACE[log a n], where a = max(de + 2f; e; f + 1; f + g). Corollary 22 There is a quasipolynomially dense hard set for P under one-sided error randomized polylog-space truth-table reductions with condence at least inversely quasipolynomial that use no more than polylogarithmically many queries i P RSPACE[polylog n]. 20