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This first item follows since the probability that more than [MATH] variables in [MATH] survive is at most [EQUATION] As for the second item, we start with the case where [MATH] Assume that more than [MATH] new variables were added to [MATH] This implies that there is a set of [MATH] good functions in step [MATH] that ... |
[EQUATION] (where we used [MATH] which finishes the case [MATH] In the case where [MATH] , we show that with high probability all good functions remain good. For each individual function, using Markov’s inequality |
[EQUATION] Thus, we can apply a union bound and show that all good functions remain good with probability at least [MATH] Say the process finished. We shall assume that [MATH] for every iteration [MATH] until the process stopped. By Claim 6.5 this happens with probability at least [MATH] by applying a union bound on th... |
1. We stopped on Step 3 at some iteration [MATH] . If [MATH] then [MATH] and at most [MATH] variables remain that affect the functions [MATH] Otherwise, since [MATH] and [MATH] , at most [MATH] variables remain that affect the functions [MATH] , and thus at most [MATH] variables remain that affect the functions [MATH] ... |
2. We stopped at Step 4 at some iteration [MATH] Certainly, [MATH] Thus, we are in the case that was handled in Section 6.1 , with [MATH] . Indeed, Lemma 6.2 guarantees that [MATH] fools the remaining function with error at most [MATH] using [MATH] random bits. |
6.3 The actual generator Algorithm described the pseudo-random generator as if we knew whether or not the condition in step 3 holds. However, a pseudorandom generator cannot depend on the function it tries to fool. To overcome this issue, we use the following general observation regarding pseudorandom generators. |
Claim 6.6 Say there are two families of functions [MATH] and [MATH] that are both closed under shifts (i.e., closed under XORing a constant string to the input). Say that [MATH] is an [MATH] -PRG for [MATH] and [MATH] is an [MATH] -PRG for [MATH] then [MATH] is an [MATH] -PRG for [MATH] |
Proof. Let [MATH] , we show that [MATH] fools [MATH] By symmetry assume [MATH] [EQUATION] where [MATH] . Since [MATH] is closed under shifts, we have that [MATH] thus [MATH] |
[MATH] -fools [MATH] and we get [MATH] The actual generator would proceed as follows. Algorithm 3 The Pseudorandom Generator [MATH] |
1: a set [MATH] of the “live” coordinates, a width [MATH] , an integer [MATH] , a parameter [MATH] 2: if [MATH] then return [MATH] for [MATH] |
3: Let [MATH] for [MATH] 4: Pick [MATH] [MATH] according to Claim 6.1 5: Let [MATH] 6: return [MATH] Claim 6.7 (Proof of Correctness) |
Let [MATH] Suppose [MATH] are functions on disjoint sets of [MATH] Suppose each function depends on at most [MATH] variables except for a total of at most [MATH] variables, and the number of non-constant functions is at most [MATH] Then, [MATH] fools [MATH] with error [MATH] |
Proof. We prove the claim by induction on [MATH] If [MATH] then Theorem 3.11 implies correctness. If [MATH] then we consider the following two cases: |
1. If there are more than [MATH] good functions, then [MATH] fools [MATH] with error [MATH] 2. Otherwise, there are at most [MATH] good functions and we apply Step 3. According to Claim 6.1 , the average acceptance probability of [MATH] is [MATH] close to that of [MATH] Furthermore, with probability at least [MATH] all... |
Since we have a pseudorandom generator fooling the function in each case, Claim 6.6 shows that [MATH] fools [MATH] with error [MATH] |
Claim 6.8 (Seed Length) The amount of random bits used to calculate [MATH] is at most [MATH] Proof. Unwrapping the recursive calls in the evaluation of [MATH] we see that there are at most [MATH] recursive calls to the procedure and that the error parameters are at least [MATH] in all of them. |
We apply the generator from Theorem 3.11 only once during these recursive calls, on a ROBP of width- [MATH] and length [MATH] . Thus, the application of Theorem 3.11 uses at most [MATH] random bits. |
The partial assignment from Claim 6.1 uses at most [MATH] each time we invoke it, and we invoke it at most [MATH] times. The generator [MATH] uses [MATH] random bits each time we invoke it, and we invoke it at most [MATH] times. |
Claims 6.7 and 6.8 complete the proof Theorem with [MATH] as the generator. 6.4 Pseudorandom generator for read-once polynomials |
Next, we restate and prove Theorem See Proof. We show that [MATH] fools any read-once polynomial with error at most [MATH] . Its seed length is [MATH] |
A read-once polynomial can be written as the XOR of AND functions on disjoint variables, i.e., as the XOR of width- [MATH] ROBPs on disjoint variables. It remains to show that these ROBPs are short. Rather, we show that any PRG that [MATH] -fools read-once polynomials of degree at most [MATH] also [MATH] -fools all rea... |
[EQUATION] be a read-once polynomial over [MATH] , where [MATH] are disjoint subsets of [MATH] Without loss of generality let [MATH] be the blocks of length bigger than [MATH] Let |
[EQUATION] be the sum over monomials of degree at most [MATH] of [MATH] Let [MATH] . By triangle inequality [EQUATION] For [MATH] , since [MATH] , we have |
[MATH] As for the distribution [MATH] , by monotonicity [EQUATION] where [MATH] is any arbitrary subset of exactly [MATH] variables from [MATH] . Since [MATH] fools degree- [MATH] read-once polynomials with error at most [MATH] , and [MATH] is such a polynomial, we get that [MATH] is at most [MATH] Plugging both bounds... |
Pseudorandom generators for width-3 ROBPs In this section, we construct pseudorandom generators fooling width-3 ROBPs ( 3ROBPs , in short) with seed-length [MATH] . For ordered width-3 ROBPs we can guarantee error [MATH] using seed-length [MATH] See |
Note that in comparison, even for constant [MATH] , the best previous generators had seed-length [MATH] for ordered 3ROBPs. We also get similar improvements for unordered 3ROBPs but with worse dependence on the error [MATH] See |
7.1 Proof overview We heavily rely on the pseudorandom restriction from Theorem that assigns [MATH] of the variables while changing the acceptance probability by at most [MATH] . As a first step we assign a constant fraction of the coordinates. |
Assigning most of the coordinates. The first step is rather simple: we apply iteratively [MATH] times the pseudorandom restriction from Theorem to get the following analog result to Claim 6.1 The proof is the same as that of Claim 6.1 and is omitted. |
Claim 7.1 Let [MATH] . For all constants [MATH] , there is a pseudorandom restriction [MATH] using [MATH] random bits, changing the acceptance probability of 3ROBPs by at most [MATH] Furthermore, [MATH] is [MATH] biased with marginals [MATH] and |
[MATH] is [MATH] biased. Let [MATH] be a 3ROBP of length- [MATH] . First, we claim that after applying the pseudorandom restriction [MATH] in Claim 7.1 , with high probability (at least [MATH] ), [MATH] has a simpler structure in that between any two width- [MATH] layers the subprogram has at most [MATH] |
colliding layers . Concretely, we use the following definitions. Definition 7.2 Given a ROBP [MATH] , we call a layer of edges colliding if either the edges marked by [MATH] and the edges marked by [MATH] collide. |
Definition 7.3 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 show that after applying the pseudorandom restriction [MATH] in Claim 7.1 , with high probability the restricting ROBP [MATH] is a [MATH] -ROBPs. Now, similar to Section , we wish to iteratively apply Claim 7.1 , making the ROBP simpler in each step. We will have one progress measures on the restricted ROBP: the max... |
7.2 Reducing the length of [MATH] -ROBPs Here, we show that [MATH] -ROBPs can be approximated by [MATH] -ROBPs for some constant [MATH] A crucial point in the analysis is that we need the approximation to hold not just under the uniform distribution but also under the pseudo-random distribution. Fortunately, we are abl... |
Lemma 7.4 (Main Structural Result) For any [MATH] the following holds. Any [MATH] -ROBP [MATH] can be written as [MATH] where [MATH] is a [MATH] -ROBP and either [MATH] or for any [MATH] [MATH] where [MATH] are non-zero events that can be computed by [MATH] -ROBPs on disjoint variables. |
We shall also show (in the next claim) that any non-zero event [MATH] that can be computed by [MATH] -ROBP, happens with probability at least [MATH] under the uniform distribution. Thus, [MATH] which is doubly-exponentially small in [MATH] provided that [MATH] is a large enough constant. |
For any vertex [MATH] in a ROBP, we denote by [MATH] the probability to reach [MATH] under a uniform random assignment to the inputs. |
Claim 7.5 In a ROBP with width [MATH] and at most [MATH] colliding layers, every vertex whose [MATH] has [MATH] We remark that this bound is sharp. |
Proof. We prove by induction (on the length of the program) that any program with width at most [MATH] , exactly [MATH] colliding layers and exactly [MATH] reachable states in the last layer, has [MATH] for any reachable vertex [MATH] Without loss of generality all nodes in the program are reachable (otherwise, we remo... |
Consider a program [MATH] of length [MATH] with parameters [MATH] Removing the last layer gives a program [MATH] of length [MATH] with parameters [MATH] . By the induction hypothesis for any [MATH] in the last layer of [MATH] we have [MATH] for [MATH] |
We perform a case analysis. The following simple bound will be used in all cases. Let [MATH] be a vertex in the last layer of [MATH] . Assume that [MATH] edges enter [MATH] from vertices in the second to last layer. Then, [MATH] . In particular, since we assumed all vertices are reachable, any vertex in the last layer ... |
If [MATH] and [MATH] , then the last layer of edges in [MATH] is regular, i.e., any node in the last layer in [MATH] has exactly two ingoing edges. In this case any vertex [MATH] in the last layer has [MATH] |
If [MATH] , then [MATH] , since there are no collisions in the last layer of edges. Since we already handled the case [MATH] , we may assume [MATH] For any vertex [MATH] in the last layer we have |
[MATH] If [MATH] , then we consider two sub-cases: if [MATH] then only one vertex is reachable in the last layer and its [MATH] equals [MATH] Otherwise, [MATH] and [MATH] thus [MATH] and for any vertex [MATH] in the last layer we have [MATH] |
We say that two vertices [MATH] and [MATH] in a ROBP are locally-equivalent if the [MATH] -edges exiting [MATH] and [MATH] reach the same vertex and the [MATH] -edges exiting [MATH] and [MATH] reach the same vertex. We say that a ROBP has no-redundant vertices if any vertex in the program is reachable, and there are no... |
Claim 7.6 (Colliding Layers [MATH] Colliding) Let [MATH] be a 3ROBP with width-2 at the start and finish, at least one colliding layer and no-redundant vertices. Let [MATH] and [MATH] be the two start nodes. Then, there exists a string on which the two paths from [MATH] and [MATH] collide. |
Proof. First consider the case that [MATH] has width [MATH] . Then, there exists a layer [MATH] and a value [MATH] such that the two edges marked by [MATH] in the [MATH] -th layer collide. Any string whose [MATH] -th bit equals [MATH] results in colliding paths. |
For the rest of the proof assume that [MATH] has a layer with width [MATH] . Let [MATH] be the layers of vertices in [MATH] Let [MATH] denote the index of the last layer in [MATH] with width [MATH] Since [MATH] has width-2 at the end, [MATH] |
There are six edges between [MATH] and [MATH] : three edges marked with [MATH] and three edges marked with [MATH] Since [MATH] , by the Pigeon-hole principle, there are two edges marked with [MATH] going to some vertex [MATH] , and two edges marked with [MATH] going to some vertex [MATH] [MATH] is not necessarily diffe... |
and the two edges from [MATH] marked with [MATH] go to [MATH] Since [MATH] is reachable, there is an input [MATH] that leads from [MATH] or [MATH] to [MATH] Without loss of generality, we assume that [MATH] is reachable from [MATH] Let [MATH] be the vertex reached by following the same input [MATH] starting from the ot... |
Claim 7.7 (“First Collisions” can be detected by 3ROBPs) Let [MATH] be a 3ROBP with 2 vertices at the first layer, denoted [MATH] Suppose there are at most [MATH] colliding layers in [MATH] and that there exists a string on which the two paths from [MATH] and [MATH] collide. Let [MATH] be the first vertex on which a co... |
Proof. To simulate whether the paths starting from [MATH] and [MATH] collide at [MATH] , we consider the 3ROBP that keeps the unordered pair corresponding to the states of the two paths during the computation. In each layer until [MATH] , we have only states corresponding to [MATH] or [MATH] When we reach the layer of ... |
We are now ready to prove the main structural lemma – Lemma 7.4 . In the following, we consider branching programs with two initial nodes [MATH] . We interpret the value of the program on input [MATH] as its average value on the two paths starting from [MATH] and [MATH] . That is, the program can get value [MATH] or [M... |
Throughout this section we think of the error terms as [MATH] -indicators (instead of the usual [MATH] -notation for other Boolean functions). We shall use [MATH] and [MATH] to denote the standard AND and negation of these Boolean values. |
Lemma 7.8 Let [MATH] be a ROBP where each [MATH] is a width- [MATH] ROBP with at most [MATH] vertices on the first and last layers. Then, for any [MATH] we can write [MATH] as the sum of [MATH] and an error term [MATH] , that is bounded in absolute value by [MATH] where [MATH] denotes the event that the two paths in [M... |
Proof. Assume without loss of generality that no layer of vertices has width- [MATH] except for maybe the first. For [MATH] , let [MATH] and [MATH] be the two nodes at the first layer of the subprogram [MATH] If [MATH] has two nodes at the first layer, then denote them by [MATH] and [MATH] , otherwise denote the single... |
Proof of Lemma 7.4 Let [MATH] be a [MATH] -ROBP [MATH] If [MATH] has no colliding layers, then there is nothing to prove since [MATH] itself is a [MATH] -ROBP. If [MATH] has colliding layers, then without loss of generality each [MATH] has at least one colliding layer (since otherwise we can merge subprograms with no c... |
7.3 PRGs for ROBPs with few colliding layers In this section we show that we can [MATH] -fool ordered ROBPs with at most [MATH] -colliding layers with [MATH] seed-length. |
Theorem 7.9 For any [MATH] , there is a log-space explicit PRG that [MATH] -fools ordered width [MATH] -ROBPs with length [MATH] and at most [MATH] colliding layers using seed length |
[EQUATION] The above relies on the PRGs for regular branching programs and generalizations of them due to Braverman, Rao, Raz, and Yehudayoff BRRY14 In the following, we say that a read-once branching program [MATH] is [MATH] -reachable if for all reachable vertices [MATH] in [MATH] we have [MATH] , where |
[EQUATION] We start by quoting a result by Braverman, Rao, Raz, Yehudayoff BRRY14 Theorem 7.10 BRRY14 There is a log-space explicit PRG that [MATH] -fools all |
[MATH] -reachable ROBPs of length- [MATH] and width- [MATH] using seed length [EQUATION] Next, we reduce the task of fooling ROBPs with at most [MATH] -colliding layers to the task of fooling [MATH] -reachable ROBPs. The reduction is similar to that in CHRT17 . The main difference is that we simulate a ROBP with width ... |
Lemma 7.11 Let [MATH] Let [MATH] be a distribution on [MATH] that [MATH] -fools all [MATH] -reachable ROBPs of length [MATH] and width [MATH] Then, [MATH] also fools width- [MATH] ROBPs with at most [MATH] colliding layers with error at most [MATH] |
Proof. Let [MATH] be a distribution on [MATH] that [MATH] -fools all [MATH] -reachable ROBPs of length- [MATH] and width- [MATH] The first observation is that [MATH] also fools prefixes of these programs. This reason is simple: to simulate the prefix of length- [MATH] of a [MATH] -reachable ROBP [MATH] , one can just r... |
Let [MATH] be a length [MATH] width- [MATH] ROBP with at most [MATH] colliding layers. Next, we introduce [MATH] , a [MATH] -reachable ROBP of length- [MATH] and width- [MATH] , that would help bound the difference between |
[EQUATION] where [MATH] is the uniform distribution over [MATH] Let [MATH] be the the following modified version of [MATH] To construct [MATH] we consider a sequence of [MATH] branching programs [MATH] where [MATH] and [MATH] Let [MATH] be the colliding layers in [MATH] For [MATH] we take [MATH] to be [MATH] except we ... |
First, we claim that any reachable vertex [MATH] in [MATH] has [MATH] Let [MATH] for convenience. We apply induction and show that for [MATH] any vertex reachable by [MATH] in layers [MATH] has [MATH] The base case holds because up to layer [MATH] the branching program has no colliding layers and we may apply Claim 7.5... |
Next, we bound [MATH] by using the triangle inequality [EQUATION] and bounding each of the three terms separately. 1. The first term is bounded by the probability of reaching one of the nodes in [MATH] in [MATH] when taking a uniform random walk. This follows since if the path defined by [MATH] didn’t pass through [MAT... |
2. The second term is at most [MATH] since the program [MATH] is [MATH] -reachable. 3. Similarly to the first term, the third term is bounded by the probability of reaching one of the nodes in [MATH] in [MATH] when taking a walk sampled by [MATH] |
[EQUATION] However since [MATH] is pseudorandom for prefixes of [MATH] , for each [MATH] the probability of reaching [MATH] when walking according to [MATH] is [MATH] -close to the probability of reaching [MATH] when walking according to [MATH] |
[EQUATION] Summing the upper bound on the three terms in Eq. ( 12 ) gives: [EQUATION] Proof of Theorem 7.9 Take [MATH] and [MATH] Take the generator from Theorem 7.10 with parameters [MATH] and [MATH] Applying Lemma 7.11 , the error of this generator on the class of ROBPs with width [MATH] length [MATH] and at most [MA... |
[MATH] By Theorem 7.10 , its seed length is [EQUATION] which is at most [MATH] 7.4 Proof of Theorem We are now ready to prove our main result on fooling 3ROBPs. Our generator is obtained by applying Claim 7.1 iteratively [MATH] times and then using a PRG fooling 3ROBPs with at most [MATH] colliding layers as in Theorem... |
Let [MATH] be a 3ROBP and let [MATH] be a pseudorandom restriction as in Claim 7.1 . We first show that with probability at least [MATH] over [MATH] [MATH] is a [MATH] -ROBP for [MATH] . Let [MATH] where each [MATH] has at most [MATH] colliding layers and begins and ends with width two layers. Let [MATH] be an independ... |
To carry the induction forward as outlined above, we need the following lemma that shows that the error terms simplify as well under the pseudorandom restrictions. |
Lemma 7.12 For any constant [MATH] , there exists [MATH] such that the following holds. Let [MATH] be sufficiently large and [MATH] . Let [MATH] where [MATH] are non-zero events on disjoint variables computed by [MATH] -ROBPs. Let [MATH] be a pseudorandom restriction as in Claim 7.1 with parameter [MATH] and error para... |
Proof. First, we show that with high probability, each [MATH] has at most [MATH] colliding layers under the pseudo-random restriction. To see it, note that any colliding layer that is restricted can be either: |
Assigned to a value that reduces the width of the original program to [MATH] , and thus the width of [MATH] to 1, in which case any previous layer in [MATH] is not affecting its value. |
Assigned to a value that applies a permutation on the states of the program, thus reducing the number of colliding layers. In either case, if [MATH] colliding layers are unassigned, then [MATH] can be a computed by a 3ROBP with at most [MATH] colliding layers. By Claim 7.1 the probability that less than [MATH] collidin... |
We move to show that with high probability at least [MATH] of the functions [MATH] are non-zero. We apply the second moment method. Denote by [MATH] for [MATH] Let [MATH] be the events that [MATH] respectively, where [MATH] is the pseudo-random restriction from Claim 7.1 and [MATH] is uniformly distributed. By Claim 7.... |
[EQUATION] and by the next lemma, whose proof is deferred to Appendix A.3 , we get [EQUATION] Lemma 7.13 Let [MATH] be 3ROBPs on disjoint sets of variables of [MATH] Let [MATH] be any Boolean function. Then, [MATH] is [MATH] -fooled by the pseudorandom restriction in Claim 7.1 |
Thus, the covariance of the two events [MATH] and [MATH] is at most [MATH] . Denote by [MATH] By Claim 7.5 we have that [MATH] (since [MATH] [MATH] and [MATH] is sufficiently large). Let [MATH] Then, [MATH] and by Chebyshev’s inequality |
[EQUATION] We bound [EQUATION] which gives [MATH] using [MATH] In the complement event, at least [MATH] of the events [MATH] occur, and in particular at least [MATH] of the restricted functions [MATH] are non-zero. |
Suppose that at least [MATH] of the restricted functions [MATH] are non-zero, and that all restricted functions has at most [MATH] colliding layers. By the above analysis this happens with probability at least [MATH] . Under this assumption, we can reduce the number of functions to be exactly [MATH] , resulting in an u... |
We are now ready to prove the main theorem, Theorem Proof of Theorem Let [MATH] Let [MATH] be a constant to be chosen later. Let [MATH] . Let [MATH] be a parameter to be chosen later and let [MATH] for [MATH] |
Our generator is as follows. First choose [MATH] independent pseudo-random restrictions as in Claim 7.1 with parameter [MATH] and [MATH] . After iteratively applying the restrictions [MATH] , we set the remaining bits using the generator from Theorem 7.9 for a parameter [MATH] and error parameter [MATH] to be chosen la... |
Let [MATH] . We first claim that [MATH] is a [MATH] -ROBP with high probability. In the following let [MATH] be uniformly random over [MATH] |
Claim 7.14 With probability at least [MATH] [MATH] is a [MATH] -ROBP and [MATH] For [MATH] , let [MATH] . We will show the following claim by induction on [MATH] |
Claim 7.15 For [MATH] , with probability at least [MATH] [MATH] can be written as [MATH] where [MATH] is a [MATH] -ROBP and the error terms [MATH] for [MATH] satisfy: Either [MATH] or [MATH] with [MATH] where [MATH] are non-zero events computed by [MATH] -ROBPs on disjoint sets of variables and [MATH] |
Furthermore, [MATH] A crucial point in the above is that the functions [MATH] bounding the error terms are conjunctions of negations of [MATH] -ROBPs and there exactly [MATH] in each of them. |
Proof. For [MATH] , the claim follows immediately by applying Lemma 7.4 to [MATH] . Now, suppose the claim is true for [MATH] . Suppose, we can write [MATH] , where [MATH] as in the claim. By the induction hypothesis, this happens with probability at least [MATH] |
Clearly, [MATH] . Let [MATH] be a decomposition where each [MATH] has at most [MATH] colliding layers, starts and ends with width- [MATH] layers and [MATH] |
Now, observe that as each [MATH] has at most [MATH] colliding layers, the probability that at least [MATH] of these colliding layers are unfixed under [MATH] is at most [MATH] by Claim 7.1 . Thus, by a union bound over [MATH] , with probability at least [MATH] (for a suitable choice of [MATH] ), over [MATH] [MATH] is a... |
[EQUATION] where [MATH] , and [MATH] satisfy the conditions of the claim. We just need to argue that [MATH] can be written in the requisite form. To this end, note that for [MATH] [MATH] . By the induction hypothesis, we either have [MATH] or we can write [MATH] where [MATH] are [MATH] -ROBPs on disjoint sets of variab... |
Adding up the failure probabilities over the choice of [MATH] , we get the desired decomposition for [MATH] with probability at least |
[EQUATION] (since [MATH] ). The furthermore part follows immediately from Claim 7.1 . The claim now follows by induction. We are now ready to prove the theorem. By the above claim, we have that with probability at least [MATH] over the choice of [MATH] , we can write |
[EQUATION] where [MATH] is a [MATH] -ROBP and [MATH] can be bounded by functions [MATH] that are conjunctions of negations of [MATH] non-zero events computed by [MATH] -ROBPs. |
Note that each such [MATH] can be written as a width- [MATH] ROBP, say [MATH] , by adding an additional layer to compute the conjunction and that the number of collisions in the width [MATH] ROBP is at most [MATH] . Therefore, if we let [MATH] be the output distribution of the generator from Theorem 7.9 with [MATH] and... |
[EQUATION] where we used Claim 7.5 to bound [MATH] . Since [MATH] [MATH] Combining the above inequalities we get that with probability at least [MATH] over the choice of [MATH] |
[EQUATION] Finally, as we also have that [EQUATION] we get [EQUATION] To get [MATH] we set [MATH] . Furthermore, setting [MATH] and [MATH] , the above error bound becomes |
[EQUATION] Finally, we estimate the seed-length of our generator. Choosing the random restrictions takes [MATH] random bits. Sampling [MATH] requires seed-length |
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