Notes on Complexty Theory Last updated: October, 2008 Jonathan Katz Lecture Space-Bounded Derandomzaton 1 Space-Bounded Derandomzaton We now dscuss derandomzaton of space-bounded algorthms. Here non-trval results can be shown wthout mang any unproven assumptons, n contrast to what s currently nown for derandomzng tme-bounded algorthms. We show frst that 1 BPL SPACElog 2 n) and then mprove the analyss and show that 2 BPL TmeSpcpolyn), log 2 n) SC. Note: we already now RL N L SPACElog 2 n) but ths does not by tself mply BPL SPACElog 2 n).) Wth regard to the frst result, we actually prove somethng more general: Theorem 1 Any randomzed algorthm wth two-sded error) that uses space S Ωlog n) and R random bts can be converted to one that uses space OS log R) and OS log R) random bts. Snce any algorthm usng space S uses tme at most 2 S by our conventon regardng probablstc machnes) and hence at most ths many random bts, the followng s an mmedate corollary: Corollary 2 For S Ωlog n) t holds that BPSPACES) SPACES 2 ). Proof Let L BPSPACES). Theorem 1 shows that L can be decded by a probablstc machne wth two-sded error usng OS 2 ) space and OS 2 ) random bts. Enumeratng over all random bts and tang majorty, we obtan a determnstc algorthm that uses OS 2 ) space. 2 BPL SPACElog 2 n) We now prove Theorem 1. Let M be a probablstc machne runnng n space S and tme 2 S ), usng R random bts, and decdng a language L wth two-sded error. Note that S, R are functons of the nput length n, and the theorem requres S Ωlog n).) We wll assume wthout loss of generalty that M always uses exactly R random bts on all nputs. Fxng an nput x and lettng l be some parameter, we wll vew the computaton of M x as a random wal on a mult-graph n the followng way: the nodes of the graph correspond to all N def 2 OS) possble confguratons of M x, and there s an edge from a to b labeled by the strng r {0, 1} l f and only f M x moves from confguraton a to confguraton b after readng r as ts next l random bts. Computaton of M x s then equvalent to a random wal of length R/l on ths graph, begnnng from the node correspondng to the ntal confguraton of M x. f x L then the probablty that ths random 1 BPL s the two-sded-error verson of RL. 2 SC stands for Steve s class, and captures computaton that smultaneously uses polynomal tme and polylogarthmc space. Space-Bounded Derandomzaton-1
wal ends up n an acceptng state s at least 2/3, whle f x L then the probablty that ths random wal ends up n an acceptng state s at most 1/3. It wll be convenent to represent ths process usng an N N transton matrx Q x, where the entry n column, row j s the probablty that M x moves from confguraton to confguraton j after readng l random bts. Vectors of length N whose entres are non-negatve and sum to 1 correspond to probablty dstrbutons over the confguratons of M x n the natural way. If we let s denote the probablty dstrbuton that places probablty 1 on the ntal confguraton of M x and 0 elsewhere), then Q R/l x s corresponds to the probablty dstrbuton over the fnal confguraton of M x ; thus: x L accept x L accept ) Q R/l x s Q R/l x s 3/4 ) 1/4. The statstcal dfference between two vectors/probablty dstrbutons s, s s SDs, s ) def 1 2 s s 1 2 s s. If Q, Q are two transton matrces meanng that all entres are non-negatve, and the entres n each column sum to 1 then we abuse notaton and defne SDQ, Q ) def max s {SDQs, Q s)}, where the maxmum s taen over all s that correspond to probablty dstrbutons. Note that f Q, Q are N N transton matrces and max,j { Q,j Q,j } ε, then SDQ, Q ) Nε/2. 2.1 A Useful Lemma The pseudorandom generator we construct wll use a famly H of parwse-ndependent functons as a buldng bloc. Defnton 1 H {h : {0, 1} l {0, 1} l } s a famly of parwse-ndependent functons f for all dstnct x 1, x 2 {0, 1} l and any y 1, y 2 {0, 1} l we have: Pr [hx 1) y 1 hx 2 ) y 2 ] 2 2l. h H It s easy to construct a parwse-ndependent famly H whose functons map l-bt strngs to l-bt strngs and such that 1) H 2 2l and so choosng a random member of H s equvalent to choosng a random 2l-bt strng) and 2) functons n H can be evaluated n Ol) space. For S {0, 1} l, defne ρs) def S /2 l. We defne a useful property and then show that a functon chosen from a parwse-ndependent famly satsfes the property wth hgh probablty. Defnton 2 Let A, B {0, 1} l, h : {0, 1} l {0, 1} l, and ε > 0. We say h s ε, A, B)-good f: Pr x A ] hx) B ρa) ρb) ε. x {0,1} l [ Space-Bounded Derandomzaton-2
Note that ths s equvalent to sayng that h s ε, A, B)-good f Pr [hx) B] ρb) ε/ρa). x A Lemma 3 Let A, B {0, 1} l, H be a famly of parwse-ndependent functons, and ε > 0. Then: ρa)ρb) Pr [h s not ε, A, B)-good] h H 2 l ε 2. Proof The proof s farly straghtforward. Consder the quantty [ ) ] 2 µ def Exp h H ρb) Pr [hx) B] x A Exp h H [ρb) 2 + Pr [hx 1) B] Pr [hx 2) B] 2ρB) Pr x 1 A x 2 A ρb) 2 [ ] + Exp x1,x 2 A; h H δhx1 ) B δ hx2 ) B 2ρB) δ hx1 ) B, ] [hx 1) B] x 1 A where δ hx) B s an ndcator random varable whch s equal to 1 f hx) B and 0 otherwse. Snce H s parwse ndependent, t follows that: For any x 1 we have Exp h H [δ hx1 ) B] Pr h H [hx 1 ) B] ρb). For any x 1 x 2 we have Exp h H [δ hx1 ) B δ hx2 ) B] Exp h H [δ hx1 ) B] ρb). For any x 1 x 2 we have Exp h H [δ hx1 ) B δ hx2 ) B] Pr h H [hx 1 ) B hx 2 ) B] ρb) 2. Usng the above, we obtan µ ρb) 2 + ρb) + ρb)2 1) 2ρB) 2 ρb) ρb)2 ρb)1 ρb)). Usng Marov s nequalty, Pr [h s not ε, A, B)-good] Pr h H h H [ ) ] 2 Pr [hx) B] ρb) > ε/ρa)) 2 x A µ ρa)2 ε 2 ρb)1 ρb))ρa) 2 l ε 2 ρb)ρa) 2 l ε 2. 2.2 The Pseudorandom Generator and Its Analyss 2.2.1 The Basc Step We frst show how to reduce the number of random bts by roughly half. Let H denote a parwsendependent famly of functons, and fx an nput x. Let Q denote the transton matrx correspondng to transtons n M x after readng l random bts; that s, the, j)th entry of Q s the Space-Bounded Derandomzaton-3
probablty that M x, startng n confguraton, moves to confguraton j after readng l random bts. So Q 2 s a transton matrx denotng the probablty that M x, startng n confguraton, moves to confguraton j after readng 2l random bts. Fxng h H, let Q h be a transton matrx where the, j)th entry n Q h s the probablty that M x, startng n confguraton, moves to confguraton j after readng the 2l random bts r hr) where r {0, 1} l s chosen unformly at random). Put dfferently, Q 2 corresponds to tang two unform and ndependent steps of a random wal, whereas Q h corresponds to tang two steps of a random wal where the frst step gven by r) s random and the second step namely, hr)) s a determnstc functon of the frst. We now show that these two transton matrces are very close. Specfcally: Defnton 3 Let Q, Q h, l be as defned above, and ε 0. We say h H s ε-good for Q f SDQ h, Q 2 ) ε/2. Lemma 4 Let H be a parwse-ndependent functon famly, and let Q be an N N transton matrx where transtons correspond to readng l random bts. For any ε > 0 we have: N 6 Pr [h s not ε-good for Q] h H ε 2 2 l. Proof For, j [N] correspondng to confguratons n M x ), defne B,j def {x {0, 1} l x taes Q from to j}. For fxed, j,, we now from Lemma 3 that the probablty that h s not ε/n 2, B,j, B j, )-good s at most N 4 ρb,j )/ε 2 2 l. Applyng a unon bound over all N 3 trples, j, [N], and notng that for any we have j ρb,j) 1, we have that h s ε/n 2, B,j, B j, )-good for all, j, except wth probablty at most N 6 /ε 2 2 l. We show that whenever h s ε/n 2, B,j, B j, )-good for all, j,, then h s ε-good for Q. Consder the, )th entry n Q h ; ths s gven by: j [N] Pr[r B,j hr) B j, ]. On the other hand, the, )th entry n Q 2 s: j [N] ρb,j) ρb j, ). Snce h s ε/n 2, B,j, B j, )-good for every, j,, the absolute value of ther dfference s )) Pr[r B,j hr) B j, ] ρb,j ) ρb j, j [N] j [N] j [N] Pr[r B,j hr) B j, ] ρb,j ) ρb j, ) ε/n 2 ε/n. It follows that SDQ h, Q 2 ) ε/2 as desred. The lemma above gves us a pseudorandom generator that reduces the requred randomness by roughly) half. Specfcally, defne a pseudorandom generator G 1 : {0, 1} 2l+R/2 {0, 1} R va: G 1 r 1,..., r R/2l ; h) r 1 hr 1 ) r R/2l hr R/2l ), 1) Space-Bounded Derandomzaton-4
where h H so h 2l) and r {0, 1} l. Assume h s ε-good for Q. Runnng M x usng the output of G 1 h, ) as the random tape generates the probablty dstrbuton R/2l Q h Q h s for the fnal confguraton, where s denotes the ntal confguraton of M x.e., s s the probablty dstrbuton that places probablty 1 on the ntal confguraton of M x, and 0 elsewhere). Runnng M x on a truly random tape generates the probablty dstrbuton R/2l Q 2 Q 2 s for the fnal confguraton. Lettng R/2l we have 2 SD Q h Q h s, ) Q 2 Q 2 s ) Q h Q h Q 2 Q 2 s 1 ) Q h Q h Q 2 Q 2 Q h Q h Q 2 Q 2 s 0 0 1 1 +1 +1 1 ) Q h Q h Q 2 Q 2 Q h Q h Q 2 Q 2 s 1 1 Q h Q h Q h Q 2 ) Q 2 Q 2 s 0 ε. Ths means that the behavor of M x when run usng the output of the pseudorandom generator s very close to the behavor of M x when run usng a truly random tape: n partcular, f x L then M x n the former case accepts wth probablty at most Pr[accepts h s ε-good for Q] + Pr[h s not ε-good for Q] 1/4 + ε/2) + N 6 /ε 2 2 l ; smlarly, f x L then M x n the former case accepts wth probablty at least 3/4 ε/2 N 6 /ε 2 2 l. Summarzng and slghtly generalzng): Corollary 5 Let H be a parwse-ndependent functon famly, let Q be an N N transton matrx where transtons correspond to readng l random bts, let > 0 be an nteger, and let ε > 0. Then except wth probablty at most N 6 /ε 2 2 l over choce of h H we have: SD Q h Q h, Q 2 Q 2 ) ε/2. Space-Bounded Derandomzaton-5
2.2.2 Recursng Fxng h 1 H, note that Q h1 s a transton matrx and so we can apply Corollary 5 to t as well. Moreover, f Q uses R random bts then Q h1 uses R/2 random bts treatng h 1 as fxed). Contnung n ths way for I def logr/2l) + 1 logr/l) teratons, we obtan a transton matrx Q h1,...,h I. Say all h are ε-good f h 1 s ε-good for Q, and for each > 1 t holds that h s ε-good for Q h1,...,h 1. By Corollary 5 we have: All h are ε-good except wth probablty at most N 6 I/ε 2 2 l. If all h are ε-good then SDQ h1,...,h I, R/2l Q 2 Q 2 ) ε 2 I 1 R 2 l ε ) R 2 l 1. Equvalently, we obtan a pseudorandom generator G I r; h 1,..., h I ) def G I 1 r; h 1,..., h I 1 ) G I 1 h I r); h 1,..., h I 1 ), where G 1 s as n Equaton 1). 2.2.3 Puttng t All Together We now easly obtan the desred derandomzaton. Recall N 2 Os). Set ε 2 S /10, and set l ΘS) so that N 6 S 1/20. Then the number of random bts used as nput to G ε 2 2 l I from the prevous secton) s Ol logr/l) + l) OS log R) and the space used s bounded by that as well usng the fact that each h H can be evaluated usng space Ol) OS)). All h are good except wth probablty at most N 6 logr/l)/ε 2 2 l N 6 S/ε 2 2 l 1/20; assumng all h are good, the statstcal dfference between an executon of the orgnal algorthm and the algorthm run wth a pseudorandom tape s bounded by 2 S /20 R 1/20. Theorem 1 follows easly. 3 BPL SC A determnstc algorthm usng space Olog 2 n) mght potentally run for 2 Olog2 n) steps; n fact, as descrbed, the algorthm from the proof of Corollary 2 uses ths much tme. For the partcular pseudorandom generator we have descrbed, however, t s possble to do better. The ey observaton s that nstead of just choosng the h 1,..., h I at random and smply hopng that they are all ε-good, we wll nstead determnstcally search for h 1,..., h I whch are each ε-good. Ths can be done n polynomal tme when S Olog n)) because: 1) for a gven transton matrx Q h1,...,h 1 and canddate h, t s possble to determne n polynomal tme and polylogarthmc space whether h s ε-good for Q h1,...,h 1 ths reles on the fact that the number of confguratons N s polynomal n n); 2) there are only a polynomal number of possbltes for each h snce l ΘS) Olog n)). Once we have found the good {h }, we then cycle through all possble choces of the seed r {0, 1} l and tae majorty as before). Snce there are a polynomal number of possble seeds, the algorthm as a whole runs n polynomal tme. Space-Bounded Derandomzaton-6
For completeness, we dscuss the case of general S Ωlog n) assumng R 2 S. Checng whether a partcular h s ε-good requres tme 2 OS). There are 2 OS) functons to search through at each stage, and OS) stages altogether. Fnally, once we obtan the good {h } we must then enumerate through 2 OS) seeds. The end result s that BPSPACES) TmeSpc2 OS), S 2 ).) Bblographc Notes The results descrbed here are due to [2, 3], both of whch are very readable. See also [1, Lecture 16] for a slghtly dfferent presentaton. References [1] O. Goldrech. Introducton to Complexty Theory July 31, 1999). [2] N. Nsan. Pseudorandom Generators for Space-Bounded Computaton. STOC 90. [3] N. Nsan. RL SC. Computatonal Complexty 4: 1 11, 1994. Prelmnary verson n STOC 92.) Space-Bounded Derandomzaton-7