The Average Sensitivity of Bounded-Depth Formulas

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1 Th Avrag Snsitivity of Boundd-Dpth Formulas Bnjamin Rossman Univrsity of Toronto July 25, 206 Abstract W show that unboundd fan-in boolan formulas of dpth d + and siz s hav avrag snsitivity O d log sd. In particular, this givs a tight 2 Ωdn/d lowr bound on th siz of dpth d+ formulas computing th parity function. Ths rsults strngthn th corrsponding 2 Ωn/d and Olog s d bounds for circuits du to Håstad 986 and Boppana 997. Our proof tchniqu studis a random procss whr th Switching Lmma is applid to formulas in an fficint mannr. Introduction W considr boolan circuits with unboundd fan-in AND and OR gats and ngations on inputs. Formulas ar th class of tr-lik circuits in which all gats hav fan-out. Siz of circuits including formulas is masurd by th total numbr of gats. Dpth is th maximum numbr of gats on an input-to-output path. Lowr bounds against boundd-dpth circuits wr first provd in th 980s, 4, 9, 5, culminating in a tight siz-dpth tradoff for circuits computing th parity function. Th tchniqu, basd on random rstrictions, applis mor gnrally to boolan functions with high avrag snsitivity. Thorm Håstad 5. Dpth d + circuits computing parity hav siz 2 Ωn/d. Thorm 2 Boppana 2. Dpth d + circuits of siz s hav avrag snsitivity Olog s d. In this papr, w prov strongr vrsions of ths rsults for boundd-dpth formulas: Thorm 3. Dpth d + formulas computing parity hav siz 2 Ωdn/d. Thorm 4. Dpth d + formulas of siz s hav avrag snsitivity O d log sd. Thorms 3 and 4 dirctly strngthn Thorms and 2 in light of th following Fact 5. Evry dpth d + circuit of siz s is quivalnt to a dpth d + formula of siz at most s d. Thorms, 2, 3, 4 ar asymptotically tight, sinc parity is computabl by dpth d+ circuits rsp. formulas of siz n2 On/d rsp. 2 Odn/d. This work was carrid out whil th author was a rsarch fllow at th Simons Institut for th Thory of Computing at UC Brkly.

2 Th main tool in th proof of Thorms and 2 is Håstad s Switching Lmma 5. Th Switching Lmma stats that vry small-width CNF or DNF simplifis, with high probability undr a random rstriction, to a small-dpth dcision tr. This yilds lowr bounds against boundd-dpth circuits via a straightforward dpth-rduction argumnt. In this papr w show how th Switching Lmma can b applid mor fficintly to boundd-dpth formulas, though in a lss straightforward mannr. In mor dtail: for indpndnt uniformly distributd random σ {0, } n assignmnt and τ 0, n timstamp, w considr th family of rstrictions {Rp } 0p i.. functions n {0,, } rprsnting partial assignmnts to input variabls x,..., x n whr Rp sts th variabl x i to σ i if τ i < p and lavs x i unst if τ i p. In th usual application of th Switching Lmma to circuits of dpth d +, all subcircuits of dpth k + ar hit with th rstriction Rp k for a fixd squnc p > > p d typically p k = n k/d+. In this papr w achiv sharpr bounds against formulas by hitting ach subformula Φ with th rstriction R qφ whr th paramtr qφ = q Φ is dfind inductivly, according to a random procss indxd by subformulas of Φ. Our tchnical main thorm is a tail bound on qφ, viwd as a random variabl dtrmind by σ and τ. Aftr prliminary dfinitions in 2, w stat and prov our tchnical main thorm in 3 and 4. As a corollaris, w driv Thorm 3 in 5 and Thorm 4 in 6. In 7 w stat a furthr corollary of our rsults on th rlativ powr of formulas vs. circuits. 2 Prliminaris N = {0,, 2,... }. n = {,..., n}. xpλ = λ. 2. Formulas A formula is a finit rootd tr whos lafs inputs ar labld by litrals i.. variabls x i or ngatd variabls x i and whos non-lafs gats ar labld by AND or OR. Gats hav unboundd fan-in. Evry formula Φ computs a boolan function on th sam st of variabls. Th siz of a formula Φ, dnotd by Φ, is th numbr of gats in Φ. Not that vry lowr bound on siz is also a lowr bound on laf-siz, i.., th numbr of lavs in a formula. Th dpth of Φ is th maximum numbr of gats on an input-to-output path. Formulas of dpth 0 ar litrals; formulas of dpth ar clauss i.. an AND or OR of litrals. W ar oftn intrstd in formulas of dpth 2 and spak of dpth d + whr d is an arbitrary positiv intgr. 2.2 Boolan functions and rstrictions A rstriction is a function ϱ : n {0,, }, viwd as a partial assignmnt of boolan input variabls x,..., x n to 0, or maning unst. For a boolan function f : {0, } n {0, }, th rstrictd function f ϱ : {0, } ϱ {0, } is dfind in th usual way. For p 0,, w writ R p for th distribution on rstrictions ϱ whr P ϱi = = p and P ϱi = 0 = P ϱi = = p/2 indpndntly for all i n. 2.3 Avrag snsitivity and dcision-tr dpth Th avrag snsitivity asf of a boolan function f is th xpctd numbr of input bits that, whn flippd, chang th output of f, starting with a random input assignmnt. 2

3 Th dcision-tr dpth Df of f is th minimum dpth of a dcision tr which computs f; in particular, Df = 0 iff f is constant. Two lmntary facts which w will us latr s 2: for vry boolan function f, 2 asf Df E asf ϱ = p asf for all 0 p. ϱ R p i.. avrag snsitivity is at most dcision-tr dpth, Håstad s Switching Lmma rlats random rstrictions and dcision-tr dpth. W giv a somwhat nonstandard statmnt th usual statmnt is in trms of width-k CNFs and width-l DNFs. Lmma 6 Switching Lmma 5. Lt k, l N. Suppos f is th AND or OR of an arbitrary family {f i } of boolan functions with Df i k for all i. Thn for all 0 p 2, P Df ϱ l 5pk l. ϱ R p 3 A random procss associatd with formulas Dfinition 7. Lt σ {0, } n assignmnt and τ 0, n timstamp b indpndnt uniformly distributd random variabls. For 0 p, lt Rp : n {0,, } b th rstriction { Rp σ i if τ i > p, i := if τ i p. W rgard th family of rstrictions {Rp } 0p as a stochastic procss whr th paramtr p rprsnts a tim which starts at and dcrass to 0. At th initial tim p =, th assignmnt σ is fully maskd i.. R is all s. As p dcrass, th valus of σ ar gradually unmaskd, until th final tim p = 0 whn σ is fully rvald i.. R 0 = σ. Of cours, for any fixd p, Rp is simply a random rstriction with distribution R p. Dfinition 8 Main Dfinition. For all formulas Φ, w dfin th stopping tim q Φ 0, by th following induction: If Φ has dpth 0 i.. Φ is a variabl or ngatd variabl, thn q Φ :=. If Φ is AND,..., m or OR,..., m, thn whr q Φ := p Φ 4 k Φ p Φ := min q i, k Φ := max{, max D i R i i p Φ }. For th sak of radability, w will supprss σ and τ whnvr possibl and simply writ qφ, pφ, kφ. Howvr, th radr should kp in mind that ths random variabls ar dtrmind, for all formulas Φ, by a singl pair of σ of τ. W will continu to writ σ and τ whn rfrring to rstrictions Rp. W viw qφ as th stopping tim for a stochastic procss indxd by formulas Φ. For Φ of dpth 0, qφ is th initial tim whn all variabls ar maskd. For Φ of dpth, qφ is dfind in trms of two auxiliary paramtrs: 3

4 pφ is th most advancd i.. minimum stopping tim q among childrn of Φ. kφ is th maximum dcision-tr dpth among childrn of Φ upon bing hit with th rstriction R pφ. For tchnical rasons, w st kφ = in th vnt that D R pφ = 0 for all. If Φ is an AND rsp. OR, thn Φ R pφ is a kφ-cnf rsp. DNF. Th choic of dfinition qφ = pφ/4 kφ allows us to apply th Switching Lmma to Φ R pφ. This is mad prcis by th following lmma. Sinc th dpndnc on σ and τ is crucial hr, w us xplicit notation: q Φ, tc. Lmma 9. Lt Φ b a formula of dpth and lt q Suppq Φ i.. q = q Φ for som σ {0, } n and τ 0, n. Thn for all 0 α and l N, P DΦ R αq l q Φ = q α l. Proof. Fix Φ and q as in th hypothsis of th lmma. Sinc Φ has dpth, it is th AND or OR of formulas i. Lt { I := p, ϱ, k : q = p/4k and thr xist σ {0, } }n and τ 0, n such that p Φ = p, R = ϱ and k. Φ = k Not that I is nonmpty and indxs a partition of th vnt {q Φ = q} into subvnts {p Φ = p, Rp = ϱ and k Φ = k}. To prov th lmma, considr any p, ϱ, k I. Conditioning on this subvnt, w can viw Rαq as th composition of ϱ and an indpndnt random rstriction θ R α/4k. Sinc Φ ϱ is an AND or OR of functions i ϱ of dcision-tr dpth k, Lmma 6 implis DΦ Rαq l p Φ = p, Rp = ϱ and k Φ = k P 4 Tail bound on qφ = P DΦ ϱ θ l θ R α/4k p α l 5 k 4k α l. Our tchnical main thorm is a tail bound on th random variabl qφ = q Φ whr th randomnss is ovr indpndnt uniform σ {0, } n and τ 0, n. W stat th rsult first with asymptotic notation. Thorm 0. For vry dpth d + formula Φ and 0 < λ, P qφ λ Φ xpωdλ /d Od. 4

5 In ordr to hav a usabl induction hypothsis, w rstat Thorm 0 with xplicit constants: Thorm 0 mor prcisly. For vry dpth d + formula Φ and l > 0, whr C = + i=0 P qφ 4 d+ l xp i i j=0 C d Φ xp 2 dl /d xpj + i i + j Proof. W first not that th thorm is trivial if l < d as th RHS is > C/ xp d > sinc C > xp. Thrfor, w assum that l d. W argu by induction on d. Considr th bas cas d = whr Φ is a dpth 2 formula. Not that q = /4 for ach dpth subformula of Φ; hnc pφ = /4. Also, ach is th AND or OR of dcision-trs of dpth ; so by Lmma 6, P D R /4 l = P D ϱ l ϱ R /4 Sinc qφ = pφ/4 kφ = /4 2 kφ, w hav P qφ 4 2 l = P kφ l = P l. D R pφ l P D R /4 l Φ xpl < Φ C d xp 2 dl /d. For th induction stp, lt d 2 and assum th thorm holds for d. Lt Φ b a formula of dpth d +. Lt rang ovr dpth-d subformulas of Φ. In particular, w hav Φ = +. W will dfin a family of vnts dnotd A and B i i N and C i,j i, j N and show that th union of ths vnts covrs th vnt {qφ }. W will thn bound th probability of ach 4 d+ l C of ths vnts and show that th infinit sum of ths probabilitis is at most Φ d xp 2 dl /d. For all i N, dfin k i and α i by k i := i l /d, α i := k i 4 d l = Evnts A and B i and C i,j i, j N ar dfind as follows: A B i C i,j df df df pφ α 0, 4 d i l d /d. q α i+ D R q k i, α i+j+ < q α i+j+2 D Rα i+ k i. 5

6 Claim: If qφ 4 d+ l, thn A i=0 B i C i,j. j=0 Proof of claim: Assum qφ /4 d+ l and furthr assum that A dos not hold. Clarly thr xists a uniqu i N such that α i < pφ α i+ sinc α i is vntually >. Sinc qφ = pφ/4 kφ, w hav kφ > α i 4 d l = k i. Not that k i k 0 = l /d using th assumption that l d. Sinc kφ = max{, max D R pφ }, it follows that thr xists a such that D R pφ k i. Fix an arbitrary choic of such that D R pφ k i. Thr ar two cass to considr: ithr q α i+ or α i+j+ < q α i+j+2 for som j N. Assum q α i+. In this cas, w hav D R pφ D R q sinc pφ q. Thrfor, D R q k i. W conclud that B i holds. Assum α i+j+ < q α i+j+2 for som j N. W hav D R pφ D R αi+ sinc pφ α i+. Thrfor, D R α i+ k i. W conclud that C i,j holds. This concluds th proof of th claim. To complt th proof of th thorm, w will bound th probabilitis of vnts A, B i and C i,j and tak a union bound. W ignor th fact that all but finitly many of ths vnts hav zro probability, sinc P B i = 0 rsp. P C i,j = 0 for all α i > rsp. α i+j+ >. Instad, w show that P B i is xponntially dcrasing in i, whil P r C i,j is xponntially dcrasing in j and doubly xponntially dcrasing in i. W first bound th probability of A: P A = P q 4 d l d /d P q 4 d l d /d C d Φ xp 2 d /d l /d C d Φ xp 2 dl /d induction hypothsis using /d d d. 6

7 W nxt bound th probability of B i : P B i = P q α i+ D R q k i P q α i+ P D R q k i q α i+ = ki P q α i+ xp i l /d P q 4 d i l d /d xp i l /d Φ xp 2 d i/d l /d C d xp i l /d Φ xp 2 d l /d i 2 l /d C d C d = xp i i + 2 l /d Φ xp 2 dl /d xp i i + 2 Φ xp 2 dl /d. C d Lmma 9 induction hypothsis i/d i d Th last inquality uss th assumption l /d as wll as th nonngativity of i i + 2 for all i N. Finally, w bound th probability of C i,j : P C i,j = P P α i+j+ < q α i+j+2 D R α i+ k i q α i+j+2 P D Rα i+ k i αi+j+ < q α i+j+2 αi+ /α ki i+j+ = xpj + i l /d P q α i+j+2 P q 4 d i+j+ l d /d C d xpj + i l /d Φ xp 2 d i+j+/d l /d xpj + i l /d Φ xp 2 d l /d i + j + 2 l /d C d = xpj + i i + j l /d Φ xp 2 dl /d C d xpj + i i + j Φ xp 2 dl /d. C d Lmma 9 ind. hyp. Th last inquality uss th assumption l /d and th nonngativity of j + i i+j +2 2 for all i, j N. 7

8 W finish th proof by taking a union bound: P qφ 5 PARITY 4 d+ l P A + i=0 P B i + j=0 P C i,j C d Φ xp 2 dl /d. W us th rsults of th last sction to prov our lowr bound for th parity function. Thorm 3 rstatd. Dpth d + formulas computing parity rquir siz xpωdn /d. Proof. Suppos Φ is a dpth d + formula computing parity. Thn P Φ ϱ is non-constant = n > ϱ R /n n. On th othr hand, by Thorm 0 and Lmma 9, P Φ ϱ is non-constant = P DΦ R ϱ R /n /n P DΦ R max{/n,qφ} P qφ /n + P DΦ R qφ Φ xpωdn /d Od +. Thrfor, Φ 2 xp Ωdn /d Od. It follows that thr xist univrsal constants c 0, c > 0 dtrmind by th constants in th Ω and O such that Φ xpc 0 dn /d in th rgim d c ln n. In th rgim d > c ln n, w hav dn /d = Θln n, mor prcisly, ln n < dn /d < c c ln n. Not that dn /d is dcrasing in d and lim d dn /d = ln n. In ordr to claim a tight lowr bound of xpωdn /d for all d and n in particular whn d = ωlog n, w rquir an n Ω lowr bound on th gat-siz of unboundd-dpth formulas computing parity. A lowr bound of n follows from Khrapchnko s classic n 2 laf-siz lowr bound 6, which complts our proof. Whil Khrapchnko s bound suffics our purposs, w rmark that an asymptotically tight Ωn 2 formula gat-siz lowr bound for parity was shown by Childs, Kimml and Kothari 3 using quantum tchniqus. 6 Avrag Snsitivity Thorm 4 rstatd. Dpth d + formulas of siz s hav avrag snsitivity O d ln sd. 8

9 Proof. Lt Φ b a formula of dpth d+ and siz s rcall that siz is th numbr of gats. Assum asφ, sinc othrwis th thorm is trivial. W furthr assum that Φ has bottom fan-in s; othrwis it is asily shown that asφ = OasΦ whr Φ is obtaind from Φ by rplacing vry bottom AND rsp. OR gat with fan-in > s with 0 rsp.. In particular, Φ has laf-siz s 2, so it dpnds on s 2 distinct variabls. Ltting p = /asφ and using facts and 2, w hav = p asφ = E ϱ Rp asφ ϱ E DΦ R p = For all k N, by Thorm 0 and Lmma 9, P Combining ths inqualitis, w hav s 2 P k= DΦ R p k P DΦ R max{p,qφ} k xpωd asφ /d Od DΦ R p k. P qφ p + P DΦ R qφ k s xpωd asφ /d Od + k. s 3 = k= k 2 s3 = Os 3. It follows that Ωd asφ /d 3 ln s + Od and thrfor asφ = O d ln sd. 7 Formulas vs. Circuits Our lowr bound for parity Thorm 3 implis a sparation btwn th powr of dpth d + formulas vs. circuits. W writ {poly-siz dpth d + circuits/formulas} for th non-uniform complxity class of languags computabl by n O -siz dpth d + circuits/formulas whr dn is an arbitrary function of n. Corollary. For all dn = olog n with lim n dn =, 3 {poly-siz dpth d + formulas} {poly-siz dpth d + circuits}. Morovr, for all d C log n log log n for som univrsal constant C > 0, 4 {poly-siz dpth d + circuits} {n od -siz dpth d + formulas}. Sparation 3 may b rgardd as th dpth d + analogu of th conjcturd sparation {poly-siz formulas} {poly-siz circuits}, also known as NC P/poly. By Spira s thorm 8, vry poly-siz formula is quivalnt to a poly-siz formula of dpth Olog n; thus, xtnding 3 from dpth olog n to dpth Olog n would imply NC P/poly in fact NC AC. For th smallr rang of d c log n log log n, w gt th strongr sparation 4. In light of Fact 5, this is th strongst possibl sparation btwn formulas and circuits of th sam dpth. W rmark that until rcntly not vn th wak sparation 3 was known to hold for any supr-constant d O. Th first progrss on this qustion was mad in 7, whr 4 was shown 9

10 to hold for all d log log log n via a lowr bound for distanc-log log n st-connctivity. In fact, th lowr bound of 7 implis a much strongr rsult: for all d log log log n, 5 {poly-siz dpth d + circuits} {n od -siz dpth log n log log n 3 It rmains an opn problm to push sparation 5 to gratr dpths. Acknowldgmnts formulas}. I would lik to thank Rahul Santhanam, Rocco Srvdio and Li-Yang Tan for hlpful discussions. I also thank th anonymous rfrs of FOCS 5 for thir fdback. This work was carrid out whil th author was a rsarch fllow at th Simons Institut. Rfrncs Miklós Ajtai. Σ 983. formula on finit structurs. Annals of Pur and Applid Logic, 24: 48, 2 Ravi B. Boppana. Th avrag snsitivity of boundd-dpth circuits. Information Procssing Lttrs, 635:257 26, Andrw M. Childs, Shlby Kimml, and Robin Kothari. Th quantum qury complxity of rad-many formulas. In Europan Symposium on Algorithms, pags , Mrrick L. Furst, Jams B. Sax, and Michal Sipsr. Parity, circuits, and th polynomial-tim hirarchy. Mathmatical Systms Thory, 7:3 27, Johan Håstad. Almost optimal lowr bounds for small dpth circuits. In 8th Annual ACM Symposium on Thory of Computing, pags 6 20, V.M. Khrapchnko. Complxity of th ralization of a linar function in th cas of Π-circuits. Math. Nots Acad. Scincs, 9:2 23, Bnjamin Rossman. Formulas vs. circuits for small distanc connctivity. In 46th Annual ACM Symposium on Thory of Computing, pags , P.M. Spira. On tim-hardwar complxity tradoffs for Boolan functions. In 4th Hawaii Symposium on Systm Scincs, pags , Andrw C.C. Yao. Sparating th polynomial-tim hirarchy by oracls. In 26th Annual IEEE Symposium on Foundations of Computr Scinc, pags 0,

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