Lecture 9: Polynomial Approximations

Transcription

1 CS 70: Complexiy Theory /6/009 Lecure 9: Polyomial Approximaios Isrucor: Dieer va Melkebeek Scribe: Phil Rydzewski & Piramaayagam Arumuga Naiar Las ime, we proved ha o cosa deph circui ca evaluae he pariy fucio. We used radom resricios o obai a boud o he complexiy of a circui evaluaig he pariy of ipus. I his lecure, we give a aleraive proof of a slighly weaker boud. Usig he radom resricio mehod, we showed ha C d ( Ω( d. This is a igh boud ad uses he propery ha he pariy fucio is sesiive o every bi of is ipu. We ca also derive a similar boud for he mod m fucio defied as follows: { 0 if x 0 (mod m mod m (x = oherwise ( where x is he umber of o-zero bis i he ipu. Pariy is a special case of mod m a m =. Aoher fucio ha we ca prove is o i AC 0 is he majoriy fucio, which reurs whe more ha half of he ipu bis are, ad 0 oherwise. This ca be proved by usig he pariy fucio as a black box ad is lef as a exercise. I his lecure, we use low degree polyomial approximaios o show ha C d ( Ω( d. Eve hough his is a weaker boud, he echique iself is ieresig. Moreover, his resul applies eve if we allow mod gaes i he circui. Fially, prove ha cosa-deph circuis are uable o approximaely evaluae pariy. Polyomial approximaio mehod Theorem. AC 0. Specifically, C d ( Ω( d Proof. We prove his i wo seps. Firs, we show ha ay cosa-deph circui ca be approximaed usig a low degree mulivariae polyomial over he field Z. Usig Z gives, for free, he abiliy o mimic mod gaes. I geeral, if we use Z p for a prime umber p, we ca hadle mod p gaes. Secod, we show ha he pariy fucio cao be approximaed usig a mulivariae polyomial of sufficiely low degree over he field Z. Sep : Cosider a circui C made of AND, OR, NOT ad MOD gaes. I ca always be represeed as a mulivariae polyomial of degree where is he size of he ipu. Our goal is o represe i usig a polyomial of lower degree, allowig errors if required. A lieral x ha is direcly passed as ipu o a gae ca be represeed usig he polyomial x. This is he base case of our cosrucio. Now, we ca assume ha a polyomial P i ca be associaed wih he i h ipu of oher gae ypes (AND, OR, NOT, MOD. The goal is o cosruc a polyomial P ha represes he oupu of he gae. Noe ha he umber of ipus o ay gae is a mos C. Cosruc each of he gaes as follows:

2 NOT: If P is he polyomial represeig he ipu of a NOT gae, he P represes he oupu. Noice ha his represeaio does icrease he degree of he polyomial or iroduce ay addiioal error. MOD : The oupu of he gae is zero whe m i= P i 0 (mod. If he summaio is or, he oupu is. Noe ha i he field Z, =. So, we ca model he gae usig he polyomial P = ( m i= P i. This polyomial accuraely models he gae ad is degree is a mos wice he degree of ay of is ipus. OR: The oupu of he OR gae is 0 if ( i P i = 0. Or, i oher words, ( i( P i =. Oherwise is oupu is oe. This ca be represeed as follows: α : P = m ( P i ( This represeaio is accurae bu he degree of P may be up o m imes he degree of he P i wih he larges degree. This ca be much higher ha he rivial boud if here are may gaes ad may levels i he circui. To ackle his, we model P as a radom liear combiaio of P i for i. Le r i Z be he coefficie associaed wih P i, chose uiformly radom. As wih MOD m, we square he liear combiaio o keep he value of P boolea. This leaves us wih: i= m β : P = ( r i P i ( i= This makes he degree of P a mos wice ha of he degree of is ipus. Bu i is defiiely o a accurae descripio of a OR gae. Evaluae he probabiliy of P i beig differe from he boolea expressio i= P i. If P i = 0 for all i, he irrespecive of he values picked for he coefficies, he m oupu is correc. If P i = for a leas oe i, he r i P i is r i. This is he wrog value, i= i P i = 0, i oe ou of hree cases for a radom assigme of he coefficies. Thus, P ca iroduce errors i he represeaio wih a probabiliy a mos. As wih ay radomized algorihm, we ca repea he above calculaio for, say, idepede rials ad see if he oupu of a leas oe of he rials is oe. (Noe: A oupu of oe will always be correc bu a oupu of zero may be wrog. This leads us o he hird, ad fial, formulaio of P. P = P α(p β ( ˆP,... P β ( ˆP (4 Here, P α is he applicaio of P as described i eq. o ipus. P β k is he k h rial usig he formulaio of P i eq.. ˆP is a shorhad for P,P,...P m. The above formulaio produces a wrog oupu if all he rials produce he wrog oupu, i.e. wih probabiliy a mos. The degree of P icreases by a facor of : a facor for he α-formulaio ad a facor of for he β-formulaio. AND: We ca hadle a AND gae i a similar way, resulig i a approximaio P wih a mos a facor of blow-up i he degree, ad givig a imprecise value wih probabiliy a mos. If he deph of he circui is d, he degree of he polyomial P represeig he eire circui will be a mos ( d. P gives he wrog value oly if he oupu of a leas oe of he gaes i C was wrog. This happes wih probabiliy a mos C. Noe, his is o a very igh upper boud

3 bu i is eough for his proof. A igher boud would deped o he umber of OR gaes i C. By averagig, he expeced umber of ipus for which P will give he wrog value is a mos C sice here are possible ipus of legh. There exiss a choice for he radom coefficies for which P is wrog i o more ha he expeced umber, derived above. More formally, Lemma. There exiss a choice of r i s such ha here exiss a se G {0,} of relaive size µ(g C such ha ( x G P(x = C(x, where P is a polyomial of degree a mos ( d cosruced as described above. Here, µ(g is he relaive size of G wih respec o he se of all possible ipus o C ad is equal o G. This cosrucio ca be geeralized o work over ay field Z p for prime p, hus allowig mod p gaes. The propery of Z we used is ha a (mod for all a 0 (mod. Thus, squarig a polyomial esures boolea values. To work over Z p, we would isead raise polyomials o he power p as a p (mod p for all a 0 (mod p. The degree of he resulig polyomial is a mos (p d raher ha ( d. Sep : I his sep, give a polyomial P of some degree ha approximaes o a subse G of ipus, we esablish a upper boud below which every fucio of ipus has a correspodig polyomial approximaig i over G. By equaig he umber of such fucios o he umber of polyomials wih degrees o greaer ha he esablished upper boud, we derive he lower boud o he deph of circui C. As a firs sep, we rasform he ipus o a slighly more coveie domai: {,} isead of {0,}. Proposiio. Suppose here exiss a polyomial P of degree a mos ha compues o a se G {0,}. The here exiss a polyomial P of degree a mos ad a se G {,} such ha µ(g = µ(g ad ( x G x i = P (x. i= The reaso is ha pariy o boolea ipus is equivale o muliplicaio over {,}. Lemma. Suppose here exiss a polyomial P of degree a mos ha represes muliplicaio i a se G {,}. The each fucio f : G Z has a mulivariae polyomial Q over Z of degree a mos + such ha i represes f, i.e. ( x G f(x = Q(x. Proof. Every fucio f has a mulivariae polyomial of degree a mos. This is rivial because we ca hardwire every possible ipu usig moomials of degree. Le us sar from oe such polyomial Q (such ha f = Q o G. Cosider a moomial i Q of he form i I x i where I is a subse of he ipu bis. Because we are oly cocered wih ± ipus, we ca rewrie i as: ( ( x i = x i i I = = i I x i = x i i I ( ( x i x i i= (5 ( x i P (x (6

4 Eq. 5 holds for ay ipu x of bis bu eq. 6 holds oly for he ipus i he se G. The LHS of 6 has degree I. The RHS has a degree a mos + Ī = + I. Averagig hese gives a miimum degree +. Thus, we ca make he degree of every moomial i Q o o exceed + Ġive he lemmas, we are ow ready o prove he heorem. Suppose here exiss a circui C of deph d compuig. From Lemma, here exiss a polyomial P of degree a mos = ( d ha compues pariy o a se G of relaive size a leas C. Cosequely, from Lemma, all fucios f : G Z for some G such ha G = G ca be represeed usig a mulivariae polyomial of degree a mos +. The oal umber of such polyomials mus be a leas he umber of fucios f from G o Z. The umber of mulivariae polyomials wih degree a mos + is exacly M where M is he umber of moomials of degree a mos +. There are ( i moomials of degree i, so M = + i ( i The umber of moomials of degree will be - half of he possible moomials. The remaiig = Θ( erms i he summaio will be lower ha ( - he maximum possible umber for ay degree. Usig Sirlig s approximaio, we ca show ha: Thus, M = + Θ ( ( = Θ = ( + Θ (. ( The umber of fucios of he form G Z is G as oe of possible values ca be assiged o each eleme of G. Because he umber of fucios of his form mus be a mos he umber of polyomials of degree a mos ( + /, G M or, i oher words, G M. This gives us he followig boud o he size of G. µ(g = G M ( + Θ From Lemma, µ(g C whe = ( d. Thus, C µ(g ( ( d + Θ [ ( ( = C d] Θ Seig ( d = O( gives a igh value for he RHS i he las equaio. Thus, = Θ( d. This gives C Ω( d. The oly par of he above aalysis ha chages whe workig over Z p raher ha Z is ha = (p d raher ha ( d. Thus he resul holds wih he same lower boud o C for 4

5 boolea circuis wih mod p gaes for ay prime p. I fac, he argume i he above proof ca be geeralized o give a lower boud for circuis wih mod p gaes o compue mod q (recall ha pariy is he special case of q =. This is achieved by viewig Sep as harmoic aalysis over Z ad he geeralizig ha o harmoic aalysis over Z q. As his geeralizaio akes a bi of work o prove, we leave i a ha. Because he lower boud for pariy was proved by viewig pariy as muliplicaio, we ge a lower boud for muliplicaio as well. Corollary. The decisio varia of biary muliplicaio is o i AC 0. We furher use he proof above o give a lower boud o circuis ha eve approximae pariy. Corollary. A deph d ubouded fa-i circui ha agrees wih pariy o a fracio a leas + of {0,} mus have size Ω(ǫ/d. ( ǫ/ Proof. Suppose we have a circui ha is correc o a leas +ρ of he ipus. Similar o Theorem, we ca prove ha here exiss a polyomial of degree = ( d ha is correc o a se G ha is a leas + ρ C of {0,}. From Sep of he proof above, + ρ C ( ( d + Θ = ρ C ( Θ (7 ( ] ( = C [ρ d Θ (8 Noe ha he ( d / erm is Ω(/, so ρ mus also be Ω(/ o esure he lower boud we ge is eve posiive. If we le ρ = / ( ǫ/, we se ( d = Θ( ǫ/ o opimize he RHS of 8. So = Θ( ǫ/(d, ad we ge ha C Ω(ǫ/(d. The above corollary proves he iapproximabiliy of he pariy fucio usig cosa deph circuis. There is aoher such resul ha ca be proved usig radom resricios. I is as follows: Theorem. A deph d ubouded fa-i circui ha agrees wih pariy o a fracio a leas + Ω(/d of {0,} mus have size Ω(/d. This is ieresig because eve rivial fucios ca guess pariy correcly o half of he ipus. This is slighly weaker ha he Ω( d boud we derived las lecure bu i disproves approximabiliy raher ha compuabiliy of he pariy fucio. We will see more such resuls of iapproximabiliy whe we discuss pseudo-radomess. Nex lecure Nex lecure, we will discuss parallelism where we disribue he compuaioal ask amog muliple processors o reduce he ime complexiy. 5