Limitations of Sums of Bounded-Read Formulas
LLimitations of Sums of Bounded Read Formulas
Purnata Ghosal
Department of Computer Science and Engineering, IIT Madras, Chennai, [email protected]
B. V. Raghavendra Rao
Department of Computer Science and Engineering, IIT Madras, Chennai, [email protected]
Abstract
Proving super polynomial size lower bounds for various classes of arithmetic circuits computingexplicit polynomials is a very important and challenging task in algebraic complexity theory. Westudy representation of polynomials as sums of weaker models such as read once formulas (ROFs)and read once oblivious algebraic branching programs (ROABPs). We prove:(1) An exponential separation between sum of ROFs and read- k formulas for some constant k .(2) A sub-exponential separation between sum of ROABPs and syntactic multilinear ABPs.Our results are based on analysis of the partial derivative matrix under different distributions.These results highlight richness of bounded read restrictions in arithmetic formulas and ABPs.Finally, we consider a generalization of multilinear ROABPs known as strict-interval ABPsdefined in [Ramya-Rao, MFCS2019]. We show that strict-interval ABPs are equivalent to ROABPsupto a polynomial size blow up. In contrast, we show that interval formulas are different from ROFsand also admit depth reduction which is not known in the case of strict-interval ABPs. Theory of computation → Algebraic complexity theory
Keywords and phrases
Algebraic Complexity Theory, Arithmetic Circuits, Lower Bounds
Digital Object Identifier
Polynomials are one of the fundamental mathematical objects and have wide applicationsin Computer Science. Algebraic Complexity Theory aims at a classification of polynomialsbased on their computational complexity. In his seminal work, Valiant [39] laid foundationsof Algebraic Complexity Theory and popularized arithmetic circuits as a natural model ofcomputation for polynomials. He proposed the permanent polynomial perm n : perm n = X π ∈ S n n Y i =1 x iπ ( i ) , as the primary representative of intractability in algebraic computation. In fact, Valiant[39]conjectured that the complexity of computing perm n by arithmetic circuits is different fromthat of the determinant function which is now known as Valiant’s hypothesis.One of the important offshoots of Valiant’s hypothesis is the arithmetic circuit lowerbound problem: prove a super polynomial lower bound on the size of an arithmetic circuitcomputing an explicit polynomial of polynomial degree. Here, an explicit polynomial is onewhose coefficients are efficiently computable. Baur and Strassen [5] obtained a super linearlower bound on the size of any arithmetic circuit computing the sum of powers of variables.This is the best known size lower bound for general classes of arithmetic circuits.Lack of improvements in the size lower bounds for general arithmetic circuits lead thecommunity to investigate restrictions on arithmetic circuits. Restrictions considered in theliterature can be broadly classified into two categories: syntactic and semantic. Syntactictic © Purnata Ghosal and B. V. Raghavendra Rao;licensed under Creative Commons License CC-BY42nd Conference on Very Important Topics (CVIT 2016).Editors: John Q. Open and Joan R. Access; Article No. 23; pp. 23:1–23:20Leibniz International Proceedings in InformaticsSchloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany a r X i v : . [ c s . CC ] O c t restrictions considered in the literature include restriction on fan-out i.e., arithmetic formulas,restriction on depth i.e., bounded depth circuits [14, 15, 35], and the related model of algebraicbranching programs. Semantic restrictions include monotone arithmetic circuits [18, 41, 37],homogeneous circuits [9], multilinear circuits [30] and noncommutative computation [26].Grigoriev and Razborov [15] obtained an exponential lower bound for the size of a depththree arithmetic circuit computing the determinant or and permanent over finite fields. Incontrast, only almost cubic lower bound is known over infinite fields [20]. Explaining thelack of progress on proving lower bounds even in the case of depth four circuits, Agrawaland Vinay [1] showed that an exponential lower bound for the size of depth four circuitsimplies Valiant’s hypothesis over any field. This lead to intense research efforts in provinglower bounds for the size of constant depth circuits, the reader is referred to an excellentsurvey by Saptharishi et al. [34] for details.Recall that a polynomial p ∈ F [ x , . . . , x n ] is said to be multilinear if every monomial in p with non-zero coefficient is square-free. An arithmetic circuit is said to be multilinear if everygate in the circuit computes a multilinear polynomial. Multilinear circuits are natural modelsfor computing multilinear polynomials. Raz [31] obtained super polynomial lower boundson the size of multilinear formulas computing the determinant or permanent. Further, hegave a super polynomial separation between multilinear formulas and circuits [30]. In fact,Raz [31] considered a syntactic version of multilinear circuits known as syntactic multilinear circuits. An arithmetic circuit C is said to be syntactic multilinear, if for every product gate g = g × g the sub-circuits rooted at g and g are variable disjoint. The syntactic versionhas an advantage that the restriction can be verified by examining the circuit whereas thereis no efficient algorithm for testing if a circuit is multilinear or not. Following Raz’s work,there has been significant interest in proving lower bounds on the size of syntactic multilinearcircuits. Exponential separation of constant depth multilinear circuits is known [11], whilethe best known lower bound for unbounded depth syntactic multilinear circuits is only almostquadratic [2].An Algebraic Branching Program (ABP) is a model of computation for polynomials thatgeneralize arithmetic formulas and were studied by Ben-Or and Cleve [6] who showed thatABPs of constant width are equivalent to arithmetic formulas. Nisan [26] proved exponentialsize lower bound for the size of an ABP computing the permanent when the variables arenon-commutative. It is known that polynomial families computed by ABPs are the same asfamilies of polynomials computed by skew circuits, a restriction of arithmetic circuits whereevery product gate can have at most one non-input gate as a predecessor [23]. Further, skewarithmetic circuits are known to characterize the complexity of determinant [38]. Despitetheir simplicity compared to arithmetic circuits, the best known lower bound for size ofABPs is only quadratic [21, 10]. Even with the restriction of syntactic multilinearity, thebest known size lower bound for ABPs is only quadratic [16]. However, a super polynomialseparation between syntactically multilinear formulas and ABPs is known [13].Proving super quadratic size lower bounds for syntactic multilinear ABPs (smABPsfor short) remains a challenging task. Given that there is no promising approach yet toprove super quadratic size lower bounds for smABPs, it is imperative to consider furtherstructural restrictions on smABPs and formulas to develop finer insights into the difficulty ofthe problem. Following the works in [27, 29, 28], we study syntactic multilinear formulasand smABPs with restrictions on the number of reads of variables and the order in whichvariables appear in a smABP. . Ghosal and B. V. Raghavendra Rao 23:3
Models and Results: (1) Sum of ROFs: A read-once formula (ROF) is a formula whereevery variable occurs exactly once as a leaf label. ROFs are syntactic multilinear bydefinition and have received wide attention in the literature. Volkovich [40] gave a completecharacterization of polynomials computed by ROFs. Further, Minahan and Volkovich [24]obtained a complete derandomization of the polynomial identity testing problem on ROFs.While most of the multilinear polynomials are not computable by ROFs [40], sum of ROFs,denoted by Σ · ROF is a natural model of computation for multilinear polynomials. Shpilka andVolkovich showed that a restricted form of Σ · ROF requires linear summands to compute themonomial x x · · · x n . Further, Mahajan and Tawari [22] obtained a tight lower bound on thesize of Σ · ROF computing an elementary symmetric polynomial. Ramya and Rao [29] obtainedan exponential lower bound on the number of ROFs required to compute a polynomial in VP . In this article, we improve the lower bound in [29] to obtain an exponential separationbetween read- k formulas and Σ · ROF for a sufficiently large constant k . Formally, we prove: (cid:73) Theorem 1.
There is constant k > and a family of multilinear polynomials f PRY computable by read- k formulas such that if f PRY = f + f + · · · + f s , where f , . . . , f s areROFs, then s = 2 Ω( n ) . (2) Sum of ROABPs: A natural generalization of ROFs are read-once oblivious branchingprograms (ROABPs). In an ROABP, a layer reads at most one variable and every variableoccurs in exactly one layer. Arguments in [26] imply that any ROABP computing thepermanent and determinant requires exponential size. Kayal et al. [19] obtain an exponentialseparation between the size of ROABPs and depth three multilinear formulas. In [27], anexponential lower bound for the sum of ROABPs computing a polynomial in VP is given.We improve this bound to obtain a super polynomial separation between sum of ROABPsand smABPs: (cid:73) Theorem 2.
There is a multilinear polynomial family ˆ f computable by smABPs of polyno-mial size such that if ˆ f = f + . . . + f s , each f i ∈ F [ X ] being computable by a ROABP ofsize poly ( n ) , then s = exp (Ω( n (cid:15) )) for some (cid:15) < / . (3) Strict-interval ABPs and Interval formulas: It may be noted that any sub-programof a ROABP computes a polynomial in an interval { x i , . . . , x j } of variables for some i < j .A natural generalization of ROABPs would be to consider smABPs where every sub-programcomputes a polynomial in some interval of variables, while a variable can occur in multiplelayers. These are known as interval ABPs and were studied by Arvind and Raja [4] whoobtained a conditional lower bound for the size of interval ABPs. Ramya and Rao [28] obtainedan exponential lower bound for a special case of interval ABPs known as strict-interval
ABPs. We show that strict-interval ABPs are equivalent to ROABPs upto polynomial size: (cid:73)
Theorem 3.
The class of strict-interval ABPs is equivalent to the class of ROABPs.
Finally, we examine the restriction of intervals in syntactic multilinear formulas. We showthat unlike ROFs, interval formulas can be depth reduced (Theorem 28).
Related Work:
To the best of our knowledge, Theorem 11 is the first exponential separationbetween bounded read formulas and Σ · ROF . Prior to this, only a linear separation betweenbounded read formulas and Σ · ROF was known [3].Ramya and Rao [29] obtain an exponential separation between Σ · ROF and multilinear VP . Our result is an extension of this result for the case of a simpler polynomial computableby bounded read formulas. Mahajan and Tawari [22] obtain tight linear lower bound forΣ · ROF computing an elementary symmetric polynomial.
C V I T 2 0 1 6
Kayal, Nair and Saha [19] obtain a separation between ROABPs and multilinear depththree circuits. The authors define a polynomial, efficiently computed by set multilinear depththree circuits, that has an exponential size ROABP computing it. This polynomial can beexpressed as a sum of three ROFs. Later, Ramya and Rao [27] obtain a sub-exponentiallower bound against the model of Σ · ROABP computing the polynomial defined by Razand Yehudayoff [33]. Dvir et al. [13] obtain a super-polynomial lower bound on the size ofsyntactic multilinear formulas computing a polynomial that can be efficiently computed bysmABPs. We use the polynomial defined by [13] and adapt their techniques to obtain aseparation between smABPs and Σ · ROABP . Organization of the Paper:
Section 2 contains basic definitions of the models of compu-tations, concepts and explicit polynomials used in the rest of the paper. The rest of thesections each describe results with respect to a particular bounded-read model. Section 3describes the lower bound on the Σ · ROF model and Section 4 describes the lower boundon the Σ · ROABP model which follows using the same arguments as in the work of Dvir etal.[13]. Section 5 shows that strict-interval ABP is a fresh way to look at ROABPs sincethe two models are equivalent. In Section 6 we see that Brent’s depth reduction result ([8])holds for the class of interval formulas.
In this section, we present necessary definitions and notations. For more details, reader isreferred to excellent surveys by Shpilka and Yehudayoff [36] and by Saptharishi et al. [34].
Arithmetic Circuits:
Let X = { x , . . . , x n } be a set of variables. An arithmetic circuit C over a field F with input X is a directed acyclic graph (DAG) where the nodes have in-degreezero or two. The nodes of in-degree zero are called input gates and are labeled by elementsfrom X ∪ F . Non-input gates of C are called internal gates and are labeled from { + , times } .Nodes of out degree zero are called output gates. Typically, a circuit has a single output gate.Every gate v in C naturally computes a polynomial f v ∈ F [ X ]. The polynomial computed by C is the polynomial represented at its output gate. The size of a circuit denoted by size( C ),is the number of gates in it, and depth is the length of the longest root to leaf path in C ,denoted by depth( C ). An arithmetic formula is a circuit where the underlying undirectedgraph is a tree. For a gate v in C , let var ( v ) denote the set of all variables that appear asleaf labels in the sub-circuit rooted at v .Multilinear polynomials are polynomials such that in every monomial, the degree of avariable is either 0 or 1. Multilinear circuits, where every gate in the circuit computes amultilinear polynomial, are natural models of computation for multilinear polynomials. Acircuit C is said to be syntactic multilinear if for every product gate v = v × v in C , wehave var ( v ) ∩ var ( v ) = ∅ . By definition, a syntactic multilinear circuit is also multilinearand computes a multilinear polynomial.An arithmetic formula F is said to be a read-once formula (ROF in short) if every inputvariable in X labels at most one input gate in F . Algebraic branching program (ABP in short) is a model of computation of polynomialsdefined as analogous to the branching program model of computation for Boolean functions.An ABP P is a layered DAG with layers L , . . . , L m such L = { s } and L m = { t } where s is the start node and t is the terminal node. Each edge is labeled by an element in X ∪ F .The output of the ABP P is the polynomial p = P ρ is a s to t path wt ( ρ ), where wt ( ρ ) is the . Ghosal and B. V. Raghavendra Rao 23:5 product of edge labels in the path ρ . Further, for any two nodes u and v let [ u, v ] P denotethe polynomial computed by the subprogram P uv of P with u as the start node and v asthe terminal node. Let X uv denote the set of variables that appear as edge labels in thesubprogram P uv . The size of an ABP P , denoted by size( P ) is the number of nodes in P .In a syntactic multilinear ABP (smABP), every s to t path reads any input variablesat most once. An ABP is oblivious if every layer reads at most one variable. A read-onceoblivious ABP (ROABP) is an oblivious syntactic multilinear ABP where every variable canappear in at most one variable i.e., for every i , there is at most one layer j i such that x i occurs as a label on the edges from L j i to L j i +1 .An interval on the set { , . . . , n } with end-points i, j ∈ [ n ], can be defined as I = [ i, j ] , i The Partial Derivative Matrix: We need the notion of partial derivative matrices introducedby Raz [31] and Nisan [26] as primary measure of complexity for multilinear polynomials. Thepartial derivative matrix of a polynomial f ∈ X defined based on a partition ϕ : X → Y ∪ Z of the X into two parts. We follow the definition in [31]: (cid:73) Definition 5. (Raz [31]) Let ϕ : X → Y ∪ Z be a partition of the input variables in twoparts. Let M Y , M Z be the sets of all possible multilinear monomials in the variables in Y and Z respectively. Then we construct the partial derivative matrix M f ϕ for a multilinearpolynomial f under the partition ϕ such that the rows of the matrix are indexed by monomials m i ∈ M Y , the columns by monomials s j ∈ M Z and entry M f ϕ ( i, j ) = c ij , c ij being thecoefficient of the monomial m i · s j in f . We denote by rank ϕ ( f ) the rank of the matrix M f ϕ . We call ϕ an equi-partition when | X | = n , n even and | Y | = | Z | = n/ rank ϕ : (cid:73) Lemma 6. Let g and h be multilinear polynomials in F [ X ] . Then, ∀ ϕ : X → Y ∪ Z , wehave the following. Sub-additivity: rank ϕ ( g + h ) ≤ rank ϕ ( g ) + rank ϕ ( h ) , and Sub-multiplicativity: rank ϕ ( gh ) ≤ rank ϕ ( g ) × rank ϕ ( h ) ,In both the cases, equality holds when var ( g ) ∩ var ( h ) = ∅ . C V I T 2 0 1 6 Two Explicit Polynomials: Polynomials that exhibit maximum rank ofthe partial derivativematrix under all or a large fraction of equi-partitions can be thought of as high complexity or hard polynomials. We need two such families found in the literature.Raz and Yehudayoff [33] defined a multilinear in VP . To define this polynomial we denotean interval { a | i ≤ a ≤ j, a ∈ N } , i, j ∈ N by [ i, j ], and consider the sets of variables X = { x , . . . , x n } , W = { w i,‘,j } i,‘,j ∈ [2 n ] . We denote it as the Raz-Yehudayoff polynomialand define it as follows. (cid:73) Definition 7 ( Raz-Yehudayoff polynomial , [33]) . Let us consider f ij ∈ F [ X, W ] definedover the interval [ i, j ] . For i ≤ j , the polynomial f ij is defined inductively as follows. If j − i = 0 , then f ij = 0 . For | j − i | > , f ij = (1 + x i x j ) f i +1 ,j − + X ‘ ∈ [ i +1 ,j − w i,‘,j f i,‘ f ‘ +1 ,j , where we assume without loss of generality, lengths of [ i, ‘ ] , [ ‘ + 1 , j ] are even and smallerthan [ i, j ] . We define f , n as the Raz-Yehudayoff polynomial f RY . Note that, f RY can be defined over any subset X ⊆ X such that | X | is even, by consideringthe induced ordering of variables in X and considering intervals accordingly. We denote thispolynomial as f RY ( X ) for X ⊆ X . Raz and Yehudayoff showed: (cid:73) Proposition 8. ([33]) Let G = F ( W ) be the field of rational functions over the field F andthe set of variables W . Then for every equi-partition ϕ : X → Y ∪ Z , rank ϕ ( f RY ) = 2 n/ . Dvir et al. [13] defined a polynomial that is hard i.e., full rank with respect to a specialclass of partitions called arc-partitions . Suppose X = { x , . . . , x n − } be identified with theset V = { , . . . , n − } . For i, j ∈ V , the set [ i, j ] = { i, ( i + 1) mod n, ( i + 2) mod n, . . . , j } is called the arc from i to j . An arc pairing is a distribution on the set of all pairings (i.e.,perfect matchings) on V obtained in n/ P , . . . , P t )constructed in t < n/ P = (0 , L t , R t ] is the interval spanned by ∪ i ∈ [ t ] P i and the random pair P t +1 is constructed such that P t +1 = ( L t − , L t − 1) with probability 1 / , ( L t − , R t + 1) with probability 1 / , ( R t + 1 , R t + 2) with probability 1 / , and therefore, [ L t +1 , R t +1 ] = [ L t , R t ] ∪ P t +1 .Given a pairing P = { P , . . . , P n/ } of V , there are exactly 2 n/ partitions of X , byassigning ϕ ( x i ) ∈ Y and ϕ ( x j ) ∈ Z or ϕ ( x i ) ∈ Z and ϕ ( x j ) ∈ Y independently for each pair( i, j ) ∈ P . An arc partition is a distribution on all partitions obtained by sampling an arcpairing as defined above and sampling a partition corresponding to the pairing uniformly atrandom. We denote this distribution on partitions by D . For a pairing P = { P , . . . , P n/ } let M P be the degree n/ Q n/ i =1 ( x ‘ i + x r i ) where P i = ( ‘ i , r i ). Dvir et al. [13]defined the arc full rank polynomial b f = P P∈D λ P M P , where λ P is a formal variable. Dviret al. [13] showed: (cid:73) Proposition 9. [13] The polynomial b f can be computed by a polynomial size smABP andfor every ϕ ∈ D , rank ϕ ( b f ) = 2 n/ over a suitable field extension G of F . Now that we are familiar with most of the definitions required for an understanding ofthe results in this paper, we proceed to discuss our results. . Ghosal and B. V. Raghavendra Rao 23:7 In [29], Ramya and Rao show an exponential lower bound for the sum of ROFs computing apolynomial in VP . While this establishes a super polynomial separation between Σ · ROF and syntactic multilinear formulas, it is interesting to see if this separation is exponential.In this section we obtain such an exponential separation. In fact, we show that there isan exponential separation between syntactic multilinear read- k formula and Σ · ROF . Webegin with the construction of a hard polynomial computable be a read- k formula for a largeenough constant k . A full rank Polynomial: Let X = { x , . . . , x n } be the set of input variables of the hardpolynomial such that 4 | n . Let f RY ( X ) to denote the Raz-Yehudayoff polynomial definedon the variable set X of even size, where X is an arbitrary subset of X .Let r = Θ(1) be a sufficiently large integer factor of n such that r and n/r are both even.For 1 ≤ i ≤ n/r , let B i = { x ( i − r +1 , . . . , x ir } and B denote the partition B ∪ B ∪ · · · ∪ B n/r of X . The polynomial f PRY is defined as follows: f PRY = f RY ( B ) · f RY ( B ) · · · f RY ( B n/r ) . (1)By definition of the polynomial f PRY , it can be computed by a constant-width ROABP ofpolynomial size as well as by a read- k formula where k = 2 O ( r ) .In order to prove a lower bound against a class of circuits computing the polynomial f PRY ,we consider the complexity measure of the rank of partial derivative matrix. Like in [30] andmany follow-up results, we analyse the rank of the partial derivative matrix of f PRY under arandom partition. The reader might have already noticed that there are equi-partitions underwhich the rank ϕ ( f PRY ) = 1. Thus, we need a different distribution on the equi-partitionsunder which f PRY has full rank with probability 1. In fact, under any partition ϕ , whichinduces an equi-partition on each of the variable blocks B i , we have rank ϕ ( f PRY ) = 2 n/ , i.e.,full rank. We define D B as the uniform distribution on all such partitions. Formally, wehave: (cid:73) Definition 10. (Distribution D B ) The distribution D B is the distribution on the set ofall equi-partitions ˆ ϕ of X obtained by independently sampling an equi-partition ϕ i of eachvariable blocks B i , for all i such that ≤ i ≤ n/r . We express ˆ ϕ as ˆ ϕ = ϕ ◦ . . . ◦ ϕ n/r . For any partition ϕ in the support of D B , we argue that the polynomial f PRY has fullrank: (cid:73) Observation 1. For any ϕ ∼ D B , rank ϕ ( f PRY ) = 2 n/ with probability . Proof. Let us fix an equi-partition function ˆ ϕ ∼ D B , ˆ ϕ : X → Y ∪ Z . Let t = r . Considering f RY ( X ) where | X | = t and t is even, we can prove the partial derivative matrix of f RY ( X )has rank 2 n/ under ˆ ϕ by induction on t . By definition of f RY , for t = 2 we have f RY = 0.So, for the higher values of t , we see the term (1 + x x t ) and f ,t − are variable disjoint,where (1 + x x t ) has rank ≤ 2, and by the induction hypothesis, f ,t − has rank 2 t/ − . Also,by induction hypothesis, for any ‘ , the ranks of partial derivative matrices of f ,‘ and f ‘ +1 ,t are 2 ‘/ and 2 ( t − ‘ ) / respectively.When ˆ ϕ ( x ) ∈ Y and ˆ ϕ ( x t ) ∈ Z , we set w ,‘,t = 0 for all ‘ ∈ [2 , t − 1] and rank ˆ ϕ ( f ,t ) = rank ˆ ϕ (1 + x x t ) · rank ˆ ϕ f ,t − = 2 · ( t/ − = 2 t/ . When ˆ ϕ ( x ) ∈ Y and ˆ ϕ ( x t ) ∈ Y , for anarbitrary ‘ ∈ [ t ] we set w ,‘,t = 1 and we have rank ˆ ϕ ( f ,t ) = rank ˆ ϕ ( f ,‘ ) · rank ˆ ϕ ( f ‘ +1 ,t ) = 2 t/ ,since ˆ ϕ is an equi-partition. C V I T 2 0 1 6 By sub-additivity of rank, and since B i , i ∈ [ n/r ] are disjoint sets of variables, we have rank ˆ ϕ ( f PRY ) = Q i ∈ [ n/r ] rank ˆ ϕ ( f RY ( B i )) = Q i ∈ [ n/r ] t/ = 2 tn/ r = 2 n/ . (cid:74) In the following, we argue that the polynomial h cannot be computed by sum of ROFs ofsub-exponential size. More formally, (cid:73) Theorem 11. Let f , . . . , f s be read-once polynomials such that f PRY = f + f + · · · + f s ,then s = 2 Ω( n ) . We use the method of obtaining an upper bound on the rank of partial derivative matrixfor ROFs with respect to a random partition developed by [29]. Though the argument in [29]works for an equi-partition sampled uniformly at random, we show their structural analysis ofROFs can be extended to the case of our distribution D B . We begin with the notations usedin [29] for the categorisation of the gates in a read-once formula F . (In this categorisation,the authors have only considered gates with at least one input being a variable.)Type- A: These are sum gates in F with both inputs variables in X .Type- B: Product gates in F with both inputs variables in X .Type- C: Sum gates in F where only one input is a variable in X .Type- D: Product gates in F where only one input is a variable in X .Thus, type-D gates compute polynomials of the form h · x i where x i ∈ X, h ∈ F [ X \ { x i } ]are the inputs to the type-D gate. Let a, b, c, d be the number of gates of type-A, B, C andD respectively. Let a be the number of Type A gates that compute a polynomial of rank 2under an equi-partition ϕ , and a be the number of Type- A gates that compute a polynomialof rank 1 under ϕ such that a = a + a .The following lemma is an adaptation, for our distribution D B , of the same lemma forthe distribution of all equi-partitions on n variables from [29]. (cid:73) Lemma 12. Let f ∈ F [ X ] be an ROP, and ϕ be an equi-partition function sampleduniformly at random from the distribution D B . Then with probability at least − − Ω( n ) , rank ϕ ( M f ) ≤ n/ − Ω( n ) . Proof. We first argue a rank upper bound for an arbitrary f i . Let Φ i be the formulacomputing f i with gates of the types described as above. Let ˆ ϕ = ϕ ◦ . . . ◦ ϕ n/r sampledfrom the distribution D B uniformly at random.We use the Lemma 3.1 from [29] which concludes that type- D gates do not contribute tothe rank of a ROF. (cid:73) Lemma 13. [29, Lemma 3.1] Let F be a ROF computing a read-once polynomial f and ϕ : X → Y ∪ Z be an partition function on n variables. Then, rank ϕ ( f ) ≤ a + a + b + c . Intuitively, Lemma 13 can be applied to a ROF F under a distribution ˆ ϕ ∼ D B as follows.If there are a large number of type D gates (say αn , for some 0 ≤ α < ϕ , rank ˆ ϕ ( f ) ≤ (1 − α ) n/ . A type C gate, too, contributes a small value (atmost 2) to the rank compared to gates of types A and B. Thus, without loss of generality,we assume that the number of type C and D gates is at most αn . Now our analysis proceedsas in [29], only differing in the estimation of a , a under an equi-partition ˆ ϕ ∼ D B .Let ( P , . . . , P t ) be a pairing induced by the gates of types A and B (i.e., the two inputsto a gate of type A or B form a pair). There can be at most n/ αn gates of type C and D for some 0 ≤ α < 1, we assume (1 − α ) n remaining type A and B . Ghosal and B. V. Raghavendra Rao 23:9 gates. Therefore, for t = ( n − αn ) / , t ≤ n/ 2, we have the pairs P , . . . , P t induced by thetype A and B gates in Φ i .Now, considering the division of X into B , . . . , B n/r , we can divide the pairs into twosets depending on whether a pair lies entirely within a block B i , i ∈ [ n/r ] or the pair hasits members in two different blocks B i and B j for i, j ∈ [ n/r ] , i = j . We define thesetwo sets as W = { P i | P i = ( x, y ) , ∃ ‘, x, y ∈ B ‘ } for pairs lying within blocks and A = { P i | P i = ( x, y ) , ∃ j, k, j = k, x ∈ B j , y ∈ B k } for pairs lying across blocks, where x, y are two arbitrary variables in X .Each pair P i can be monochromatic or bichromatic under the randomly sampled equi-partition ˆ ϕ with the probability . Presence of monochromatic edges will give us a reductionin the rank of f i under ˆ ϕ . The analysis on W and A is done separately as follows. Analysing W , | W | > t/ : Let B i , . . . , B i ‘ be the blocks containing at least one pair from W , ‘ ≤ n/r . We want toestimate ‘ and count how many of these ‘ blocks have at least one monochromatic pair underˆ ϕ from W .For each B i , i ∈ [ t ], we define the Bernoulli random variable X i such that, X i = ( , if ∃ P ∈ W, P = ( x, y ) , x, y ∈ B i , , otherwise . Let Pr [ X i = 1] = Pr [ ∃ P ∈ W, P = ( x, y ) , x, y ∈ B i ] = (cid:15) , for some (cid:15) > E [ X i ] = (cid:15) , and for X = X + . . . + X n/r , E [ X ] = (cid:15) · n/r . By the Chernoff’sbound defined in [25], we have, Pr [ X > (cid:15)n/r ] < exp( − (cid:15)n r ) . Now we estimate (cid:15) as follows: (cid:15) = Pr [ X i = 1] = Pr [ ∃ P ∈ W, P = ( x, y ) , x, y ∈ B i ]= Pr [ x, y ∈ B i |∃ P ∈ W, P = ( x, y )]= Pr [ x, y ∈ B i ] Pr [ ∃ P ∈ W, P = ( x, y )] ≥ Pr [ x, y ∈ B i ] since Pr [ ∃ P ∈ W, P = ( x, y )] ≤ 1= 1 r . Therefore, Pr [ X > (cid:15)n/r ] < exp( − (cid:15)n r ) ≤ exp( − Ω( n )) , when r is a constant. This impliesthat at least 2 /r fraction of the blocks have a pair entirely within them with probability1 − exp( − Ω( n )) and each of these pairs is monochromatic under ˆ ϕ with the constant probability1 / 2. This gives an upper bound on the rank of f i ,rank ˆ ϕ ( f i ) ≤ n/ − n/r = 2 n/ − Ω( n ) . Analysing A , | A | > t/ : Since each pair of variables in A lies across two blocks, we create a graph G = ( V, E ) whereeach v i ∈ V represents the block B i and E = { ( v i , v j ) | ( x, y ) ∈ A, x ∈ B i , y ∈ B j , i = j } .The graph G has maximum degree r since there can be at most r pairs with one memberin a fixed block B i . If the edges in E form a perfect matching M in G , then under ˆ ϕ , the C V I T 2 0 1 6 edges in E can be either bichromatic or monochromatic. We need to show there will besufficient number of monochromatic edges to give a tight upper bound for rank ˆ ϕ ( f i ).By a result in [7], any graph with maximum degree r has a maximal matching of size m/ (2 r − | E | = m . Since | A | ≥ t/ m ≥ t/ t/ r − 1) = Ω( n ) when r is a suitable constant. With probability 1 / 2, an edge in themaximal matching is bichromatic. Hence, ≤ t/ / t/ = O (exp( n − )). So, with the high probability of1 − O (exp( n − )), more than half of the edges in the maximal matching are monochromatic,thus giving us the rank bound,rank ˆ ϕ ( f i ) ≥ n/ − t/ = 2 n/ − Ω( n ) . (cid:74) Given an upper bound on the rank of ROFs under a random partition from D B , we nowproceed to prove the Theorem 11 by showing a lower bound on the size of ROFs computingour hard polynomial h . Proof. (Proof of Theorem 11) By Observation 1, the upper bound on the rank of ROFsgiven by Lemma 12 and the sub-additivity of rank, we have: s · n/ − Ω( n ) ≤ n/ = ⇒ s = 2 Ω( n ) . (cid:74) With this result, the relationship between the classes of polynomials computable bypolynomial size ROFs, ROABPs and depth-3 multilinear circuits is clear. Since the class ofsmABPs of polynomial size is strictly smaller than the class of polynomial size multilinearcircuits (as in the non-multilinear setting), in the next section we obtain a lower bound on thesum of ROABPs computing the explicit polynomial in [13], which is efficiently computableby smABPs. In this section we prove a sub-exponential lower bound against the size of sum of read-once oblivious ABPs computing the hard polynomial constructed in [13]. This shows asub-exponential separation between syntactically multilinear ABPs and sum of ROABPs.We prove the following theorem in this section: (cid:73) Theorem 2. There is a multilinear polynomial family ˆ f computable by smABPs of polyno-mial size such that if ˆ f = f + . . . + f s , each f i ∈ F [ X ] being computable by a ROABP ofsize poly ( n ) , then s = exp (Ω( n (cid:15) )) for some (cid:15) < / . Our aim is to give an upper bound on the maximum rank of ROABPs under an arcpartition. We refer to the rank of the coefficient matrix of the sum of ROABPs against anarc-partition as the arc-rank . We analyze the arc-rank of the sum of ROABPs against anarc-partition to give a lower bound on the size of the sum necessary to compute b f .Let us assume that n is even. In order to prove the lower bound, we need to estimate anupper bound on the arc-rank computed by a ROABP. We define the notion of F -arc-partition, F being a ROABP, as follows: . Ghosal and B. V. Raghavendra Rao 23:11 (cid:73) Definition 14. Let us consider an arc partition Q constructed from a ROABP F in the fol-lowing manner: Let the order of variables appearing in the ROABP be x σ (1) , x σ (2) , . . . , x σ ( n ) ,where σ ∈ S n is a permutation on n indices. Then, Q = { ( x σ ( i ) , x σ ( i +1) ) | i ∈ [ n ] , i is odd } is a F -arc-partition. We assume 2 K | n . Let S , . . . , S K be a K -coloring of the variable set X , where x , . . . , x n are ordered according to the ROABP and for every i ∈ [ k ], S i contains the variables x ( i − n/K +1 , . . . , x in/K according to that ordering. Then S , . . . , S K is a K -partitioning ofthe pairs in the F -arc-partition Q . So pairs in Q are monochromatic, whereas the pairs( P , . . . , P n/ ) on which a random arc-partition Π sampled from D is based, might crossbetween two colors.Our analysis for the ROABP arc-rank upper bound follows along the lines of the analysisfor the arc-rank upper bound given by [13] for syntactic multilinear formulas. For thisanalysis we define the set of violating pairs for each color c , V c (Π), that is defined as: V c (Π) = { Π t | | Π t ∪ S c | = 1 , t ∈ [ n/ } , where Π , . . . , Π n/ are pairs in Π. The quantity G (Π) = |{ c | | V c (Π) | ≥ n }| , representingthe number of colors with many violations, is similarly defined. We use the following lemmadirectly from [13]: (cid:73) Lemma 15. Let K ≤ n , Π be the sampled arc-partition, and G (Π) be as defined above.Then, we have, Pr Π ∈D [ G (Π) ≤ K/ ≤ n − Ω( K ) . The following measure is used to compute the arc-rank upper bound for ROABPs. (cid:73) Definition 16. (Similarity function) Let ϕ be a distribution on functions S × S → N ,such that S is the support of the distribution on arc-partitions, D . Let P, Q be arc-partitionssampled independently and uniformly at random from D . Then, ϕ ( Q, P ) : S × S → N is thetotal number of common pairs between two arc-partitions Q and P . We assume Q to be the F -arc-partition for the ROABP F . For a pair that is not commonbetween Π and Q , we show both the variables in the pair is in the same partition, Y or Z with high probability. (cid:73) Theorem 17. Under an arc-partition Π sampled from D uniformly at random, if p ∈ F [ X ] is the polynomial computed by a ROABP P , then, for the similarity function ϕ and δ > , Pr Π ∼D [ ϕ (Π , Q ) ≥ n/ − n δ ] ≤ − o ( n ) . Proof Outline: Our argument is the same as [13]. It is being included here for completenessfor the parameters here being somewhat different than [13].In order to analyse the number of common pairs counted by ϕ , we consider the K -coloringof F and show that under a random arc-partition Π, the number of crossing pairs are largein number using Lemma 15. Then, we show, this results in large number of pairs havingboth elements in Y . In order to identify the colors with the high number of crossing pairs, agraphical representation of the color sets is used. Proof. [13] construct the graph H (Π), where each vertex is a color c such that | V c (Π) | ≥ n ,and vertices c and d have an edge connecting them if and only if | V c (Π) ∩ V d (Π) | ≥ n .We know for any two colors c, d ∈ [ K ], | V c (Π) ∩ V d (Π) | ≤ n . So, by definition of H (Π),the least degree of a vertex in H (Π) is 1. Using this, [13] prove the following claim: C V I T 2 0 1 6 (cid:66) Claim 18. Let the size of the vertex set of H (Π), V ( H (Π)), be M . For any subset U of V ( H (Π)) size N ≥ M/ − 1, there is some color h j +1 , j ∈ [ N − 1] such that in the graphinduced on all vertices except { h , . . . , h j } , the degree of h j +1 is at least 1.By Claim 18, we have U ⊆ V ( H (Π)), U = { c , . . . , c M/ − } such that this is the set ofcolours having high number of crossing pairs common with colors not in U . Considering thecolors sequentially, given Π, we first examine the pairs crossing from color c to other colors,then c and so on. Therefore, to examine the event E i for color c i , we have to estimate Pr Π ∼D [ E i | E , . . . , E i − , Π].Here, E i is the event | Y c i − | S c i | / | ≤ n , equivalently expressed as | S c i | / − n ≤ Y c i ≤ | S c i | / − n . But for an upper bound, it suffices to analyse the n crossingpairs from S c i to S c j instead of considering the entire set. Let the subset of Y c i constitutedby one end of crossing pairs going to color c j be P ij . Each element x in a crossing pair P t = ( x, w ) is a binomial random variable in a universe of size ≥ n = s with probability1 / Y of the universe. This event is independent of how the c i colored element of other crossing pairs P t are allotted. So, | B ij | = b j is a hypergeometricrandom variable where B ij contains all such x ∈ Y . By the properties of a hypergeometricdistribution, Pr b j [ b j = a ] = O ( s − ) = O ( n − ), where a is a specific value taken by the sizeof B ij .Applying the union bound over all colors c j for the crossing pairs, and taking b = P j ∈ U \{ i } b j , we have: Pr b [ s/ − n ≤ b ≤ | S c i | / − n ] ≤ n O ( n − ) = n − Ω(1) . Therefore, Pr Π ∼D [ E i | E , . . . , E i − , Π] = n − Ω( δ ) .We want an upper bound for Pr [ | Y c − | S c | / | ≤ n ∀ c ∈ [ K ]]. We have calculated anupper bound for the colors in [ K ] that were highly connected to each other in H (Π). So, wecan now estimate the total probability as follows: Pr [ | Y c − | S c | / | ≤ n ∀ c ∈ [ K ]]= E [ n − Ω( G ( P )) | G ( P ) > K/ E [ n − Ω( G ( P )) | G ( P ) ≤ K/ E [ n − Ω( G ( P )) | G ( P ) > K/ n − Ω( K ) by Lemma 15 ≤ n − Ω( K ) . If we consider δ = 1 / Pr Π ∼D [ ϕ (Π , Q ) ≥ n/ − n δ ] ≤ Pr [ | Y c − | S c | / | ≤ n ∀ c ∈ [ K ]] ≤ n − Ω( K ) Now, in Lemma 15, K ≤ n .Hence, Pr Π ∼D [ rank ϕ ( M ( p Π )) ≥ n/ − n δ ] ≤ − cn log n = 2 − o ( n ) . (cid:74) Now, using the above Theorem 17, we can prove the lower bound on the size of the sumof ROABP, s . Proof. (of Theorem 2) Since the polynomial f is such that each multiplicand is of the form λ e ( x u + x v ), if x u , x v are both mapped to the same partition Y or Z , it will reduce the rankof the partial derivative matrix by half. Hence, we have the following: Pr Π ∼D [ rank ϕ ( M ( f Π )) ≥ n/ − n δ ] = Pr Π ∼D [ ϕ (Π , Q ) ≥ n/ − n δ ] , . Ghosal and B. V. Raghavendra Rao 23:13 for some suitable δ > Pr [ rank ( M ( f Π )) = 2 n/ ] ≤ Pr [ ∃ i ∈ [ s ] , rank ( M (( f i ) Π )) ≥ n/ /s ] ≤ s X i =1 Pr [ rank ( M (( f i ) Π )) ≥ n/ /s ] ≤ s X i =1 Pr [ rank ( M (( f i ) Π )) ≥ n/ − n δ ] for some δ > ≤ s · n − Ω( n ) = ⇒ s = 2 Ω( n log n ) = 2 Ω( n ) . (cid:74) The difference in computational power of ROABPs and smABPs highlights the power ofreads of variables. From their definition, strict-interval ABPs generalise ROABPs by readingan interval of variables in every sub-program instead of reading a subset of variables in afixed order. However, in the following section, we note that reading in intervals do not lendmore computational power, and that ROABPs and Strict-interval ABPs in fact compute thesame class of polynomials. A strict-interval ABP, defined in [28] (See Definition 4), is a restriction of the notion ofinterval ABPs introduced by [4]. In the original definition given by [28], every sub-programin a strict-interval ABP P is defined on a π -interval of variables for some order π , however,without loss of generality, we assume π to be the identity permutation on n variables.Therefore, an interval of variables [ i, j ] , i < j here is the set { x i , . . . , x j } . In this section weshow that strict-interval ABPs are equivalent to ROABPs upto a polynomial blow-up in size. (cid:73) Theorem 19. The class of strict-interval ABPs is equivalent to the class of ROABPs. The proof of Theorem 19 involves a crucial observation that in a strict-interval ABP,variables are read in at most two orders and the nodes that correspond to paths that read indifferent orders can be isolated. We start with some observations on intervals in [1 , n ] andthe intervals involved in a strict interval ABP.Let P be a strict-interval ABP over the variables X = { x , . . . , x n } . For any two nodes u and v in P , let I u,v be the interval of variables associated with the sub-program of P with u as the start node and v as the terminal node. For two intervals I = [ a, b ] , J = [ c, d ] in[1 , n ], we say I (cid:22) J , if b ≤ c . Note that any two intervals I and J in [1 , n ] are comparableunder (cid:22) if and only if either they are disjoint or the largest element in one of the intervals isthe smallest element in the other. This defines a natural transitive relation on the set of allintervals in [1 , n ]. The following is a useful property of (cid:22) : (cid:73) Observation 2. Let I, J and J be intervals over [1 , n ] such that I (cid:22) J and J ⊆ J . Then I (cid:22) J . Proof. Let I = [ a, b ] , J = [ c, d ] and J = [ c , d ]. As I (cid:22) J , we have b ≤ c . Further, since J ⊆ J , we have c ≤ c and d ≤ d . Therefore, b ≤ c and hence I (cid:22) J . (cid:74) We begin with an observation on the structure of intervals of the sub-programs of P . Let v be a node in P . We say v is an ascending node, if I s,v (cid:22) I v,t and a descending node if I v,t (cid:22) I s,v . C V I T 2 0 1 6 (cid:73) Observation 3. Let P be a strict-interval ABP and v any node in P . Then, v is eitherascending or descending and not both. Proof. Let I = I s,v and J = I v,t . Since P is a strict-interval ABP, the intervals I and J aredisjoint and hence either I (cid:22) J or J (cid:22) I as required. (cid:74) Consider any s to t path ρ in P . We say that ρ is ascending if every node in ρ except s and t is ascending. Similarly, ρ is called descending if every node in ρ except s and t is descending. (cid:73) Lemma 20. Let P be a strict interval ABP and let ρ any s to t path in P . Then either ρ is ascending or descending. Proof. We prove that no s to t path in P can have both ascending and descending nodes.For the sake of contradiction, suppose that ρ has both ascending and descending nodes.There are two cases. In the first, there is an edge ( u, v ) in ρ such that u is an ascending nodeand v is a descending node. Let I = I s,u , J = I u,t , I = I s,v and J = I v,t . Since P s,u is asub-program of P s,v , we have I ⊆ I , similarly J ⊆ J . By the assumption, we have I (cid:22) J and J (cid:22) I . By Observation 2, we have I (cid:22) J and J (cid:22) I . By transitivity, we have I (cid:22) I .However, by the definition of (cid:22) , I and I are incomparable, which is a contradiction. Thesecond possibility is u being a descending node and v being an ascending node. In this case, J (cid:22) I and I (cid:22) J . Then, by Observation 2, we have J (cid:22) I as J ⊆ J . Therefore, J (cid:22) J bythe transitivity of (cid:22) , a contradiction. This completes the proof. (cid:74) Lemma 20 implies that the set of all non-terminal nodes of P can be partitioned into twosets such that there is no edge from one set to the other. Formally: (cid:73) Lemma 21. Let P be an interval ABP. There exist two strict-interval ABPs P and P such that All non-terminal nodes of P are ascending nodes and all non-terminal nodes of P aredescending nodes; and P = P + P . Proof. Let P be the sub-program of P obtained by removing all descending nodes from P and P be the sub-program of P obtained by removing all ascending nodes in P . ByLemma 20, the non-terminal nodes in P and P are disjoint and every s to t path ρ in P iseither a s to t path in P or a s to t path in P but not both. Thus P = P + P . (cid:74) Next we show that any strict-interval ABP consisting only of ascending or only ofdescending nodes can in fact be converted into an ROABP. (cid:73) Lemma 22. Let P be a strict-interval ABP consisting only of ascending nodes or only ofdescending nodes. Then the polynomial computed by P can also be computed by a ROABP P of size polynomial in size ( P ) . The order of variables in P is x , . . . , x n if P has onlyascending nodes and x n , . . . , x if P has only descending nodes. Proof. We consider the case when all non-terminal nodes of P are ascending nodes. Let ρ be any s to t path in P . We claim that the edge labels in ρ are according to the order x , . . . , x n . Suppose that there are edges ( u, v ) and ( u , v ) occurring in that order in ρ suchthat ( u, v ) is labelled by x i and ( u , v ) is labelled by x j with j < i . Let I = I s,u and J = I u ,t . Since i ∈ I , j ∈ J and I ∩ J = ∅ , it must be the case that J (cid:22) I and hence u must be a descending node, a contradiction. This establishes that P is an one ordered ABP.By the equivalence between one ordered ABPs and ROABPs ([16], [17]), we conclude that . Ghosal and B. V. Raghavendra Rao 23:15 the polynomial computed by P can also be computed by a ROABP of size polynomial in thesize of P .The argument is similar when all non-terminal nodes of P are descending. In this case,we have i < j in the above argument and hence I (cid:22) J , making u an ascending node leadingto a contradiction. This concludes the proof. (cid:74) A permutation π of [1 , n ] naturally induces the order x π (1) , . . . , x π ( n ) . The reverse of π isthe order x π ( n ) , x π ( n − , . . . , x π (1) . Since branching programs are layered, any multilinearpolynomial computed by a ROABP where variables occur in the order given by π can alsobe computed by a ROABP where variables occur in the reverse of π . (cid:73) Observation 4. Let P be a ROABP where variables occur in the order induced by apermutation π . The polynomial computed by P can also be computed by a ROABP of samesize as P that reads variables in the reverse order corresponding to π . Proof. Let P be the ROABP obtained by reversing the edges of P and swapping the startand terminal nodes. Since P is a layered DAG, there is a bijection between the set of all s to t paths in P and the set of all s to t paths in P , where the order of occurrence of nodes andhence the edge labels are reversed. This completes the proof. (cid:74) The above observations immediately establish Theorem 19. Proof of Theorem 19. Let P be a strict-interval ABP of size S computing a multilinearpolynomial f . By Lemma 21 there are strict interval ABPs P and P such that P hasonly ascending non-terminal nodes and P has only descending non-terminal nodes suchthat f = f + f where f i is the polynomial computed by P i , i ∈ { , } . By Lemma 22 andObservation 4, f and f can be computed by a ROABPs that read the variables in theorder x , . . . , x n . Then f + f can also be computed by an ROABP. It remains to boundthe size of the resulting ROABP. Note that size ( P i ) ≤ S . A ROABP for f i can be obtainedby staggering the reads of P i which blows up the size of the ABP by a factor of n ([16], [17]).Therefore size of the resulting ROABP is at most 2 nS ≤ O ( S ). (cid:74) Using Theorem 19, we can design the following white-box PIT for strict-interval ABPs. (cid:73) Corollary 23. Given a strict-interval ABP P of size s , we can check whether the polynomialcomputed by P is identically zero in time O ( poly ( S )) . Proof. The proof follows from Theorem 19 and the polynomial time white-box PIT algorithmgiven by [32] for non-commutative ABPs, since the variables in X are read only once, in afixed order, in a ROABP. (cid:74) The notion of intervals of variables corresponding to every sub-program can be appliedto formulas in the form of Interval Formulas, where every sub-formula corresponds to aninterval. In the following section we explore how such a model can be used to generalize themodel of ROFs, and in what ways it differs from ROFs. We saw that strict-interval ABPs have the same computational power as ROABPs despitebeing seemingly a non-trivial generalization. It is naturally tempting to guess that a similargeneralization of ROFs might yield a similar result. However, we observe that such ageneralization of ROFs yields a class different from ROFs. C V I T 2 0 1 6 We introduce interval formulas as a generalization of read-once formulas. An interval onvariable indices, [ i, j ] , i < j , is an interval corresponding to the set of variables X ij ⊆ X = { x , . . . , x n } , where X ij = { x p | x p ∈ X, i ≤ p ≤ j } . Polynomials are said to be defined onthe interval [ i, j ] when the input variables are from the set X ij . When there is no ambiguity,we refer to X ij as an interval of variables [ i, j ]. Gates in a read-once formula F can alsobe viewed as reading an interval of variables according to an order π on the variables i.e.,there is a permutation π ∈ S n such that every gate v in F is a sub-formula computing apolynomial on a π -interval of variables. Thus, interval formulas are a different generalizationof read-once formulas where every gate v in the formula F reads an interval of variables in afixed order.We formally define interval formulas as follows: (cid:73) Definition 24. (Interval Formulas) An arithmetic formula F is an interval formula if forevery gate g in F , there is an interval [ i, j ] , i < j such that g computes a polynomial in X ij and for every product gate g = h × h , the intervals corresponding to h and h must benon-overlapping. Thus, if a product gate g in F defined on an interval I = [ i, j ] takes inputs from gates g , . . . , g t , then the gates g , . . . , g t compute polynomials on disjoint intervals [ i, j ] , [ j +1 , j ] , . . . , [ j t − + 1 , j ] respectively, where ∀ p, j p < j p +1 and i ≤ j p ≤ j . If g , g , defined onintervals I , I are input gates to a sum gate g , then the interval I associated with g is I = I ∪ I .A quick observation is that interval formulas are different from ROFs: (cid:73) Proposition 25. The set of all polynomials computable by interval formulas is differentfrom that of ROFs Proof. By [40], the polynomial x x + x x + x x is not an ROF. However, the expression x x + x x + x x is itself an interval formula. (cid:74) In fact, interval formulas are universal, since any sum of monomials can be representedby an interval formula.Our next observation is that the polynomial f PRY defined in Section 3 can be computedby an interval formula. (cid:73) Proposition 26. The polynomial family f PRY is computable by an interval formula ofpolynomial size. Proof. Recall that f PRY ( X ) = f RY ( B ) · f RY ( B ) · · · f RY ( B n/r ). Since each of the f RY ( B i )is a constant variate polynomial and the sum of product representation of any multilinearpolynomial is an interval formula by definition, we have that f RY ( B i ) is computable by aninterval formula of constant size. This f PRY ( X ) has a polynomial size interval formula. (cid:74) It is not known if every ROF can be converted to a ROF of logarithmic depth. However, weargue, in the following that interval formulas can be depth-reduced efficiently.We have the following depth reduction result for general arithmetic formulas given by [8],who showed that depth of any arithmetic formula can be reduced by allowing its size to beincreased by a polynomial factor. (cid:73) Theorem 27. [8] Any polynomial p computed by an arithmetic formula of size s and depth d , can also be computed by a formula of size poly ( s ) and depth O (log s ) . . Ghosal and B. V. Raghavendra Rao 23:17 We know that this reduction preserves multilinearity. However, we don’t know if The-orem 27 can be modified to preserve the read- k property. We show that the depth reductionalgorithm given by Theorem 27 preserves the interval property. (cid:73) Theorem 28. Let f ∈ F [ X ] be a polynomial computed by an interval formula F of size s and depth d . Then f can also be computed by an interval formula of size poly ( s ) and depth O (log s ) . Proof. We know that the underlying structure of any arithmetic formula is a tree. The proofby Brent crucially uses the fact that by the tree-separator lemma [12], we are guaranteedthat there exists a tree-separator node g such that the sub-tree Φ of a formula Φ of totalsize s , rooted at the node g , has size ≤ s/ g by a new formal variable y . Let the resulting polynomial computed by F be f ( x , . . . , x n , y ), where f ( x , . . . , x n ) = f ( x , . . . , x n , g ) under the new substitution y = g . As f is linear in y , we have f ( x , . . . , x n , y ) = yf ( x , . . . , x n ) + f ( x , . . . , x n ) , where f = f | y =0 and f = f | y =1 − f | y =0 . Thus, f , f can be computed by multilinearformulas of size less than size( F ). Now, recursively obtaining small-depth formulas for f , f ,we obtain a O (log s ) depth formula computing f .However, the above construction does not necessarily preserve the interval property, sincethe intervals of variables on which f , f and g are defined, can be overlapping. We overcomethis problem by expressing f , f as products of polynomials over disjoint intervals, each ofthe intervals being disjoint to the interval corresponding to g .We assume, without loss of generality, that the interval formula F corresponds to theinterval [1 , n ]. Let the interval corresponding to g be I g = [ i, j ] , i < j . Now, by definition of f and f , they are defined on the same interval of variables. We consider the intervals I , I such that I ∪ I = [1 , n ] \ [ i, j ], I = [ j + 1 , n ] and I = [1 , i − f , f as products of two polynomials on I and I respectively. As f and g are multiplicativelyrelated in F , we show that f = f , × f , where f , is a polynomial on the interval I and f , is a polynomial on the interval I .We consider the root to leaf ( g ) path ρ in the original formula F containing the node g . All the paths meeting ρ at a sum gate represent polynomials additively related to y i.e.,contributing towards the computation of f and not f . For f , we will analyze only thepaths meeting ρ at product gates. Let us consider a product gate on ρ computing h × h ,such that h lies on ρ . Since I is contained in the interval corresponding to h , the intervalcorresponding to h , I h must be either fully contained in I or I . Constructing an interval formula for f : We ignore all sum gates on ρ computing p + p ,with p on ρ , by substituting p to zero. The resulting formula is F . In any product gatecomputing h × h , where h is on ρ , if I h ⊂ I , we substitute h by 1. We also substitute g by 1. The remaining formula F computes the polynomial f (1) .We repeat this process above, but this time, we substitute h by 1 only when I h ⊂ I .This remaining formula F computes f (2) . By definition of f , f = f (1) · f (2) . The intervalcorresponding to F is contained in I , the interval corresponding to F is contained in I . Constructing an interval formula for f : We ignore all product gates on ρ computing h × h , with h on ρ , by substituting h by 1. The resulting formula is ˆ F . C V I T 2 0 1 6 In any sum gate computing p + p , where p is on ρ , if I p ⊂ I , we substitute p by 0.We also substitute g by 0. The remaining formula ˆ F computes the polynomial p (1) .We repeat this process from the beginning, but substitute p by 0 only when I p ⊂ I .This remaining formula ˆ F computes p (2) . By definition of f , f = p (1) + p (2) . 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