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It is natural to ask whether this hardness result extends to FMEI, in the sense that proving FMEI for [MATH] will imply FMEI for [MATH] . The proof fails because the min-entropy of the original function [MATH] vanishes, as [MATH] becomes the largest coefficient of [MATH] . Furthermore, FMEI is easy for functions with [... |
Lemma 20 Let [MATH] such that [MATH] . Then [MATH] Proof. [EQUATION] The third inequality makes use of the fact that [MATH] for [MATH] , and the fourth inequality is due to the fact that [MATH] |
FEI for Functions With Entropy Linear in n In the previous section, we proved FEI for functions with exponentially low influence. We have also matched this with a “hardness result”, showing that proving FEI for a class of functions with slightly higher influence will imply FEI for all Boolean functions. |
These results raise the question of the other non-trivial extremal case - proving FEI for functions with high entropy. As [MATH] , a natural interpretation of large entropy could be [MATH] for some constant [MATH] . In this section, we prove FEI for the class of functions with entropy linear in [MATH] , and show that i... |
The class of functions with linearly-high entropy. For every constant [MATH] , define [MATH] The class of functions with “almost” linearly-high entropy. For every function [MATH] such that [MATH] , define [MATH] |
Formally, we show that for any [MATH] FEI holds for the class [MATH] with constant [MATH] . We then show that improving on this result by proving FEI for any [MATH] will imply FEI for [MATH] , i.e. Conjecture |
4.1 Proving FEI for LHE We restate and prove Theorem as follows: Theorem 21 Let [MATH] . For all [MATH] [MATH] , where [MATH] is the inverse of the binary entropy function. |
Proof. We use the concentration method presented by CKLS16 , where our partition of the coefficients is of the form [MATH] . Obviously, we have: |
[EQUATION] Recall that [MATH] is the entropy of the normalized-to-1 distribution of the squared coefficients of sets in [MATH] Intuitively, we upper-bound [MATH] by the fact that there are not too many subsets of size [MATH] or less. For a constant [MATH] and [MATH] we can approximation the volume of the Hamming ball o... |
[EQUATION] Therefore, [MATH] For the second and third term of equation 4.1 , we trivially have: [MATH] and [MATH] So combining all of these, we obtain: |
[EQUATION] We start by focusing on functions [MATH] with [MATH] , and later extend our proof to all functions in [MATH] , with [MATH] . Observing that [MATH] , we obtain: |
[EQUATION] We remove the [MATH] term for simplicity, as it is negligible compared to the the other terms (to formalize this, we can replace [MATH] by [MATH] ). Now, dividing equation 4.3 by [MATH] and rearranging it: |
[EQUATION] [EQUATION] and finally, [EQUATION] Therefore, when picking [MATH] we get [EQUATION] Note that by this we picked [MATH] which is linear in [MATH] for our constant [MATH] , and therefore the Hamming ball volume approximation in equation 4.2 is valid. Continuing the computation, it follows that [MATH] |
We can lower-bound the influence by [MATH] , which is [EQUATION] All in all, we get: [EQUATION] So far we proved the inequality for functions with [MATH] . For functions with [MATH] , we have the same inequality with the bound [MATH] . This function is monotone (decreasing) for the relevant range of [MATH] , and theref... |
In DPV11 , the authors prove FEI for random functions (w.h.p.) with constant [MATH] , arguing the influence of a random function is strongly concentrated around its mean [MATH] . It is known that a random function also has (w.h.p.) entropy linear in [MATH] . It is even true that w.h.p. [MATH] for any constant [MATH] . ... |
4.2 Proving FEI for ALHE Implies FEI Completely Theorem 22 Let [MATH] such that [MATH] . Suppose that FEI holds for some class [MATH] with universal constant [MATH] , Then FEI holds for the class [MATH] with constant [MATH] |
Proof. Define the [MATH] function on two variables [MATH] . It is easy to see that [MATH] and [MATH] Also, we recall that the influence, the entropy and the number of variables tensorize nicely: for any [MATH] on [MATH] variables and [MATH] on [MATH] variables, the tensor [MATH] is a function on [MATH] variables, with ... |
Let [MATH] Consider the function [MATH] defined as [EQUATION] [MATH] is a function on [MATH] variables, and using tensorization iteratively [MATH] times we obtain [MATH] and [MATH] |
By taking [MATH] we have [MATH] and [MATH] is on less than [MATH] variables because [MATH] . So [MATH] , and therefore by our assumption [MATH] |
[EQUATION] If [MATH] we are done. If [MATH] , we make use of the fact that [MATH] , and apply the tensorization technique for the class BF to get that [MATH] |
We remark that this hardness result extends to FMEI as well. As opposed to the classes discussed in the last section (functions with low influence), in this case the reduction works for FMEI. Multiplying a function by [MATH] copies the spectral distribution 4 times, with all (squared) coefficients multiplied by [MATH] ... |
Theorem 23 Let [MATH] such that [MATH] . Suppose that FMEI holds for some class [MATH] with universal constant [MATH] , Then FMEI holds for the class [MATH] with constant [MATH] |
FEI for Functions With Constant L1 Fourier Norm As mentioned previously, it is shown in CKLS16 that the Fourier entropy of a function is bounded by the logarithm of [MATH] . Specifically they show that [MATH] . We mention two other ways in which this can be seen that provide a better constant: |
Lemma 6.9 in GSTW16 taken with [MATH] , states that for any non-negative [MATH] that sum up to [MATH] , we get [MATH] . Plugging in the distribution [MATH] , we get exactly [MATH] |
Recalling the definition of the Renyi entropy over the distribution [MATH] of the squared Fourier coefficients [MATH] , we get [MATH] , and [MATH] . By the fact that for [MATH] we get [MATH] , we obtain [MATH] |
The main caveat of these results is that they do not show FEI for the class of functions with constant [MATH] , denoted [MATH] , because the influence of functions in this class can be arbitrarily small. By making subtle changes to the proof given by CKLS16 we overcome this and prove FEI for [MATH] with universal const... |
Theorem 24 Let [MATH] be a Boolean function with [MATH] . Then [MATH] , and in particular [MATH] Proof. Let [MATH] . We divide the Fourier coefficients into three sets: |
1. [MATH] 2. [MATH] 3. [MATH] Now, we look separately at the terms of the entropy according to this partition: [EQUATION] To bound the first term of equation 5.1 , we note that [MATH] |
1. If [MATH] we can bound [MATH] 2. If [MATH] , then [MATH] , recalling that for [MATH] , we have [MATH] Accounting for the two possibilities, we get |
[MATH] To bound the second term of equation 5.1 , we note that for [MATH] [MATH] [EQUATION] To bound the third term of equation 5.1 , note that for a Boolean function it holds that [MATH] and [MATH] , hence [MATH] . Also note that for [MATH] [MATH] , and therefore [MATH] |
[EQUATION] Plugging these three bounds into equation 5.1 [EQUATION] rearranging the inequality and applying Lemma 12 we get: [EQUATION] |
in particular, we obtain: [EQUATION] Fact 25 For a Boolean function [MATH] , let [MATH] be the size of a minimal sub-cube partition of [MATH] |
[MATH] be the degree of [MATH] as a real valued polynomial, [MATH] be the granularity of [MATH] [MATH] be the size of [MATH] [MATH] be the minimal depth of a decision tree computing [MATH] , and |
[MATH] be the minimal number of leaves of a decision tree computing [MATH] . Then: [MATH] , (Lemma 4.8(i) in CKLS16 [MATH] (Proposition 3.16 in O’D14 ). This is still true when allowing parity queries at each node. |
[MATH] CKLS16 [MATH] (from Parseval’s identity) [MATH] (from Parseval’s identity) This means that functions with constant degree, for instance, are a subclass of the functions with constant L1 spectral norm, and FEI holds for them as well with the appropriate constant. |
Corollary 26 FEI holds for functions with constant sub-cube partition, constant degree, constant decision tree depth, constant decision tree size, constant granularity or constant sparsity. |
Protocol Based Approach to FEI In WWW14 , the authors suggest an insightful perspective on FEI: they note that [MATH] is true for a given function if there exists a communication protocol, that given a random subset [MATH] sampled according to the spectral distribution of [MATH] , can communicate the value of S using a... |
It is unclear how this method could be harnessed to prove FEI for less structured classes of functions than decision trees. For example, If we consider symmetric functions there is no obvious structure to the Fourier coefficients we can exploit - even if we know the size of the set we need to send, [MATH] , all coeffic... |
That being said, we are hopeful this method could be applied to more classes of functions with useful structure. Promising candidates could be circuits and formulas, and specifically DNFs. This will have the additional application of proving a version of Mansour’s conjecture, as explained in OWZ11 . A first natural ste... |
6.1 Towards FEI for Read-k DNFs As an example to the intuitive power of the protocol method, we examine a natural protocol that works for the Tribes function [MATH] , which is a read-once DNF. We denote by [MATH] the number of variables in a tribe, [MATH] the number of tribes, and [MATH] the total number of variables. ... |
Given [MATH] 1. If [MATH] , output nothing. 2. For each tribe with non-empty intersection with [MATH] , output the tribe index ( [MATH] ) and 0-1 string of length [MATH] , denoting for each variable in [MATH] whether it is in [MATH] or not. |
3. Terminate with a [MATH] We recall (see Section 4.2 in O’D14 ) that for [MATH] that has non-empty intersection with [MATH] tribes, |
[MATH] . Furthermore, it is easy to see that for [MATH] intersecting with [MATH] tribes, [MATH] The following calculation shows that the expected protocol length is indeed [MATH] , by summing over the sets [MATH] according to the number of tribes they intersect: |
[EQUATION] It is well known that [MATH] (see Proposition 4.13 in O’D14 ), so indeed [MATH] We note (without proof) that this protocol can be extended to general read-once DNFs, which FEI is already known for, by assigning variable-length encodings to each of the tribes, based on their size. In fact, this is the protoco... |
Definition Let [MATH] . We say [MATH] is a [MATH] -regular DNF (or just “regular”), if there exists some [MATH] s.t. [MATH] s.t. the number of variables in each clause respects [MATH] , and the number of clauses is [MATH] |
For example, the (essentially unbiased) Tribes function [MATH] is a regular read-once DNF with [MATH] [MATH] 6.1.1 A Suggested Protocol for Read-k DNFs |
Recall the strategy of WWW14 for decision trees: for read-once decision tree, they note that every [MATH] with non-zero weight has exactly one path from the root that contains its variables (and maybe additional variables), and encode the path efficiently. For read-k decision tree, they note that every [MATH] with non-... |
Analogously, we can view our protocol for Tribes as sending a “set cover” of tribes, in the sense that [MATH] is a subset of their union. For a read-once DNF, [MATH] has exactly one cover, and from the Fourier coefficients of [MATH] , it is obvious that sets [MATH] with large Fourier coefficients have small covers that... |
Given [MATH] 1. If [MATH] , output nothing. 2. Let [MATH] be the smallest cover of [MATH] . For each tribe [MATH] output the tribe index and a 0-1 string of length [MATH] , denoting for each variable in [MATH] whether it is in [MATH] or not. |
3. Terminate with a [MATH] To specify this protocol completely, we still need to define the size of the cover in order for the term “smallest” cover to be meaningful. Natural options can be the number of tribes involved - [MATH] , the combined sizes of the tribes [MATH] , or the number of unique variables involved, [MA... |
Conjecture 3 The expected price of the above protocol for a read-k DNF [MATH] is [MATH] 6.1.2 FMEI for Regular Read-k DNFs As a modest first step, we prove FMEI for the class of regular read-k DNFs. When considering min-entropy, only the weight of the largest Fourier coefficient matters, and as can be derived from the ... |
Lemma 27 For any read-k DNF [MATH] there is a family [MATH] with [MATH] such that [MATH] The family of sets the lemma refers to is [MATH] where [MATH] is the set of variable in clause [MATH] and [MATH] is the power set of [MATH] . In words, these are sets that can be covered by at most one tribe that is not too large (... |
Lemma 28 Let [MATH] be a regular read-k DNF with width [MATH] . Then [MATH] Proof. By Lemma , we know there is a set [MATH] of size less than [MATH] s.t. [MATH] . Hence, |
[EQUATION] and it follows that [EQUATION] [EQUATION] In the last step (and only there) we use the fact that [MATH] is a regular DNF, and therefore [MATH] |
Lemma 29 Let [MATH] be a regular read-k DNF with width [MATH] . Then [MATH] Proof. We show that for a regular read-k DNF the influence of any single variable must be small, and then by using KKL we lower bound the total influence. For any variable [MATH] , to be influential for some input, it must change the value of a... |
Combining Lemmas 28 and 29 , and viewing [MATH] as constant, we obtain FMEI for regular read-k DNFs. Corollary 30 Let [MATH] be a regular read-k DNF, then [MATH] |
It is very probable that with some finer arguments, this can be generalized to read-k DNF for constant [MATH] 6.2 On the Covariance of Read-k Decision Trees |
In WWW14 , the authors define the following covariance measure on a decision tree [MATH] computing [MATH] [EQUATION] where [MATH] and [MATH] are the left and right sub-trees of [MATH] , with the corresponding functions [MATH] on the variables [MATH] , assuming w.l.o.g that at the root the variable that is queried is [M... |
Theorem 31 Let [MATH] be computed by a read-k decision tree. Then [MATH] Our proof will follow the lines of the original proof of WWW14 , which is a structural induction. To show that [MATH] , something stronger is actually shown: |
[EQUATION] where [MATH] , and [MATH] is the number of appearances of [MATH] in [MATH] . It is obvious that [MATH] , and their theorem follows. |
To show the [MATH] , it would make sense to refine [MATH] into a more suitable measure. For example, to define: [EQUATION] A technical note - this is not defined if any variable in [MATH] has [MATH] , but that means it doesn’t appear in [MATH] and therefore [MATH] , so this will not be a problem. As [MATH] , it would b... |
[EQUATION] Sadly, the proof did not follow through with [MATH] , so we had to compromise and use the following definition: [EQUATION] |
The 2 factor is needed for technical reasons. Our proof is by structural induction on [MATH] that [EQUATION] and then it will follow that [MATH] |
Proof. Base case: [MATH] is a tree with one variable, and the left and right sub-trees are constant. Therefore the left-hand side of Inequality 6.4 is [MATH] and the right-hand side is always non-negative. |
Inductive step: Suppose w.l.o.g the root variable of [MATH] is [MATH] . We use the recursive definition of [MATH] given by Equation 6.1 |
[EQUATION] We focus on the first term. Let [MATH] be the set of coordinates which appear in both sub-trees [MATH] . Because [MATH] is the root variable, it doesn’t appear in either [MATH] , so [MATH] is a subset of [MATH] |
[EQUATION] This is exactly the same bound as in WWW14 , but we stop before their last step (they bounded [MATH] by [MATH] ), which is potentially wasteful. |
To bound the second term, we apply the inductive hypothesis: [EQUATION] For [MATH] , we do not get anything added from the first term, so it is enough to notice that [MATH] and we get |
[EQUATION] For [MATH] , we need to add the [MATH] term. So we would like to show that [EQUATION] We pause our proof for a short intuitive discussion regarding the last inequality. If we could show that [MATH] for both [MATH] that would be enough, by upper bounding [MATH] as done by WWW14 . Note that if the appearances ... |
The challenging case is when we have all variables in [MATH] appear almost exclusively in one sub-tree. The most extreme instance of this case is when we have [MATH] appearances of every variable from [MATH] in [MATH] , only one appearance for each variable in [MATH] , and [MATH] appearances in [MATH] . Note that even ... |
Assuming for a moment Inequality 6.3 , we finish the proof exactly as done in WWW14 [EQUATION] where the last step is due to the fact [MATH] , and the one before is due to Proposition 2.3 of WWW14 stating that for a decision tree of a function [MATH] with root [MATH] and sub-trees computing [MATH] the following holds: ... |
So all that is left to prove is Inequality 6.3 for [MATH] . By simple calculation, [MATH] , and in general [MATH] , and for [MATH] , then obviously [MATH] |
If [MATH] has a variable [MATH] with [MATH] , then [MATH] , and hence by definition of [MATH] , we get [MATH] . If [MATH] has such variable too, we can use the bound [MATH] and we are done. |
We are left with the case where (w.l.o.g) all variables in [MATH] “tend to [MATH] ”, but also appear at least once in [MATH] . Formally, for any [MATH] |
[MATH] [MATH] We write Inequality 6.3 (which we need to prove) a bit differently: [EQUATION] [MATH] The first inequality is because in this case, [MATH] . The second inequality is correct if all [MATH] are larger than 16 - this is merely a technical detail, that can be fixed for smaller [MATH] , for example, by definin... |
[MATH] The first inequality comes from the fact [MATH] Now we split the weight of [MATH] , but not necessarily to [MATH] . For any non-zero [MATH] , we have [MATH] . This is due to [MATH] . We want to pick [MATH] such that |
[EQUATION] As we cannot bound the ratio of [MATH] , we satisfy the two separate inequalities: [EQUATION] Which is equivalent to: |
[EQUATION] Picking [MATH] , the last inequality is held due to the “arithmetic mean is larger than harmonic mean” theorem. This covers all the cases of the induction step, therefore we have |
[EQUATION] and then [MATH] , concluding the proof. Improved Bound on the L1 Norm of Decision Trees It is known that for any Boolean function [MATH] that is computed by a decision tree [MATH] [MATH] (see Proposition 3.16 in O’D14 ), where [MATH] is the number of leaves of the tree. We provide the following stronger boun... |
Proposition 32 For a Boolean function [MATH] that is computed by a decision tree [MATH] [EQUATION] where [MATH] is the number of nodes that have at least one child that is a leaf. The sum of [MATH] is over all inner nodes of T, i.e. nodes that have two non-leaf children. |
This is stronger than the original bound in two senses: the first, [MATH] , and can be as small as [MATH] for a “full” binary tree. The second is that we subtract a non-negative term that can be significant. Examine the standard and the new bounds for the parity function on [MATH] variables, [MATH] . Obviously, [MATH] ... |
Proof. Let us examine a function [MATH] computed by a tree with root [MATH] , with left function [MATH] and right function [MATH] . We can write |
[MATH] . It is easy to see that for any [MATH] [MATH] , and [MATH] We also note that for any two numbers, [MATH] So we get: [EQUATION] |
Summing over all [MATH] , we get: [EQUATION] In other words, [EQUATION] We can bound the last term as follows: [EQUATION] So in conclusion, we get: |
[EQUATION] We can now prove [MATH] using structural induction on [MATH] The base case is where we have a function with a root and two leaves. [MATH] [MATH] , and there are no inner nodes so the claim holds. |
The semi induction step is where we have a function with a root, one leaf child and one non-leaf sub-tree [MATH] computing a function [MATH] . In this case, [MATH] [MATH] , and [MATH] , as the inner nodes in [MATH] and [MATH] are the same. Using the inductive hypothesis on [MATH] is enough to finish this case. |
The induction step is where we have a function with a root and two non-leaf sub-trees [MATH] computing [MATH] . We use the inductive hypothesis for the two sub-trees and inequality 7.1 |
[EQUATION] These three are the only possible cases so we are done. Acknowledgements The author wishes to thank Amir Shpilka for advising him throughout this research, and also Dor Minzer and Ben Lee Volk for fruitful discussions and suggestions. |
# Source: arxiv 1806.03703 # Title: Towards Completely Characterizing the Complexity of Boolean Nets Synthesis # Sections: all # Downloaded: 2026-03-03T01:45:09.053921+00:00 |
\Copyright Ronny Tredup and Christian Rosenke Towards Completely Characterizing the Complexity of Boolean Nets Synthesis Abstract. |
Boolean Petri nets, which tolerate at most one token per place, are widely regarded as a fundamental model for concurrent systems. They are differentiated into types of nets by the variety of applications that define their individual interaction set between places and transitions. Taking a step back, one observes that ... |
Elementary net systems nop inp out ), for instance, have a connection to prime event structures and applications in workflow management systems like milano |
Contextual nets nop inp out used free ), as a second example, implement reading without consuming common to database systems, concurrent constraint programming, and shared memory systems |
Other known classes are event/condition nets nop inp out used inhibitor nets nop inp out free set nets nop inp set used trace nets nop inp out set res used free |
, and flip flop nets nop inp out swap This paper is devoted to a computational complexity analysis of the boolean net synthesis problem subject to a target class [MATH] The challenge is to translate given finite automata [MATH] , called transition systems (TSs, for short), into boolean [MATH] -nets having a state trans... |
or even considerably restricted TSs can be synthesized, and for flip flop nets, which can be synthesized in polynomial time Our main result is a generic reduction scheme for NP-hardness proofs of boolean net synthesis that works for 77 different net classes allowing nop We significantly generalize preliminary methods u... |
for the hardness proof of synthesizing elementary net systems from heavily restricted TSs. Unlike these premature approaches, the present solution covers all supersets of ( nop inp out ) that exclude swap , all supersets of either ( nop inp set ) or ( nop out res ), and extensions of ( nop swap ) by at least one intera... |
We also identify seven classes with hard synthesis where the reduction does not fit into the general scheme. This comprises ( nop set res ) extended with at least one of used and free as well as ( nop inp free , [ used ]) and ( nop out used , [ free ]), where [ [MATH] ] marks optional interactions. |
Aside from this, we find 36 tractable cases for boolean net synthesis. Firstly, like flip flop nets, the 16 extensions of ( nop swap ) with a subset of ( inp out used free ) can be synthesized in polynomial time by a version of Schmitt’s approach |
Secondly, we presents a new polynomial time synthesis algorithm that works for the 16 classes combining ( nop set ) with a subset of ( out used free ) and combining ( nop res ) with a subset of ( inp used free ). Four rather simple cases of polynomial synthesis are found in extending ( nop ) with subsets of ( used free... |
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