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2. Treating the number of colliding layers in a branching program as a progress measure and showing that it reduces significantly under pseudorandom restrictions. |
In addition, we achieve nearly optimal seed-length [MATH] for the classes of: (1) read-once polynomials on [MATH] variables, (2) locally-monotone ROBPs of length [MATH] and width [MATH] (generalizing read-once CNFs and DNFs), and (3) constant-width ROBPs of length [MATH] having a layer of width [MATH] in every consecut... |
Introduction A central challenge in complexity theory is to understand the trade-off between space and randomness as resources and in particular, whether [MATH] . One of the main techniques we have for approaching this question is to design pseudorandom generators that fool tests computable in small space. The latter q... |
Definition 1.1 For [MATH] , a read-once branching program (ROBP) of width [MATH] and length [MATH] is a layered directed graph [MATH] with [MATH] layers where all but the first layer have at most [MATH] nodes, the first layer has a single vertex designated the start vertex, and the vertices in the last layer are either... |
Given a ROBP as above, it defines a function [MATH] naturally where on input [MATH] starting from the start vertex, you follow the edges labeled by [MATH] for [MATH] and output [MATH] if the last vertex reached is accepting and [MATH] otherwise. |
Derandomizing space-bounded computations is fundamentally related to designing pseudorandom generators (and hitting set generators ) for ROBPs as above. |
Definition 1.2 Given a class of functions [MATH] , a function [MATH] is a pseudorandom generator (PRG) with error [MATH] (or [MATH] fools [MATH] if for every [MATH] |
[EQUATION] We say the generator is log-space explicit if [MATH] can be computed in space logarithmic in the output length [MATH] and refer to [MATH] as the seed-length of the generator. |
It is well-known by now that if there exists a log-space explicit PRG (or even a hitting set generator) with constant error that fools ROBPs of width [MATH] and length [MATH] with seed-length [MATH] , then [MATH] . In this vein, a seminal result of Nisan Nis92 gave a log-space explicit PRG that [MATH] -fools ROBPs of w... |
Theorem 1 (Main Theorem) For any [MATH] , there exists a log-space explicit PRG that [MATH] -fools width- [MATH] ROBPs with seed-length |
[MATH] We in fact also obtain PRG’s with nearly optimal dependence for constant error for the bigger class of unordered width- [MATH] ROBPs, which are functions computable by ROBPs under some unknown permutation (see Section 3.5 for the formal definition). In this regime, we improve the results of SVW17 that gave a PRG... |
Theorem 2 For any [MATH] , there exists a log-space explicit PRG that [MATH] -fools unordered width- [MATH] ROBPs with seed-length [MATH] |
A special class of unordered width- [MATH] ROBPs that have received recent attention are read-once polynomials (see Tre10 LV17 ) for which we give a PRG with nearly optimal seed-length both in terms of the error and input length (up to [MATH] factors): |
Theorem 3 There exists a log-space explicit [MATH] -PRG for the class of read-once polynomials on [MATH] variables with seed-length [MATH] |
In comparison, the best previous PRG for read-once polynomials had seed-length [MATH] , thus in particular needed [MATH] seed-length to fool read-once polynomials with polynomially small error. |
Our results rely on several new conceptual ideas as well as technical ingredients, including PRGs fooling other interesting intermediate classes of ROBPs, that we believe could be useful for other applications especially in the context of obtaining PRGs for constant-width ROBPs. Our results rely on the framework of ite... |
1.1 The Ajtai-Wigderson framework The Ajtai-Wigderson AW85 framework, that was revived and refined for ROBPs in the work of Gopalan, Meka, Reingold, Trevisan, Vadhan GMR 12 , provides a “recipe” for constructing PRGs for classes of functions that simplify under (pseudo)random restrictions. Roughly speaking, in order to... |
Achieving Near-Logarithmic Seed-Length. In the work of GMR 12 the Ajtai-Wigderson approach was used to construct [MATH] -PRGs for read-once CNFs (and read-once DNFs) with seed length [MATH] In order to achieve nearly-logarithmic seed-length GMR 12 showed that one can assign a constant fraction of the coordinates while ... |
We describe the approach more precisely. A [MATH] -pseudorandom restriction against a class of functions [MATH] specifies a set [MATH] of roughly [MATH] of the coordinates, and an assignment [MATH] to these coordinates, such that for any [MATH] |
[EQUATION] where [MATH] denotes the string whose [MATH] -coordinates are taken from [MATH] and other coordinates are taken from [MATH] The main observation of GMR 12 is that given [MATH] , it suffices that [MATH] would fool the Bias-function , defined as |
[EQUATION] This is due to the fact that [EQUATION] The observation that it suffices to fool the bias-function instead of just fooling the restricted functions, enabled GMR 12 to use “mild” restrictions with [MATH] for the class of CNFs/DNFs. They show that in this case, the average of the restricted functions (i.e., th... |
Proof Overview Similarly to GMR 12 , in order to achieve a PRG with nearly-logarithmic seed-length fooling width-3 ROBPs, we show that: |
1. We can pseudorandomly assign half the input coordinates while preserving the acceptance probability (on average) of every width-3 ROBP up to error [MATH] , using seed-length [MATH] |
2. After pseudorandomly assigning all but [MATH] of the coordinates any width-3 ROBP simplifies enough so that it can be fooled using additional [MATH] random bits. |
Both steps are involved and explained in greater detail in the next two sections. 2.1 Pseudorandomly assigning half of the coordinates |
In Sections and we prove the following theorem showing that we can pseudorandomly assign [MATH] of the coordinates while changing the acceptance probability by at most [MATH] |
Theorem 4 Let [MATH] . There exists a log-space explicit pseudorandom restriction assigning [MATH] fraction of the variables using [MATH] random bits, that maintains the acceptance probability of any unordered width- [MATH] length- [MATH] ROBP up to error [MATH] |
Given Theorem , we can assign half of the coordinates by iteratively applying the pseudorandom restriction [MATH] times. This ultimately uses [MATH] random bits to assign half of the coordinates, as promised. |
We describe the techniques that go into the proof of Theorem The proof proceeds in two steps. The first step (described in Section ) reduces the task of generating a pseudorandom restriction for width-3 ROBPs to the task of generating a pseudorandom restriction for the XOR of short (logarithmic-length) width-3 ROBPs. T... |
2.1.1 Reducing width-3 ROBPs to the XOR of short width-3 ROBPs Next, we explain how we reduce fooling width-3 ROBPs to fooling the XOR of short width-3 ROBPs. Let [MATH] be a ROBP of length- [MATH] and width- [MATH] We pick a set [MATH] of size [MATH] using an almost [MATH] -wise independent distribution. We wish to sh... |
Recall that every layer of edges in a ROBP contains two sets of edges, one corresponding to the transition made when the input bit equals [MATH] and similarly one corresponding to the input bit equaling [MATH] Observe that if the two sets of edges are the same, then the layer is redundant and the value of the input bit... |
First, suppose (ideally) that all layers in a width-3 ROBP are colliding. Then, under the pseudorandom restriction, with high probability, in every [MATH] consecutive layers we will have a layer of edges whose corresponding variable is fixed to a value for which the edges in the layer collide, leaving at most [MATH] ve... |
The assumption that all layers in a width-3 ROBP are colliding is not necessarily true. In fact, it can be the case that in every layer of edges both the [MATH] -edges and the [MATH] -edges form a permutation on the state space with no collisions. Indeed, such ROBPs are known in the literature as permutation-ROBPs. (Fo... |
Relabeling Under The Bias Function: In the following, we consider relabeling of a ROBP. Recall that in a ROBP every vertex has a pair of outgoing edges: one labeled [MATH] and the other labeled [MATH] . A relabeling of a ROBP [MATH] is any ROBP [MATH] that can be achieved from [MATH] by swapping the labels for some of ... |
Our key observation is that the bias function [MATH] of a program [MATH] does not depend on the labels of the edges associated with the variables outside [MATH] This is due to the fact that the value of [MATH] on a given partial input [MATH] is the probability of acceptance of [MATH] on a random assignment to the varia... |
To sum up, since the bias-function is the average over all restricted functions of [MATH] , it also equals the average over all restricted functions of [MATH] , and these restricted functions are simple enough for us to fool. |
Relabeling was previously used in BV10 Ste13 CGR14 to show that the best ROBPs distinguishing between certain distributions and the uniform distribution must be “locally-monotone” (see Section 3.5 for the formal definition). In general, it is unclear how to argue locally monotone programs are the hardest ROBPs to fool.... |
2.1.2 Pseudorandom restrictions for the XOR of short width-3 ROBPs Our main result in Section is the following: Theorem 5 Let [MATH] [MATH] . There exists a log-space explicit pseudorandom restriction assigning [MATH] fraction of [MATH] variables using [MATH] random bits, that maintains the acceptance probability of an... |
Recall that in the previous section, we reduced the case of width-3 ROBPS to this case with [MATH] and [MATH] Our proof for Theorem follows previous strategies by GMR 12 GY14 GKM15 LV17 Indeed, the functions we are trying to fool are a special case of product-functions that were recently studied in HLV17 LV17 Product-f... |
PRGs for product-functions were constructed in previous work, however none achieve the parameters we need. Haramaty, Lee and Viola HLV17 and Lee and Viola LV17 constructed PRGs with seed length [MATH] and [MATH] respectively for such functions. While the latter is nearly optimal for constant [MATH] , we require [MATH] ... |
The main reason we are able to achieve better seed-length is due to the fact that we further assume that the functions [MATH] are computed by constant-width ROBPs. We rely on the previous work of Chattopadhyay, Hatami, Reingold, Tal CHRT17 . They constructed PRGs for constant-width length- [MATH] ROBPs with seed-length... |
2.2 Simplification under pseudorandom restrictions Recall that our proof strategy is similar to that of GMR 12 1. For [MATH] : assign half of the remaining coordinates pseudorandomly using [MATH] random bits, while changing the acceptance probability by at most [MATH] |
2. Pseudorandomly assign the remaining coordinates using [MATH] random bits. The first step was overviewed in the previous section. In order to carry on the second step, we wish to find some progress measure, that would decrease in each iteration of the first step. For the case of CNFs the CNF-width (i.e., the maximal ... |
Our analogous progress measure is the number of colliding layers. We recall that in a ROBP, some layers of edges form permutations on the state space, while others are colliding. |
We show that after the first application of step 1, with high probability the restricted ROBP can be written as a composition of [MATH] subprograms [MATH] where each [MATH] has at most [MATH] vertices in the first and last layers and at most [MATH] colliding layers. Intuitively, this happens since every colliding layer... |
Definition 2.1 We call a ROBP [MATH] [MATH] -ROBP if [MATH] can be written as [MATH] , with each [MATH] being a width [MATH] ROBP with the first and last layers having at most two vertices and each [MATH] having at most [MATH] colliding layers. |
We wish to show that the parameter [MATH] (that bounds the maximal number of colliding layer in a subprogram [MATH] with width- [MATH] in the first and last layers) reduces by a constant factor under any iteration of step 1. That is, to show that after iteration [MATH] of step 1 we get with high probability a [MATH] -R... |
Applying the restriction and the structure result [MATH] times, we end up with a [MATH] ROBP where [MATH] As a last step, we show that [MATH] -ROBPs are fooled by the INW generator INW94 with seed-length [MATH] . This follows from the results of BRRY14 For the unordered case, we use the generator from the recent work o... |
2.3 The proof of Theorem Theorem is a special case of the following theorem Theorem 6 Let [MATH] [MATH] . There exists a log-space explicit pseudorandom generator that [MATH] -fools any XOR of ROBPs of width- [MATH] and length- [MATH] (defined on disjoint sets of variables), using seed-length [MATH] |
We consider [MATH] as the progress measure, and wish to show that this parameter reduces under pseudorandom restrictions. This is analogous to the the number of colliding layers [MATH] in the previous section. However, here, in some cases, we cannot guarantee that the application of the pseudorandom restriction from Th... |
2.4 Organization In Section , we state useful definitions and results from previous work to be used throughout the paper. The rest of paper is organized such that each section starts with an overview that highlights one or two main results proved in it. Section proves the reduction from 3ROBPs to the XOR of short 3ROBP... |
Preliminaries Denote by [MATH] the uniform distribution over [MATH] , and by [MATH] for [MATH] the uniform distribution over [MATH] Denote by [MATH] the logarithm in base [MATH] . For any function [MATH] , we shorthand by [MATH] and by [MATH] For an event [MATH] we denote by [MATH] its indicator function. |
3.1 Restrictions For a set [MATH] and two strings [MATH] [MATH] we denote by [MATH] the string with [EQUATION] Definition 3.1 (Restriction) |
Let [MATH] be a function. A restriction is a pair [MATH] where [MATH] and [MATH] . We denote by [MATH] the function [MATH] restricted according to [MATH] , defined by [MATH] |
Definition 3.2 (Random Valued Restriction) Let [MATH] A random variable [MATH] , distributed over restrictions of [MATH] is called random-valued if conditioned on [MATH] , the variable [MATH] is uniformly distributed over [MATH] |
Definition 3.3 [MATH] -Random Restriction) [MATH] -random restriction is a random-valued restriction over pairs [MATH] sampled in the following way: For every [MATH] , independently, pick [MATH] to [MATH] with probability [MATH] Sample [MATH] uniformly from [MATH] We denote this distribution of restrictions by [MATH] |
Definition 3.4 (The Bias-Function) Let [MATH] Let [MATH] We denote by [MATH] the function defined by [MATH] When [MATH] is clear from the context, we shorthand [MATH] as [MATH] |
3.2 Fourier analysis of Boolean functions Any function [MATH] has a unique Fourier representation: [EQUATION] where the coefficients [MATH] are given by [MATH] We have [MATH] We denote the spectral-norm of [MATH] by [MATH] For any functions [MATH] it holds that [MATH] where equality holds if [MATH] and [MATH] depends o... |
Fact 3.5 O’D14 , Proposition 4.17] Let [MATH] and [MATH] . Then, [MATH] 3.3 Small-biased distributions We say that a distribution [MATH] over [MATH] is [MATH] -biased if for any non-empty [MATH] it holds that |
[MATH] NN93 AGHP92 ABN 92 BT13 Ta-17 show that [MATH] -biased distributions can be sampled using [MATH] random bits. Let [MATH] . We say that a distribution [MATH] over subsets of [MATH] is [MATH] -biased with marginals [MATH] if for any non-empty [MATH] it holds that |
[MATH] Claim 3.6 Let [MATH] for some integer [MATH] let [MATH] be an [MATH] -biased distribution over [MATH] Define [MATH] to be a distribution over subsets of [MATH] as follows: Sample [MATH] Output |
[MATH] Then [MATH] is [MATH] -biased with marginals [MATH] Proof. For any fixed [MATH] the probability that [MATH] is exactly the probability that [MATH] In an [MATH] -biased distribution, the latter event happens with probability [MATH] (See AGHP92 ). |
Claim 3.7 If [MATH] is [MATH] -biased with marginals [MATH] , then for any disjoint [MATH] it holds that [MATH] Proof. By inclusion-exclusion |
[EQUATION] 3.4 Standard tail bounds for [MATH] -wise independence Lemma 3.8 SSS95 , Thm. 4, restated] Let [MATH] be an even positive integer. Let [MATH] be some [MATH] -wise independent random variables bounded in [MATH] with expectation [MATH] Let [MATH] Then, |
[MATH] 3.5 Branching programs read-once branching program (ROBP) [MATH] of length [MATH] and width [MATH] is a directed layered graph with [MATH] layers of vertices denoted [MATH] . Each [MATH] consists of [MATH] vertices [MATH] , and between every two consecutive layers [MATH] and [MATH] there exists a set of directed... |
A branching program [MATH] and an input [MATH] naturally describes a computation path in the layered graph: we start at node [MATH] in [MATH] For [MATH] , we traverse the edge going out from [MATH] marked by [MATH] to get to a node [MATH] The resulting computation path is [MATH] We say that [MATH] accepts [MATH] iff th... |
Unordered branching programs are defined similarly, expect that there exists a permutation [MATH] such that in step [MATH] the computation path follows the edge marked by [MATH] , for [MATH] We also consider unordered branching programs on [MATH] of shorter length [MATH] . In such case, the program stops after reading ... |
For two programs [MATH] and [MATH] defined over disjoint sets of variables and having the end width of [MATH] equal the start width of [MATH] , we denote by [MATH] the concatenation of [MATH] and [MATH] , defined in the natural way. |
Locally Monotone Branching Programs. Let [MATH] be a width- [MATH] length- [MATH] ROBP. For any vertex [MATH] in the ROBP, denote by [MATH] the probability to accept a uniformly random input starting from the vertex [MATH] Since renaming the vertices in each layer does not affect the functionality of [MATH] , we may as... |
For [MATH] , denote by [MATH] the set of edges in [MATH] marked by [MATH] and similarly define [MATH] . We say that [MATH] is a identity layer if [MATH] (in which case [MATH] does not affect the output of of [MATH] ). We say that [MATH] is a permutation layer if both [MATH] and [MATH] form a matching between [MATH] and... |
Lemma 3.9 (Collision Lemma BV10 In a locally monotone branching program, every permutation layer is an identity layer. To see it, note that if we think of the vertices in each layer [MATH] as written from top to bottom according to [MATH] , then in a locally monotone program for any vertex [MATH] the [MATH] -edge leads... |
The following is a restatement of a result from CHRT17 . We give its proof for completeness in Appendix A.1 Theorem 3.10 Let [MATH] be an unordered oblivious read-once branching programs with width- [MATH] and length- [MATH] . Let [MATH] [MATH] [MATH] , and [MATH] be a [MATH] -biased distribution over subsets of [MATH]... |
[EQUATION] Theorem 3.11 (Implied by CHRT17 , Thm. 2] and SVW17 , Thm. 4.1] Let [MATH] be the class of all unordered oblivious read-once branching programs on [MATH] of length at most [MATH] and width at most [MATH] Then, there exists a log-space explicit pseudorandom generator |
[EQUATION] that [MATH] -fools [MATH] where [MATH] 3.6 Helpful lemmas Lemma 3.12 Let [MATH] If [MATH] is a real-valued random variable bounded in [MATH] with mean [MATH] , then [MATH] |
Proof. [MATH] since [MATH] . As [MATH] is convex and [MATH] domain is bounded, the maximal value that [MATH] can get is if all of [MATH] ’s probability mass is on the boundary. Denote by [MATH] Since [MATH] we get [MATH] , i.e., [MATH] , thus |
[EQUATION] Theorem 3.13 (Hyper-contractivity of Variance) Let [MATH] be a Boolean function. Then, [MATH] . Furthermore, if [MATH] , then |
[MATH] Proof. First, observe that using Fact 3.5 and [MATH] , we have [EQUATION] For the first item, we get [MATH] For the second item, we use the Hyper-contractivity Theorem Bon70 (cf. O’D14 , Ch. 9] ) stating that [MATH] for any function [MATH] (where [MATH] is the noise operator that satisfies [MATH] for all [MATH] ... |
[EQUATION] We analyze the RHS. Let [MATH] Then, [MATH] [MATH] , and under the uniform distribution [MATH] gets value [MATH] with probability [MATH] and value [MATH] with probability [MATH] We get |
[EQUATION] where the last inequality follows by concavity of [MATH] Overall if [MATH] , then [MATH] Lemma 3.14 Suppose [MATH] is [MATH] -biased distribution with marginals [MATH] Let [MATH] Let [MATH] be real valued functions, not necessarily distinct. Then, |
[EQUATION] Proof. Using Fact 3.5 , for any fixed [MATH] , we have [EQUATION] Thus, [EQUATION] and [EQUATION] The difference between the two is at most |
[EQUATION] which completes the proof. From width-3 ROBPs to the XOR of short ROBPs In Section , we prove the following theorem. See |
The pseudorandom restriction assigns [MATH] fraction of the variables as follows: 1. Choose a set of coordinates [MATH] according to a [MATH] -biased distribution with marginals [MATH] , for [MATH] |
2. Assign the variables in [MATH] according to a [MATH] -biased distribution, for [MATH] Known constructions of small-biased distributions NN93 AGHP92 ABN 92 BT13 Ta-17 show that it suffices to use [MATH] random bits to sample the restriction. |
In this section, we show how to design pseudorandom restrictions for unordered width-3 ROBPs from pseudorandom restrictions to the XOR of many width-3 ROBPs of length [MATH] We get the following theorem. See |
Proof Sketch. In this section, we shall show that under pseudorandom restrictions leaving each variable alive with probability [MATH] , with high probability, the bias function of a ROBP [MATH] can be written as a linear combination (up to a small error) over functions of the form [MATH] where each [MATH] is a short su... |
The reduction will first establish that with high probability (over the choice of the set of coordinates that are left alive) the bias function of a ROBP [MATH] can be written as the average of width- [MATH] length- [MATH] ROBPs, whose vast majority have at most [MATH] layers between every two layers with width- [MATH]... |
We focus on the first part of the reduction. First, consider the case when [MATH] is locally-monotone. In this case, every layer of edges is either the identity layer or a colliding layer (Lemma 3.9 ). Assume without loss of generality that there are no identity layers. Then, under a pseudorandom restriction, with high... |
However, in the case that [MATH] is not locally-monotone (e.g., when [MATH] is a permutation ROBP) it could the case that the widths of all layers of vertices remain [MATH] under the random restriction. Our main observation is that since the bias function takes the average over all assignments to the restricted variabl... |
Essentially, the bias function allows us to imagine as if we are taking the average over restrictions of [MATH] rather than restrictions of [MATH] , and restrictions of [MATH] are “simpler” to fool than restrictions of [MATH] since they have many layers with width- [MATH] |
The formal argument follows. Theorem 4.1 (From width- [MATH] to almost width- [MATH] Let [MATH] be a ROBP of width- [MATH] and length- [MATH] Let [MATH] Let [MATH] be a [MATH] -biased distribution over subsets of [MATH] with marginals [MATH] Let [MATH] be a random variable. Let [MATH] be the branching program [MATH] wh... |
[EQUATION] and with probability at least [MATH] over the choice of [MATH] and [MATH] [MATH] can be computed by a ROBP of the form [MATH] where [MATH] are defined over disjoint sets of at most [MATH] variables, and each [MATH] is a width- [MATH] ROBP with at most [MATH] vertices on the first and last layers. |
Proof. We first observe that [MATH] Indeed, for any fixed [MATH] [MATH] equals the probability that the following random-path in [MATH] accepts: |
Initiate [MATH] to be the start node of [MATH] . For [MATH] if [MATH] , take the edge exiting [MATH] marked by [MATH] , otherwise (i.e., if [MATH] ) pick a random edge out of the two edges exiting [MATH] . Denote by [MATH] the node at the end of the edge taken in the [MATH] -th step. Accept if and only if [MATH] is an ... |
Observe that the following random process is oblivious to the labels of edges in layers [MATH] , thus it would yield the same probability for [MATH] and for [MATH] . Overall, we got that [MATH] and [MATH] are equal as functions. |
In the remainder of the proof, we analyze [MATH] Let [MATH] and [MATH] denote the set of edges in the [MATH] -layer of [MATH] marked by [MATH] and [MATH] respectively. We assume without loss of generality that in all layers of edges [MATH] , as otherwise the [MATH] -th layer is redundant and may be eliminated. (Observe... |
Next, we consider the program [MATH] for a pseudorandom [MATH] and a random [MATH] For [MATH] we say that the [MATH] -th layer of edges is “good” under the choice of [MATH] and [MATH] , if [MATH] and layer [MATH] of [MATH] has at most [MATH] end-vertices. Let [MATH] For [MATH] let [MATH] be the event that none of layer... |
[EQUATION] Under the event that all [MATH] are false, we get that [MATH] has width [MATH] in every [MATH] layers. In such a case, we may write the restricted function [MATH] as [MATH] where each [MATH] is a width- [MATH] and length at most [MATH] ROBP with at most [MATH] vertices on the first and last layer. |
Theorem 4.2 (from almost width-2 to the XOR of short ROBPs - restatement of BDVY13 , Thm. 2.1] Let [MATH] be a ROBP of the form [MATH] where [MATH] are defined over disjoint sets of variables, and each [MATH] is a width- [MATH] ROBP with at most [MATH] vertices on the first and last layers. Then, (as a real-valued func... |
[MATH] where [MATH] are subprograms of [MATH] and [MATH] Proof of Theorem We prove that the following pseudorandom restriction maintains the acceptance probability of ROBPs of width- [MATH] and length- [MATH] up to error [MATH] Let [MATH] [MATH] |
1. Pick [MATH] using a [MATH] -biased distribution with marginals [MATH] 2. (a) Pick [MATH] using a [MATH] -biased distribution with marginals [MATH] |
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