Derandomizing the Isolation Lemma and Lower Bounds for Circuit Size

Derandom izing the Isolation Lemma and Lo wer Bounds for Circuit Size V . Arvind and Partha Mukhopadhyay Institute of Mathematical Sciences C.I.T Campus,Chennai 600 113, India { arvind,pa rtham } @imsc .res.in Abstract The isolation lemma of Mulmuley et al [MVV87] is an importan t tool in the design of ran domized algorithm s and has played an impor tant role in several nontrivial complexity upper bounds. On the other hand, polynomial identity testing is a well-studied a lgorithmic pro blem with efficient rand omized algo- rithms and the p roblem of ob taining effi cient d eterministic iden tity tests has r eceiv ed a lot of attentio n recently . The goal of this note is to compare the isolation lemma with polyn omial iden tity testi ng : 1. W e show that dera ndomizin g reasona bly restricted versions of the isolation lemma implies circuit size lower boun ds. W e d eriv e the circuit lower bounds by examin ing the connection between th e isolation lemma and p olynomia l identity testing. W e gi ve a rando mized polynomial- time identity test f or non commutative circuits of p olynom ial degree b ased on the isolation lemma. Using th is result, we show that derand omizing the iso lation lemm a implies no ncommu tati ve circuit size lower bound s. The r estricted version s of the isolatio n lemm a we consider are natural an d would suf fice for the standard application s of the isolation lemm a. 2. From the re sult o f Kliv ans-Sp ielman [KS01] we o bserve that th ere is a ran domized poly nomial- time identity test for commutati ve circuits of polyno mial degree, also based o n a more g eneral isolation lem ma for linear fo rms. Consequ ently , de random ization of (a su itable version of) this isolation lem ma implies that either NEXP 6⊂ P / poly or the Perman ent over Z does no t have polyno mial-size arithm etic circuits. 1 Introd uction W e recall the Isolation Lemma [MVV87]. Let [ n ] denote the set { 1 , 2 , · · · , n } . L et U be a set of size n and F ⊆ 2 U be any family of subsets of U . Let w : U → Z + be a weight function that assigns positi ve inte ger weights to the elements of U . For T ⊆ U , define its weight w ( T ) as w ( T ) = P u ∈ T w ( u ) . Then Isolati on Lemma guarantees that for any family of subsets F of U and for any random w eight assign ment w : U → [2 n ] , with high probab ility there will be a unique minimum weight set in F . Lemma 1.1 (Isola tion L emma) [MVV87] L et U be an uni verse of size n and F be any family of subsets of U . Let w : U → [2 n ] denot e a w eight assignmen t function to elements of U . Then, Prob w [ There e xists a unique minimum weight set in F ] ≥ 1 2 , where the weight functio n w is picked uniformly at random. 1 In the seminal paper [MVV87] Mulmuley et al apply the isolati on lemma to gi ve a randomized NC algori thm for computin g maximum cardina lity m atchin gs for general graphs (also see [AR Z99]). Since then the isolation lemma has found sev eral other applicatio ns. For exa mple, it is crucially used in the proof of the result that NL ⊂ UL / poly [AR00] and in designin g rand omized NC algorith ms for linear repres entable matroid prob lems [NSV94]. It is a lso kno wn th at the isolati on lemma can be used to pro ve th e V aliant- V azirani lemma that SA T is many-o ne reducible via randomized reductions to USA T . Whether the matching proble m is in deterministic NC, and whether NL ⊆ UL are outstandin g open proble ms. Thus, the questi on whether the isolation lemma can be derando mized is clearly important . As noted in [Agr07], it is easy to see by a counting argumen t that the isolation lemma can not be derand omized, in general, because there are 2 2 n set syst ems F . More formall y , the foll owing is ob served in [Agr07]. Observ ation 1.2 [Agr07] The Isolat ion Lemma can not be fully de randomiz ed if we al low weight functi ons w : U → [ n c ] for a constant c (i.e. weight functions with a polynomial range ). Mor e pr ecis ely , for any polyno mially bounded collect ion of weight assig nments { w i } i ∈ [ n c 1 ] with weight ra nge [ n c ] , ther e e xists a family F of [ n ] suc h that for all j ∈ [ n c 1 ] , ther e e xists two m inimal weight subs ets with r espect to w j . Ho wev er that does not rule out the derandomiz ation of any special usage of the isolation lemma. In- deed, for all application s of the isolatio n lemma (mentioned above , for instanc e) we are interested only in exp onentially many set systems F ⊆ 2 U . W e make the setti ng more precis e by gi ving a g eneral fra mewo rk. Fix th e uni ve rse U = [ n ] and consider an n -input boolean circuit C where size ( C ) = m . The set 2 U of all subsets of U is in a natural 1 - 1 corres pondence with the length n -binary strings { 0 , 1 } n : each subset S ⊆ U corresp onds to its characteri stic binary s tring χ S ∈ { 0 , 1 } n whose i th bit is 1 iff i ∈ S . Thus th e n -input boolean circ uit C implicitly defines the set system F C = { S ⊆ [ n ] | C ( χ S ) = 1 } . As an easy consequ ence of Lemma 1.1 we ha ve the follo wing. Lemma 1.3 Let U be an un iverse of size n and C be an n -in put boolean cir cui t of size m . Let F C ⊆ 2 U be the family of subsets of U defined by cir cuit C . Let w : U → [2 n ] denote a weight assign ment function to elements of U . Then, Prob w [ Ther e e xists a unique minimum weight set in F C ] ≥ 1 2 , wher e the weight functio n w is pic ked unifo rmly at ra ndom. F urthe rmor e, ther e is a collec tion of weight functi ons { w i } 1 ≤ i ≤ p ( m,n ) , wher e p ( m, n ) is a fixed polynomia l, suc h that for eac h F C ther e is a w eight functi on w i w .r .t. which the r e is a unique minimum weight set in F C . Lemma 1.3 allo ws us to formu late two natural and reasonabl e deran domization hypothes es for the isolati on lemma. Hypothesis 1. There is a determinis tic algorith m A 1 that tak es as input ( C , n ) , where C is an n -input boolea n circuit, and outpu ts a collectio n of weight functio ns w 1 , w 2 , · · · , w t such that w i : [ n ] → [2 n ] , with the proper ty that for some w i there is a unique minimum weight set in the set system F C . Furthermore, A 1 runs in time sube xponent ial in size ( C ) . 2 Hypothesis 2. T here is a deterministi c algorithm A 2 that takes as input ( m, n ) in unary and outputs a collec tion of weig ht function s w 1 , w 2 , · · · , w t such th at w i : [ n ] → [2 n ] , with the p roperty that fo r each size m b oolean circuit C w ith n inputs there is some weight func tion w i w .r .t. which F C has a uniqu e minimum weight set. F urthermo re, A 2 runs in time polyn omial in m . Clearly , Hypot hesis 2 is stronger than Hypothesis 1. It demands a “black- box” derandomiza tion in the sense that A 2 ef ficiently computes a collect ion of weight functions that will work for any set system in 2 U specified by a boolean circui t of size m . Notice that a ra ndom colle ction w 1 , · · · , w t of weight functio ns will fulfil the requ ired proper ty of e ither hypot heses with high proba bility . T hus, t he deran domization hypoth eses are plausib le. Indee d, it is not ha rd to see that suitable standard hardness assumption s that yield pseudo random genera tors for derando mizing BPP would imply these hypotheses . W e do not elaborate on this here. In this paper w e sho w the follo wing conseq uences of Hypothes es 1 and 2. 1. Hypothes is 1 implies that eithe r NEXP 6⊂ P / poly or the Permanent doe s not ha ve poly nomial size nonco mmutativ e arithmetic circuits. 2. Hypothes is 2 impli es that for each n ther e is an exp licit polyn omial f n ( x 1 , x 2 , · · · , x n ) ∈ F { x 1 , x 2 , · · · , x n } in noncommuti ng varia bles x i (where by expl icit we mean that the coef ficients of the polyno mial f n are computable by a uniform algorith m in time expon ential in n ) that does not ha ve nonco mmutativ e arithmetic circuits of size 2 o ( n l g n ) (where the fiel d F is ei ther the rationa ls or a finite field). These two results are a conseque nce of an identity testing algorith m for noncommutati v e circuits that is based on the isolation lemma. This algorithm is based on ideas from [AMS08] where we used automata theory to pick matrices from a suitable m atrix ring and e valu ate the gi ven arithmetic circuit on these matrices. In the ne xt section, w e descr ibe the back ground and then giv e the identity test in the follo wing section. Remark 1.4 Notice that derando mizing the isolation lemma in specifi c applicati ons like the RNC algo- rithm for matching s [MVV87] and the containment N L ⊆ UL / poly [AR00 ] might still be possible without implying suc h cir cuit size lower bounds. Noncommutat iv e polynomial identity testing has been the focus of recent research [RS05, BW05, AMS08]. O ne re ason to be liev e that it could be e asier than th e commutati ve case t o derand omize is because lo wer bounds are some what easier to prov e in the noncommutat iv e setting as sho wn by Nisan [N91]. Using a rank argumen t Nisan has sho wn expon ential size lower bounds for noncommutat iv e formulas (and non- commutati ve algebraic branch ing programs) that compute the noncommutati ve permanen t or determinan t polyn omials in the ring F { x 1 , · · · , x n } where x i are noncommutin g v ariables. Howe ve r , no superpoly no- mial lo wer bounds are kno wn for the size of noncommutati v e circuits for explic it polynomials. Our result i n this paper is similar in flav or to the Impaglia zzo-Kabanet s result [ KI03 ], where f or commu- tative polynomial identity testing they sho w that derandomizin g poly nomial identity testing implies circuit lo wer bou nds. Specifically , it implies that either NEXP 6⊂ P/poly or the integ er Permanent does not ha ve polyn omial-size arithmetic circuits . In [AMS08] we hav e observ ed that an analogous result also holds in the noncommutati ve setting. I.e., if noncommutati ve PIT has a determin istic polyn omial-time algorithm then either NEXP 6⊂ P/poly or the nonco mmutative Permanent functi on does not ha ve polyno mial-size noncommutat iv e circuits . 3 The conn ection that we sho w here between derand omizing the isolation lemma and noncommutat iv e circuit size lower bounds is based on the abo ve observ ation and our nonc ommutati ve polyno mial identity test based on the isolat ion lemma. Commutative c ircuits Kli vans and Spielman [KS01] apply a m ore gene ral form of the isolat ion lemma to obtain a polyno mial identi ty test (in the commutati ve ) case. This lemma is sta ted belo w . Lemma 1.5 [KS01, L emma 4] Let L be any collection of linear forms ove r varia bles z 1 , z 2 , · · · , z n with inte ger coeffic ients in the range { 0 , 1 , · · · , K } . If each z i is pic ked independent ly and uniformly at ran dom fr om { 0 , 1 , · · · , 2 K n } then with pr obabil ity at least 1 / 2 ther e is a unique linear form fr om C that attains minimum value at ( z 1 , · · · , z n ) . W e can formula te a restricted versio n of this lemma similar to Lemma 1.3 that will apply only to se ts of linear forms L accepted by a boolean circuit C . M ore precisel y , an integ er v ector ( α 1 , · · · , α n ) such that α i ∈ { 0 , · · · , K } is in L if and only if ( α 1 , · · · , α n ) is ac cepted by the boolean circuit C . Thus, for this form of the isolati on lemma we can formulate anoth er deran domization hypo thesis analog ous to Hypothesis 1 as follo ws. Hypothesis 3. Ther e is a deterministic algorith m A 3 that takes as input ( C , n, K ) , where C is a boolean circuit that tak es as input ( α 1 , · · · , α n ) such that α i ∈ { 0 , · · · , K } , and outputs a collect ion of weight functi ons w 1 , w 2 , · · · , w t such that w i : [ n ] → [2 K n ] , with the property that for some weight vector w i there is a uni que linear form ( α 1 , · · · , α n ) acc epted by C w hich at tains the minimum va lue P n j = 1 w i ( j ) α j . Furthermor e, A 3 runs in time sube xponent ial in size ( C ) . 2 A utomata Theory backgr ound W e recall some standard automata theo ry [HU78]. Fix a finite automaton A = ( Q, δ, q 0 , q f ) which takes inputs in { 0 , 1 } ∗ , Q is the set of states, δ : Q × { 0 , 1 } → Q is the transit ion function, and q 0 and q f are the initial and final states respecti v ely (we only consider automata w ith unique acceptin g states). For each b ∈ { 0 , 1 } , let δ b : Q → Q be defined by: δ b ( q ) = δ ( q , b ) . T hese functio ns gene rate a submonoid of the monoid of all functions from Q to Q . This is the transition monoid of the automaton A and is well-studi ed in automa ta theory [Str94 , page 55]. W e no w define the 0 - 1 matrix M b ∈ F | Q |×| Q | as follo ws: M b ( q , q ′ ) =  1 if δ b ( q ) = q ′ , 0 otherwis e. The m atrix M b is the adjacenc y matrix of the graph of δ b . As M b is a 0 - 1 matrix, we can consid er it as a matrix ov er any field F . For a string w = w 1 w 2 · · · w k ∈ { 0 , 1 } ∗ we define M w to be the matrix produc t M w 1 M w 2 · · · M w k . If w is the empty string, define M w to be the i dentity matrix of dimens ion | Q | × | Q | . Let δ w denote the natura l ext ension of the transi tion function to w ; if w is the empty string, δ w is simply the identity function. W e ha ve M w ( q , q ′ ) =  1 if δ w ( q ) = q ′ , 0 otherwis e. (1) Thus, M w is also a m atrix of zeros and ones for any string w . A lso, M w ( q 0 , q f ) = 1 if and only if w is accept ed by the automato n A . 4 2.1 Noncommutativ e a rithmetic cir cuits and automata This subse ction is reprod uced from [AMS08] to make this pape r self-conta ined. Consider the ring F { x 1 , · · · , x n } of polynomials w ith nonco mmuting variab les x 1 , · · · , x n ov er a fi eld F . Let C be a noncommutati ve arithmetic circuit computing a polynomial f ∈ F { x 1 , · · · , x n } . L et d be an upper bou nd on the degr ee of f . W e can consid er monomials ov er x 1 , · · · , x n as strin gs over an alphab et of size n . For our const ruction, it is more con ve nient to encode each x i as a string over { 0 , 1 } . W e e ncode the v ariable x i by th e string v i = 01 i 0 . Clearly , each monomial o ver the x i ’ s of de gree at most d maps unique ly to a binary string of length at most d ( n + 2) . Let A = ( Q, δ , q 0 , q f ) be a finit e automaton ov er the alphabet { 0 , 1 } . W e ha ve matric es M v i ∈ F | Q |×| Q | as defined in S ection 2, where v i is the binary string that encodes x i . W e are interested in the output matrix obtain ed when the inputs x i to the cir cuit C are replaced by the matric es M v i . This outpu t m atrix is d efined in the obvi ous way: the inputs are | Q | × | Q | matrices and w e do m atrix addition and matrix m ultipl ication at each addition gate (respecti ve ly , multiplica tion gate) of the circuit C . W e define the output of C on the automato n A to be this outp ut matrix M out . Clearly , giv en circuit C and automaton A , the matrix M out can be computed in time poly ( | C | , | A | , n ) . W e observe the follo wing proper ty: the m atrix output M out of C on A is determined completely by the polyn omial f computed by C ; the structure of the cir cuit C is otherwise irrele v ant. This is import ant for us, since we are only intere sted in f . In particular , the output is alway s 0 when f ≡ 0 . More speci fically , consider what happen s when C computes a polynomial with a single term, say f ( x 1 , · · · , x n ) = cx j 1 · · · x j k , with a non-zero coef ficient c ∈ F . In this case, the output matrix M out is clearly the matrix cM v j 1 · · · M v j k = cM w , where w = v j 1 · · · v j k is the binary string represe nting the monomial x j 1 · · · x j k . Thus, by Equation 1 abov e, w e see that the entry M out ( q 0 , q f ) is 0 when A rejects w , and c when A accepts w . In general, suppose C computes a polynomial f = P t i =1 c i m i with t nonzero terms, where c i ∈ F \ { 0 } and m i = Q d i j = 1 x i j , where d i ≤ d . Let w i = v i 1 · · · v i d i denote the binary string repres enting monomial m i . Finally , let S f A = { i ∈ { 1 , · · · , t } | A ac cepts w i } . Theor em 2.1 [AMS 08] Given a ny arithmetic c ir cuit C computing polyno mial f ∈ F { x 1 , · · · , x n } and any finite automaton A = ( Q, δ, q 0 , q f ) , then the output M out of C on A is such tha t M out ( q 0 , q f ) = P i ∈ S f A c i . Pr oof . The proof is an easy conseque nce of the definitions and the properties of the matrices M w stated in Section 2. Note that M out = f ( M v 1 , · · · , M v n ) . But f ( M v 1 , · · · , M v n ) = P s i =1 c i M w i , where w i = v i 1 · · · v i d i is the binary string representin g monomial m i . By Equation 1, we know that M w i ( q 0 , q f ) is 1 if w i is accep ted by A , and 0 othe rwise. Adding up, we obt ain the result. W e no w ex plain the role of th e automaton A in testing if th e polynomial f computed by C is identic ally zero. O ur basic idea is to design an automaton A that accepts exactly one word from among all the words that cor respond to the nonzer o terms in f . T his w ould ensure that M out ( q 0 , q f ) is th e nonzero coef ficient of the m onomial filtered out. More precisely , w e w ill use the above theorem primarily in the followin g form, which we state as a corollar y . Cor ollary 2.2 [AMS08] Given any arithmetic cir cuit C computin g polynomial f ∈ F { x 1 , · · · , x n } and any finite automaton A = ( Q, δ, q 0 , q f ) , then the output M out of C on A satisfi es: (1) If A r ejec ts every string corr esp onding to a monomial in f , then M out ( q 0 , q f ) = 0 . (2) If A accepts exact ly one string corr espondin g to a monomial in f , then M out ( q 0 , q f ) is the nonzer o coef ficient of that monomial in f . 5 Mor eov er , M out can be computed in time poly ( | C | , | A | , n ) . Pr oof . Both points ( 1 ) and ( 2 ) are immediate conse quences of the abov e theorem. The comple xity of computin g M out easily follo ws from its definition. Another interest ing corollary to the abov e theorem is the follo wing. Cor ollary 2.3 [AMS08] Given any arithmetic cir cuit C over F { x 1 , · · · , x n } , and any monomial m of de- gr ee d m , we can compute the coef ficient of m in C in time poly ( | C | , d m , n ) . Pr oof . A pply Corollary 2.2 with A being any standard automaton that accepts the string correspon ding to monomial m and reject s eve ry other string. Clearly , A can be chose n so that A has a unique acceptin g state and | A | = O ( nd m ) . Remark 2.4 Cor ollary 2.3 is very unli kely to hold in the commutative ring F [ x 1 , · · · , x n ] . F or , it is easy to see that in the commutative case computi ng the coef ficient of the monomia l Q n i =1 x i in even a pr oduc t of linear forms Π i ℓ i is at leas t as har d as comput ing the perman ent over F , w hic h is # P -complete when F = Q . 3 Noncommutative identity tes t based on isolation lemma W e no w describe a ne w ide ntity test for nonco mmutativ e circuits based on the isolati on lemma. It is directly based on the results from [AMS08]. This is concep tually quite differe nt from the randomized identi ty test of Bogdan ov and W ee [BW05]. Theor em 3.1 L et f ∈ F { x 1 , x 2 , · · · , x n } be a polynomia l given by an arithmetic cir cuit C of size m . Let d be an upper boun d on the de gr ee of f . Then ther e is a ran domized algori thm which runs in time poly ( n, m, d ) and can test whether f ≡ 0 . Pr oof . Let [ d ] = { 1 , 2 , · · · , d } and [ n ] = { 1 , 2 , · · · , n } . Consider the set of tuples U = [ d ] × [ n ] . Let v = x i 1 x i 2 · · · x i t be a nonzero monomial of f . Then the monomial can be identified with the follo wing subset S v of U : S v = { (1 , i 1 ) , (2 , i 2 ) , · · · , ( t, i t ) } Let F denotes the family of subse ts of U corresp onding to the nonzero monomials of f i.e, F = { S v | v is a nonzero monomial in f } By the Isolation Lemma we kno w that if we assign random weights from [2 dn ] to the elements of U , with probabilit y at least 1 / 2 , there is a unique minimum weight set in F . Our aim will be to construct a family of small size automatons which are inde xed by weights w ∈ [2 nd 2 ] and t ∈ [ d ] , such that the automata A w ,t will precis ely accept all the str ings (correspond ing to the monomials ) v of length t , such tha t the weight of S v is w . Then from the isolation lemma we will arg ue that the automata correspon ding to the minimum weight w ill precis ely accept only one string (monomial). Now for w ∈ [2 nd 2 ] , and t ∈ [ d ] , w e descri be the construc tion of the automat on A w ,t = ( Q, Σ , δ, q 0 , F ) as foll ows: Q = [ d ] × [2 nd 2 ] ∪ { (0 , 0) } , Σ = { x 1 , x 2 , · · · , x n } , q 0 = { (0 , 0) } and F = { ( t, w ) } . W e define the transit ion function δ : Q × Σ → Q , δ (( i, V ) , x j ) = ( i + 1 , V + W ) , 6 where W is the random w eight assig n to ( i + 1 , j ) . Our automata family A is simply , A = { A w ,t | w ∈ [2 nd 2 ] , t ∈ [ d ] } . No w for each of the automaton A w ,t ∈ A , we mimic the run of the automaton A w ,t on the circuit C as descr ibed in Section 2. If the outpu t matrix correspo nding to any of the automaton is nonz ero, our algori thm declares f 6 = 0 , otherwise declares f ≡ 0 . The correctn ess of the algorithm follo ws easily from the Isolation Lemma. By the Isolat ion Lemma we know , on random assignment, a unique set S in F gets the minimum w eight w min with probab ility at least 1 / 2 . Let S correspond s to the monomial x i 1 x i 2 · · · x i ℓ . Then the automaton A w min ,ℓ accept s the string (monomial) x i 1 x i 2 · · · x i ℓ . Furthermore , as no othe r set in F get th e same minimum weight , A w min ,ℓ rejects all the other m onomial s. So the ( q 0 , q f ) entry of the output matrix M o , that w e get in runnin g A w min ,ℓ on C is nonz ero. Hence with proba bility at least 1 / 2 , our algorith m correc tly decid e that f is nonzero. The succes s probab ility can be bo osted to any constan t by standard in dependen t repetition of the same algorithm. Finally , it is tri vial to see that the algorithm always decid es correct ly if f ≡ 0 . 4 Noncommutative identity tes ting and cir cuit lower bound s For commutati ve circuits, I mpagliazzo and K abane ts [KI03] hav e shown that derando mizing PIT implies cir- cuit lo wer bo unds. It implie s that either NEXP 6⊂ P/poly or the in teger Permanent d oes not hav e polyn omial- size arithmet ic circuits. In [AMS 08] we hav e observed that this also holds in the noncommutati ve setting. I.e., if noncommuta- ti ve PIT has a determini stic polynomial-time algorithm then either NEXP 6⊂ P/poly or the noncommuta tive Permanent fu nction does not ha ve p olynomial- size noncommutat iv e circuits. W e note here th at noncommu- tati ve circuit lower bound s are sometimes easier to prov e than for commutati ve circuits. E.g. Nisan [N91] has shown expo nential-s ize lo wer bounds for noncommutati ve formula size and further results are known for pure noncommut ativ e circuits [N91, RS 05]. Howe ver , pro ving superpolyn omial size lo wer bounds for genera l noncommutati ve circuits computing the Permanent has remained an open problem. T o keep this pape r self contained, w e bri efly recall the discussion from [AMS 08]. The nonco mmutativ e P ermanen t function P er m ( x 1 , · · · , x n ) ∈ R { x 1 , · · · , x n } is defined as P er m ( x 1 , · · · , x n ) = X σ ∈ S n n Y i =1 x i,σ ( i ) , where the coefficien t ring R is any commutativ e ring with unity . Specifically , for the next theorem we choose R = Q . Let SUBEX P denote ∩ ǫ> 0 DTIME(2 n ǫ ) and NSUBEXP den ote ∩ ǫ> 0 NTIME(2 n ǫ ) . Theor em 4.1 [AMS 08] If PIT for noncommutative cir cuits of polynomia l de gr ee C ( x 1 , · · · , x n ) ∈ Q { x 1 , · · · , x n } is in SUB EXP , then either NEXP 6⊂ P/poly or the noncommuta tiv e P ermanent function does not have polyno mial-size nonco mmutative cir cuits. Pr oof . Suppose NEXP ⊂ P/poly. T hen, by the main result of [IKW02] we hav e NEXP = MA. Furthermore, by T oda’ s theorem MA ⊆ P P er m Z , where the oracle computes the integer permanent. Now , assuming PIT for n oncommutati ve circu its of pol ynomial degr ee is in determin istic polynomia l-time we will sho w that the 7 (nonco mmutati ve) Permanent function does not ha v e polynomial- size noncommutat iv e circuits . Suppose to the contrary that it does ha ve polynomial-s ize noncommuta tiv e ci rcuits. Clearly , we can use it to compute the inte ger permanent as w ell. Furthermor e, as in [KI03] we notice that the noncommutati ve n × n Permanent is also uniquely characteri zed by the identities p 1 ( x ) ≡ x and p i ( X ) = P i j = 1 x 1 j p i − 1 ( X j ) for 1 < i ≤ n , where X is a matrix of i 2 nonco mmuting v ariables and X j is its j -th minor w .r .t. the first ro w . I.e. if ar bitrary polyn omials p i , 1 ≤ i ≤ n satisfies these n identities over noncommutin g variab les x ij , 1 ≤ i, j ≤ n if and only if p i compute s the i × i permanent of noncommuting variab les. The rest of the proof is exactly as in Impaglia zzo-Kabanet s [KI03]. W e can easily describe an NP machine to simulate a P P er m Z computa tion. The NP machine guess es a polyn omial-size noncommuta tiv e circuit for P er m on m × m matrices, wher e m is a po lynomial bound o n the matrix size of the q ueries made. Then t he NP verifies that t he circu it computes the permanent by checking the m noncommuta tive identitie s it must satisfy . This can be done in S UBEXP by assumption . Finally , the NP machines uses the circuit to answer all the inte ger permanent queries. Putting it together , we get NEXP = N SUBEXP w hich contradicts the nondeterminis tic time hie rarchy theo rem. 5 The Results W e are now ready to prov e our fi rst result. Suppose the derandomizat ion H ypoth esis 1 holds (as stated in the introductio n): i.e. suppos e there is a determinis tic algorithm A 1 that takes as input ( C , n ) where C is an n -input boolean circuit and in sube xponent ial time computes a set of weight functions w 1 , w 2 , · · · , w t , w i : [ n ] → [2 n ] such that the se t system F C defined by the circuit C has a uniqu e minimum weight set w .r .t. at least one of the weight functi ons w i . Let C ′ ( x 1 , x 2 , · · · , x n ) be a non commutati ve arithmetic circui t of degre e d boun ded by a polyn omial in size ( C ′ ) . By Corollary 2.3, there is a deterministic polyno mial-time algorith m that takes as input C ′ and a monomial m of de gree at m ost d and accepts if and only if the m onomial m has nonzero coef ficient in the polynomial computed by C ′ . Thus, we ha ve a boolean circuit C of size polynomial in si ze ( C ′ ) that accept s only the (binary en codings of) mono mials x i 1 x i 2 · · · x i k , k ≤ d that ha ve nonz ero coef ficients in the polyn omial computed by C ′ . No w , as a consequen ce of Theorem 3.1 and its proof w e hav e a determini stic sube xponent ial algo rithm for checking if C ′ ≡ 0 , assuming algorith m A 1 exi sts. Namely , we compute the boolea n circuit C from C ′ in polyno mial time. T hen, in v oking algorith m A 1 with C as input we compute at most sub expone ntially many weight functi ons w 1 , · · · , w t . Then, follo wing the proof of Theore m 3.1 we constr uct the automata co rrespondi ng to these weight f unctions and e v aluate C ′ on the matr ices that each of these automata define in the prescri bed manner . B y assumption about algorithm A 1 , if C ′ 6≡ 0 then one of these w i will giv e matrix inputs for the variab les x j , 1 ≤ j ≤ n on w hich C ′ e valu ates to a nonzero matrix. W e can no w show the follo wing theorem. Theor em 5.1 If the sube xponentia l time algorithm A 1 satisfy ing H ypoth esis 1 ex ists then noncommuta tive identi ty testing is in SUBE XP which implies that either NEXP 6⊂ P / poly or the P ermanent does not have polyno mial size noncommuta tive cir cuits. Pr oof . The resul t is a direct consequ ence of the discussion preceding the theore m state ment and Theo- rem 4.1. W e no w turn to the result under the str ong er derand omization Hypothesi s 2 (stated in the introduct ion). More precisely , suppose there is a deterministic algorithm A 2 that takes as input ( m, n ) and in time poly- nomial in m comput es a set of weight functions w 1 , w 2 , · · · , w t , w i : [ n ] → [2 n ] such that for each n -inpu t 8 boolea n circuit C of size m , the set system F C defined by the circuit C has a unique minimum weight set w .r .t. at least one of the weight function s w i . W e sho w that there is an ex plicit polynomial 1 f ( x 1 , · · · , x n ) in nonc ommuting va riables x i that does not ha ve subex ponentia l size noncommutat iv e circuits . Theor em 5.2 Suppose ther e is a polynomial-ti me algorit hm A 2 satisfy ing Hypothesis 2. Then for all b ut finitel y many n ther e is an exp licit polyno mial f ( x 1 , · · · , x n ) ∈ F { x 1 , x 2 , · · · , x n } (wher e the field F is either rational s or any finite field) in noncommuting variab les x i that is computabl e in 2 n O (1) time (by a unifor m algorith m) and does not have noncommuta tive arithmetic cir cuits of size 2 o ( n lg n ) . Pr oof . Let T n denote the set of all sequences ( i 1 , i 2 , · · · , i n ) , for i j ∈ [ n ] , 1 ≤ j ≤ n . For each such sequen ce α = ( i 1 , i 2 , · · · , i n ) ∈ T n let m α denote the monomial x i 1 x i 2 · · · x i n . Now , we w rite f ( x 1 , x 2 , · · · , x n ) = X α ∈ T n c α m α , where we will pick the scalars c α approp riately so that the polynomial f has the claimed prope rty . S uppo se A 2 runs in t ime m c for consta nt c > 0 , where m denotes the size bound of the bool ean circuit C defining set system F C . Notice that the number t of w eight functio ns is bounded by m c . As explain ed in T heore m 3.1, each w eight function will giv e ri se to a collecti on of 2 n 4 automata A k , each of w hich w ill prescribe matri ces of dimension at most r = poly ( n ) to be assigned for the input variab les x j , 1 ≤ j ≤ n . Call these matrices M ( k ) i,j . For e ach weight function w i write do wn linear equations for each k ∈ [2 n 4 ] . f ( M ( k ) i, 1 , M ( k ) i, 2 , · · · , M ( k ) i,n ) = 0 . This will actually gi ve us a system of at most 2 n 4 r 2 linear equations in the unkno wn scalars c α . Since there are t ≤ m c weight functions in all, all the linear constrain ts put togeth er gi ve us a system of at most 2 n 4 r 2 m c linear equatio ns. N o w , the number of distinct (noncommuting ) monomials m α is n n = 2 n lg n which asymptotic ally ex ceeds 2 n 4 r 2 m c for m = 2 o ( n l g n ) , since r is polynomia lly bound ed. Thus, the system of linear equati ons has a nontri vial solution in the c α ’ s that can be computed using G aussia n eliminati on in time expo nential in n . Notice that the polynomial f ( x 1 , · · · , x n ) , defined by the solution to the c α ’ s , is a nonzero polyno - mial. W e claim that f cannot hav e a noncommutati ve circuit of size 2 o ( n l g n ) . Assume to the contrary that C ′ ( x 1 , · · · , x n ) is a noncommut ativ e circuit of size s = 2 o ( n l g n ) for f . Then, by Corollary 2.3 ther e is an n ′ -input boolean circuit C of size m = s O (1) = 2 o ( n l g n ) that accepts precisely the (binary encodings) of those monomials that are nonzero in C ′ . Let w 1 , · · · , w t be the weight functions output by A 2 for input ( m, n ′ ) . By H ypoth esis 2, for some weight function w i and some k ∈ [2 n 4 ] the circuit C ′ must be nonzero on matrices M ( k ) i,j . Ho wev er , f e va luates to zero, by construction , on the matrix inputs prescribed by all the weight functi ons w 1 , · · · , w t . This is a cont radiction to the assumption and it completes the proof. Remark 5.3 W e can formulate both Hypothesis 1 and Hypothesis 2 mor e gener ally by letting the running time of algorith ms A 1 and A 2 be a function t ( m, n ) . W e will then obtain suitabl y quantified circ uit lower bound r esults as conse quence . 1 By explicit we mean that the coef ficients of f are computable in time expone ntial in n . 9 Commutative c ircuits W e now sho w that under the derandomizat ion Hypothesis 3 (stated in the intro duction) we can obtain a strong er consequ ence than Theorem 5.1. Theor em 5.4 If a subexp onential -time algorith m A 3 satisfy ing Hypoth esis 3 exist s then identity testing ove r Q is in SU BEXP which implies that either NEXP 6⊂ P / poly or the inte ger P ermanent does not have polyno mial size arithmeti c circ uits. Pr oof . Using Lemma 1.5 it is shown in [KS01, Theorem 5] that there is a randomiz ed identity test for small degree polyno mials in Q [ x 1 , · · · , x n ] , where the poly nomial is giv en by an arith metic circuit ˆ C of polyn omially bounded deg ree d . The idea is to pick a random weight vec tor w : [ n ] → [2 nd ] and replace the indete rminate x i by y w ( i ) , where d is the total degree of the inpu t polynomial . As the circ uit ˆ C ha s small degree, after this uni vari ate substituti on the circuit can be ev aluate d in determinist ic polyno mial time to expli citly fi nd the polyno mial in y . By Lemma 1.5 it will be nonz ero with proba bility 1 / 2 if ˆ C computes a nonze ro polynomia l. Coming to the proo f of this theore m, if NEXP 6⊂ P / poly then we are done. S o, suppose NEXP ⊂ P / poly. Notice that gi ven any monomial x d 1 1 · · · x d n n of total deg ree bounde d by d we can test if it is a nonzero monomial of ˆ C in exponen tial time ( explici tly listing down the monomials of the polyn omial computed by ˆ C ). Therefore, since NEXP ⊂ P / poly there is a polynomial-si ze boolean circuit C that accepts the vector ( d 1 , · · · , d n ) iff x d 1 1 · · · x d n n is a nonzero monomial in the giv en polynomial (as required for application of Hypothe sis 3). No w , we in v oke the derandomiz ation Hypothesis 3. W e can apply the K li van s-Spielman polynomial identi ty test, explained abo ve, to the arithmetic circuit ˆ C for each of the t w eight v ectors w 1 , · · · , w t gener - ated by algori thm A 3 to obtain a sube xponen tial deterministi c identity te st for the cir cuit ˆ C by the propert ies of A 3 . No w , follo wing the argu ment of Impagl iazzo-Kaban ets [KI03] it is easy to deri ve that the inte ger Permanent does not ha ve polyno mial size arithmetic circuit s. Remark 5.5 W e formulate a str onge r versio n of Hypo thesis 3 to obt ain a conclu sion similar to Theor em 5.2 for commutative cir cuits . F or exa mple we can formulate the hypothesis : Ther e is a deterministic algorithm A 4 that takes as input ( m, n , K ) and outputs a collection of weight functi ons w 1 , w 2 , · · · , w t suc h that w i : [ n ] → [2 n ] , with the pr oper ty that for each size m , n -input oracle boolea n cir cuit C A (wher e A is EX P-complete) that tak es as input ( α 1 , · · · , α n ) such that α i ∈ { 0 , · · · , K } , ther e is some weight vector w i for which ther e is a unique linear form ( α 1 , · · · , α n ) accep ted by C A which attain s the minimum value P n j = 1 w i ( j ) α j . Furthermor e , A 4 runs in time polyn omial in m . It is easy to see that, similar to Theor em 5.2, as a consequen ce of this hypothesis ther e is some exp licit polyno mial f ( x 1 , · · · , x n ) (i.e. computable in E XP) which does not have commutative cir cuits of sube xpo- nentia l size. 6 Discussion An interesti ng open question is whether derandomizin g similar restricted ve rsions of the V aliant-V azira ni lemma also implies circuit lo wer bound s. W e recall the V aliant-V azirani lemma as stated in the origi nal paper [VV86]. 10 Lemma 6.1 Let S ⊆ { 0 , 1 } t . Suppose w i , 1 ≤ i ≤ t ar e pic ked u niformly at r andom fr om { 0 , 1 } t . F or eac h i , let S i = { v ∈ S | v .w j = 0 , 1 ≤ j ≤ i } and let p t ( S ) be the pr ob ability that | S i | = 1 for some i . T hen p t ( S ) ≥ 1 / 4 . Analogo us to our discussi on in Section 1, here too we can consider the restricted versio n w here we consid er S C ⊆ { 0 , 1 } n to be the set of n -bit vectors accepted by a boolean circu it C of size m . W e can similarly formulat e derandomizatio n hypoth eses similar to H ypoth eses 1 and 2. W e do not know if there is another randomized polynomial identity test for noncommutati v e arithmetic circuit s based on the V alia nt-V azirani lemma. The automata-th eoretic techn ique of Section 3 doe s not appear to work. Specifically , giv en a m atrix h : F n 2 → F k 2 , there is no deterministic finite automaton of size poly ( n, k ) that acce pts x ∈ F n 2 if and only if h ( x ) = 0 . Acknowledgements. W e are grateful to Manindra Agrawal for interestin g discu ssions and his suggestio n that Theorem 5.2 can be obtained from the stronge r hypothesis. W e also thank S rikanth Srini v asan for discus sions. Refer ences [Agr07] M . A G R A W A L . R ings and Integer Lattices in C omputer Science. Barbados W or kshop on Compu- tation al Complex ity , Lecture no 9, 2007. [AR00] K L AU S R E I N H A R D T A N D E R I C A L L E N D E R . Making Nondeterminism Unambiguous . SIAM J. Comput. 29(4): 1118-11 31 (2000) . [ARZ99] E R I C A L L E N D E R , K L AU S R E I N H A R D T , A N D S H I Y U Z H O U . Isolation, matchin g and counting unifor m and nonu niform upper bounds. Journa l of Computer and System Sciences , 59(2 ):164–181 , 1999. [AMS08] V . A RV I N D , P . M U K H O P A D H Y A Y , S . S R I N I V A S A N Ne w results on Noncommutati ve and Com- mutati ve P olynomia l Identi ty T esting. In Proceeding s of the 23rd IEEE Conferen ce on Computational Complex ity , June 2008 , to appear . T echn ical report versio n in ECCC report TR08-025, 2008. [BW05] A . B O G D A N O V A N D H . W E E More on Non commutati ve Polynomial I dentity T esting . In Proc . of the 20th Annual Conferenc e on Computation al Complexit y , pp. 92-99 , 2005. [HU78] J . E . H O P C R O F T A N D J . D . U L L M A N Introduct ion to Automata T heory , L anguag es and Computa- tion, Addison -W esley , 1979. [IKW02] R . I M P A G L I A Z Z O , V . K A B A N E T S A N D A . W I G D E R S O N . In search of an easy witness: Expo- nentia l time vs. probabilist ic polynomial time. Journa l of Computer and System S cience s 65(4)., pages 672-6 94, 2002. [KI03] V . K A B A N E T S A N D R . I M P A G L I A Z Z O . Derandomiz ation of polynomial identity tests means prov- ing circuit lo wer bounds. In Proc. of the thirty-fifth annual A CM Sym. on T heory of computing ., pages 355-3 64, 2003. [KS01] A D A M R . K L I V A N S , D A N I E L A . S P I E L M A N . Randomness Efficien t Identity T estin g. Proceedi ngs of the 33rd Symposium on Theory of Computing (STOC), 216-2 23, 2001. 11 [MVV87] K . M U L M U L E Y , U . V A Z I R A N I A N D V . V A Z I R A N I . Matching is as easy as matrix in versi on. In Proc. of the nineteenth annual A CM confer ence on Theory of Computing., pages 345-3 54. A CM P ress, 1987. [N91] N . N I S A N . Lower bounds for n on-commutati v e computatio n. In Proc. of th e 23rd annual A CM Sym. on Theory of comput ing., pages 410-41 8, 1991. [NSV94] H . N A R AY A N A N , H U Z U R S A R A N , V . V . V A Z I R A N I . Randomiz ed Paralle l Algorithms for Ma- troid Union and Intersection , W ith Applications to Arboresence s and Edge-Disjoi nt Spanning Tre es. SIAM J. Comput. 23(2): 387-397 (1994 ). [RS05] R . R A Z A N D A . S H P I L K A . Deterministic polynomial identit y testing in non commutati ve models. Computatio nal Complex ity ., 14(1): 1-19, 2005. [Sch80] J AC O B T. S C H W A R T Z . Fast Probabilis tic algori thm for verificat ion of polynomial identities . J. A CM., 27(4), pages 701-7 17, 1980. [Str94] H O W A R D S T R AU B I N G . Finite automata, formal logic, and circuit complexit y . Progress in Theoret- ical Computer Science . Birkhuse r B oston Inc., Bosto n, M A, 19 94. [VV86] L . G . V A L I A N T A N D V . V . V A Z I R A N I . NP is as Easy as D etecti ng Unique Solutions . Theor . Comput. Sci. 47(3): 85-93 (1986). [Zip79] R . Z I P P E L . Probabilistic algorithms for sparse polynomials. In Proc. of the Int. S ym. on Symbolic and Algebrai c Computatio n., pages 216-22 6, 1979. 12