1 Introduction
1.1 Background
Is there an efficient computational task that cannot be perfectly parallelized? Equivalently, is ? The answer is still unknown. The question can be rephrased as follows: is there a function inΒ that does not have a (DeΒ Morgan) formula of polynomial size?
The history of formula lower bounds for functions inΒ goes back to the 1960s, with the seminal result of SubbotovskayaΒ [Sub61] that introduced the technique of random restrictions. Subbotovskaya showed that the Parity function on variables requires formulas of size at least . KhrapchenkoΒ [Khr72], using a different proof technique, showed that in fact the Parity function on variables requires formulas of size . Later, AndreevΒ [And87] came up with a new explicit function (now known as the Andreev function) for which he was able to obtain an size lower bound. This lower bound was subsequently improved by [IN93, PZ93, HΓ₯s98, Tal14] to .
The line of work initiated by Subbotovskaya and Andreev relies on the shrinkage of formulas under random restrictions. A random restriction is a randomly chosen partial assignment to the inputs of a function. Set a parameter . We fix each variable independently with probability to a uniformly random bit, and we keep the variable alive with probability . Under such a restriction, formulas shrink (in expectation) by a factor more significant than . Subbotovskaya showed that DeΒ Morgan formulas shrink to at most times their original size, whereas subsequent works ofΒ [PZ93, IN93] improved the bound to and , respectively. Finally, HΓ₯stadΒ [HΓ₯s98] showed that the shrinkage exponent of De Morgan formulas is , or in other words, that DeΒ Morgan formulas shrink by a factor of under random restrictions. TalΒ [Tal14] improved the shrinkage factor to β obtaining a tight result, as exhibited by the Parity function.
In a nutshell, shrinkage results are useful to proving lower bounds as long as the explicit function being analyzed maintains structure under such restrictions and does not trivialize. For example, the Parity function does not become constant as long as at least one variable remains alive. Thus any formula that computes Parity must be of at least quadratic size, or else the formula under restriction, keeping each variable alive with probability , would likely become a constant function, whereas Parity would not. Andreevβs idea is similar, though he manages to construct a function such that under a random restriction keeping only of the variables, the formula size should be at least (in expectation). This ultimately gives the nearly cubic lower bound.
The KRW Conjecture.
Despite much effort, proving , and even just breaking the cubic barrier in formula lower bounds, have remained a challenge for more than two decades. An approach to solve the versus problem was suggested by Karchmer, Raz and Wigderson [KRW95]. They conjectured that when composing two Boolean functions, and , the formula size of the resulting function, , is (roughly) the product of the formula sizes of and .^{2}^{2}2More precisely, the original KRW conjecture [KRW95] concerns depth complexity rather than formula complexity. The variant of the conjecture for formula complexity, which is discussed above, was posed in [GMWW17]. We will refer to this conjecture as the βKRW conjectureβ. Under the KRW conjecture (and even under weaker variants of it), [KRW95] constructed a function in with no polynomialsize formulas. It remains a major open challenge to settle the KRW conjecture.
A few special cases of the KRW conjecture are known to be true. The conjecture holds when either or is the AND or the OR function. HΓ₯stadβs resultΒ [HΓ₯s98] and its improvementΒ [Tal14] show that the conjecture holds when the inner functionΒ is the Parity function and the outer functionΒ is any function. This gives an alternative explanation to the lower bound for the Andreev function. Indeed, the Andreev function is at least as hard as the composition of a maximallyhard function on bits and , where the formula size of is and the formula size of is . Since the KRW conjecture holds for this special case, the formula size of the Andreev function is at least . In other words, the stateoftheart formula size lower bounds for explicit functions follow from a special case of the KRW conjecture β the case in which is the Parity function. Moreover, this special case follows from the shrinkage of DeΒ Morgan formulas under random restrictions.
BottomUp versus TopDown Techniques.
Whereas random restrictions are a βbottomupβ proof technique [HJP95], a different line of work suggested a βtopdownβ approach using the language of communication complexity. The connection between formula size and communication complexity was introduced in the seminal work of Karchmer and Wigderson [KW90]. They defined for any Boolean function a twoparty communication problem : Alice gets an input such that , and Bob gets an input such that . Their goal is to identify a coordinate on which , while minimizing their communication. It turns out that there is a onetoone correspondence between any protocol tree solving and any formula computing the function . Since protocols naturally traverse the tree from root to leaf, proving lower bounds on their size or depth is done usually in a topdown fashion. This framework has proven to be very useful in proving formula lower bounds in the monotone setting (see, e.g., [KW90, GH92, RW92, KRW95, RM99, GP18, PR17]) and in studying the KRW conjecture (see, e.g., [KRW95, EIRS01, HW93, GMWW17, DM18, KM18, Mei20, dRMN20, MS20]). Moreover, a recent work by Dinur and MeirΒ [DM18] was able to reprove HΓ₯stadβs cubic lower bound using the framework of Karchmer and Wigderson. As Dinur and Meirβs proof showed that topdown techniques can replicate HΓ₯stadβs cubic lower bound, a natural question (which motivated this project) arose:
Are topdown techniques superior to bottomup techniques?
Towards that, we focused on a candidate problem: prove a cubic lower bound for an explicit function in .^{3}^{3}3Recall that is the class of functions computed by constant depth polynomial size circuits composed of AND and OR gates of unbounded fanin, with variables or their negation at the leaves. Based on the work of Dinur and Meir [DM18], we suspected that such a lower bound could be achieved using topdown techniques. We were also certain that the problem cannot be solved using the random restriction technique. Indeed, in order to prove a lower bound on a functionΒ using random restrictions, one should argue that Β remains hard under a random restriction, however, it is wellknown that functions inΒ trivialize under random restrictions [Ajt83, FSS84, Yao85, HΓ₯s86]. Based on this intuition, surely random restrictions cannot show that a function in requires cubic size. Our intuition turned out to be false.
1.2 Our results
In this work, we construct an explicit function in which requires DeΒ Morgan formulas of size . Surprisingly, our proof is conducted via the bottomup technique of random projections, which is a generalization of random restrictions (more details below).
Theorem 1.1.
There exists a family of Boolean functions for such that

can be computed by uniform depth unbounded fanin formulas of size .

The formula size of is at least .
Prior to our work, the best formula size lower bounds on an explicit function in were only quadratic [Nec66, CKK12, Juk12, BM12].
Our hard function is a variant of the Andreev function. More specifically, recall that the Andreev function is based on the composition , where is a maximallyhard function and is the Parity function. Since Parity is not in , we cannot take to be the Parity function in our construction. Instead, our hard function is obtained by replacing the Parity function with the Surjectivity function ofΒ [BM12].
As in the case of the Andreev function, we establish the hardness of our function by proving an appropriate special case of the KRW conjecture. To this end, we introduce a generalization of the complexity measure of KhrapchenkoΒ [Khr72], called the minentropy Khrapchenko bound. We prove the KRW conjecture for the special case in which the outer functionΒ is any function, and Β is a function whose formula complexity is bounded tightly by the minentropy Khrapchenko bound. We then obtain Theorem 1.1 by applying this version of the KRW conjecture to the case where Β is the Surjectivity function. We note that our KRW result also implies the known lower bounds in the cases where is the Parity functionΒ [HΓ₯s98] and the Majority functionΒ [GTN19].
Our KRW result in fact applies more generally, to functions whose formula complexity is bounded tightly by the βsoftadversary methodβ, denoted , which is a generalization of Ambainisβ unweighted adversary methodΒ [Amb02] (seeΒ Section 6.2).
Our proof of the special case of the KRW conjecture follows the methodology of HΓ₯stadΒ [HΓ₯s93], who proved the special case in which is Parity on variables. HΓ₯stad proved that DeΒ Morgan formulas shrink by a factor of (roughly) under random restrictions. Choosing shrinks a formula for by a factor of roughly , which coincides with the formula complexity of . On the other hand, on average each copy of simplifies to a single input variable, and so simplifies to . This shows that .
Our main technical contribution is a new shrinkage theorem that works in a far wider range of scenarios than just random restrictions. Given a function with softadversary bound , we construct a random projection^{4}^{4}4A projection is a mapping from the set of the variables to the set , where are formal variables. which, on the one hand, shrinks DeΒ Morgan formulas by a factor of , and on the other hand, simplifies to . We thus show that , and in particular, if , then , just as in HΓ₯stadβs proof. Our random projections are tailored specifically to the structure of the function , ensuring that simplifies to under projection. This enables us to overcome the aforementioned difficulty. In contrast, random restrictions that do not respect the structure of would likely result in a restricted function that is much simpler than and in fact would be a constant function with high probability.
Our shrinkage theorem applies more generally to two types of random projections, which we call fixing projections and hiding projections. Fixing projections are random projections in which fixing the value of a variable results in a projection which is much more probable. Hiding projections are random projections in which fixing the value of a variable hides which coordinates it appeared on. We note that our shrinkage theorem for fixing projections captures HΓ₯stadβs result for random restrictions as a special case.
The proof of our shrinkage theorem is based on HΓ₯stadβs proofΒ [HΓ₯s98], but also simplifies it. In particular, we take the simpler argument that HΓ₯stad uses for the special case of completely balanced trees, and adapt it to the general case. As such, our proof avoids a complicated case analysis, at the cost of slightly worse bounds. Using our bounds, it is nevertheless easy to obtain the lower bound for the Andreev function. Therefore, one can see the specialization of our shrinkage result to random restrictions as an exposition of HΓ₯stadβs cubic lower bound.
An example: our techniques when specialized to .
To illustrate our choice of random projections, we present its instantiation to the special case of , where is nonconstant and
for some odd integer
. In this case, the input variables to are composed of disjoint blocks, , each containing variables. We use the random projection that for each block , picks one variable in the block uniformly at random, projects this variable to the new variable , and fixes the rest of the variables in the block in a balanced way so that the number of zeros and ones in the block is equal (i.e., we have exactly zeros and ones). It is not hard to see that under this choice, simplifies to . On the other hand, we show that this choice of random projections shrinks the formula complexity by a factor of . Combining the two together, we get that . Note that in this distribution of random projections, the different coordinates are not independent of one another, and this feature allows us to maintain structure.1.3 Related work
Our technique of using tailormade random projections was inspired by the celebrated result of Rossman, Servedio, and Tan [RST15, HRST17] that proved an averagecase depth hierarchy. In fact, the idea to use tailormade random restrictions goes back to HΓ₯stadβs thesis [HΓ₯s87, ChapterΒ 6.2]. Similar to our case, in [HΓ₯s87, RST15, HRST17], random restrictions are too crude to separate depth from depth circuits. Given a circuit of depth , the main challenge is to construct a distribution of random restrictions or projections (tailored to the circuit ) that on the one hand maintains structure for , but on the other hand simplify any depth circuit .
Paper outline
The paper starts with brief preliminaries inΒ Section 2. We prove our shrinkage theorem for fixing projections in Section 3, and our shrinkage theorem for hiding projections in Section 4. In Section 5 we provide a brief interlude on concatenation of projections. Khrapchenkoβs method, the quantum adversary bound and their relation to hiding projections are discussed in Section 6. Finally, Section 7 contains a proof of Theorem 1.1, as a corollary of a more general result which is a special case of the KRWΒ conjecture. In the same section we also rederive the cubic lower bound on Andreevβs function, and the cubic lower bound on the Majoritybased variant considered inΒ [GTN19].
2 Preliminaries
Throughout the paper, we use bold letters to denote random variables. For any
, we denote by the set . Given a bitΒ , we denote its negation by . We assume familiarity with the basic definitions of communication complexity (see, e.g., [KN97]). All logarithms in this paper are baseΒ .Definition .
A (DeΒ Morgan) formula (with bounded fanin) is a binary tree, whose leaves are labeled with literals from the set , and whose internal vertices are labeled as AND () or OR () gates. The size of a formulaΒ , denoted , is the number of leaves in the tree. The depth of the formula is the depth of the tree. A formula with unbounded fanin is defined similarly, but every internal vertex in the tree can have any number of children. Unless stated explicitly otherwise, whenever we say βformulaβ we refer to a formula with bounded fanin.
Definition .
A formula computes a Boolean functionΒ in the natural way. The formula complexity of a Boolean functionΒ , denoted , is the size of the smallest formula that computesΒ . The depth complexity ofΒ , denotedΒ , is the smallest depth of a formula that computes . For convenience, we define the size and depth of the constant function to be zero.
A basic property of formula complexity is that it is subadditive:
Fact .
For every two functions it holds that and .
The following theorem shows that every small formula can be βbalancedβ to obtain a shallow formula.
Theorem 2.1 (Formula balancing, [Bb94], following [Spi71, Bre74]).
For every , the following holds: For every formulaΒ of sizeΒ , there exists an equivalent formulaΒ of depth at mostΒ and size at mostΒ .
Notation .
With a slight abuse of notation, we will often identify a formulaΒ with the function it computes. In particular, the notation denotes the formula complexity of the function computed byΒ , and not the size ofΒ (which is denoted by ).
Notation .
Given a Boolean variable , we denote by and the literals and , respectively. In other words, .
Notation .
Given a literal , we define to be the underlying variable, that is, .
Notation .
Let be a deterministic communication protocol that takes inputs from , and recall that the leaves of the protocol induce a partition of to combinatorial rectangles. For every leafΒ ofΒ , we denote by the combinatorial rectangle that is associated withΒ .
We use the framework of KarchmerβWigerson relationsΒ [KW90], which relates the complexity ofΒ to the complexity of a related communication problemΒ .
Definition ([Kw90]).
Let be a Boolean function. The KarchmerβWigderson relation ofΒ , denoted , is the following communication problem: The inputs of Alice and Bob are strings and , respectively, and their goal is to find a coordinate such that . Note that such a coordinate must exist since and hence .
Theorem 2.2 ([Kw90], see also [Raz90]).
Let . The communication complexity ofΒ is equal toΒ , and the minimal number of leaves in a protocol that solvesΒ is .
We use the following two standard inequalities.
Fact (the AMGM inequality).
For every two nonnegative real numbers it holds that .
Fact (special case of CauchySchwarz inequality).
For every Β nonnegative real numbers it holds that .
Proof.
It holds that , as required. β
3 Shrinkage theorem for fixing projections
In this section we prove our main result on the shrinkage of DeΒ Morgan formulas under fixing projections, which we define below. We start by defining projections and the relevant notation.
Definition .
Let and be Boolean variables. A projectionΒ from to is a function from the set to the set . Given such a projectionΒ and a Boolean functionΒ over the variables , we denote by the function obtained fromΒ by substituting each input variableΒ with in the natural way. Unless stated explicitly otherwise, all projections in this section are from to , and all functions fromΒ to are over the variables . A random projection is a distribution over projections.
Notation .
Let be a projection. For every and bit , we denote by the projection that is obtained from by substituting withΒ .
Notation .
With a slight abuse of notation, if a projectionΒ maps all the variables to constants inΒ , we will sometimes treat it as a binary string inΒ .
We use a new notion of random projections, which we call fixing projections. Intuitively, a fixing projection is a random projection in which for every variableΒ , the probability that maps a variableΒ to a literal is not much larger than the probability that Β fixes that literal to a constant, regardless of the values thatΒ assigns to the other variables. This property is essentially the minimal property that is required in order to carry out the argument of HΓ₯stad [HΓ₯s98]. Formally, we define fixing projections as follows.
Definition .
Let . We say that a random projection is a fixing projection if for every projectionΒ , every bit , and every variableΒ , it holds that
(1) 
For shorthand, we say that is a fixing projection, for .
If needed, one can consider without loss of generality only variables such that , as otherwise Equation 1 holds trivially with the lefthand side equaling zero.
Example .
In order to get intuition for the definition of fixing projections, let us examine how this definition applies to random restrictions. In our terms, a restriction is a projection from to that maps every variable either to itself or to . Suppose that is any distribution over restrictions, and that is some fixed restriction. In this case, the condition of being fixing can be rewritten as follows:
Denote by the restrictions obtained from by truncating (i.e., ). Using this notation, we can rewrite the foregoing equation as
Now, observe that it is always the case , and therefore the probability on the lefthand side is nonzero only if . Hence, we can restrict ourselves to the latter case, and the foregoing equation can be rewritten again as
Finally, if we divide both sides by , we obtain the following intuitive condition:
This condition informally says the following: is a fixing projection if the probability of leaving unfixed is at most times the probability of fixing it to , and this holds regardless of what the restriction assigns to the other variables.
In particular, it is now easy to see that the classic random restrictions are fixing projections. Recall that a random restriction fixes each variable independently with probability to a random bit. Due to the independence of the different variables, the foregoing condition simplifies to
and it is easy to see that this condition is satisfied for .
We prove the following shrinkage theorem for fixing projections, which is analogous to the shrinkage theorem ofΒ [HΓ₯s98] for random restrictions in the case of balanced formulas.
Theorem 3.1 (Shrinkage under fixing projections).
Let be a formula of sizeΒ and depthΒ , and let be a fixing projection. Then
Our shrinkage theorem has somewhat worse parameters compared to the theorem ofΒ [HΓ₯s98]: specifically, the factor of does not appear inΒ [HΓ₯s98]. The reason is that the proof ofΒ [HΓ₯s98] uses a fairlycomplicated caseanalysis in order to avoid losing that factor, and we chose to skip this analysis in order to obtain a simpler proof. We did not check if the factor ofΒ in our result can be avoided by using a similar caseanalysis. By applying formula balancing (Theorem 2.1) to our shrinkage theorem, we can obtain the following result, which is independent of the depth of the formula.
Corollary .
Let be a function with formula complexityΒ , and let be a fixing projection. Then
Proof.
By assumption, there exists a formulaΒ of sizeΒ that computesΒ . We balance the formulaΒ by applying Theorem 2.1 with , and obtain a new formulaΒ that computesΒ and has sizeΒ and depth . The required result now follows by applying Theorem 3.1 toΒ . β
3.1 Proof of Theorem 3.1
In this section, we prove our main shrinkage theorem, Theorem 3.1. Our proof is based on the ideas ofΒ [HΓ₯s98], but the presentation is different. Fix a formula of sizeΒ and depthΒ , and let be a fixing projection. We would like to upperbound the expectation ofΒ . As inΒ [HΓ₯s98], we start by upperbounding the probability that the projectionΒ shrinks a formula to sizeΒ . Specifically, we prove the following lemma inΒ Section 3.2.
Lemma .
Let be a Boolean function, and let be a fixing projection. Then,
Next, we show that to upperbound the expectation ofΒ , it suffices to upperbound the probability that the projectionΒ shrinks two formulas to sizeΒ simultaneously. In order to state this claim formally, we introduce some notation.
Notation .
Let be a gate ofΒ . We denote the depth ofΒ inΒ by (the root has depthΒ ), and omitΒ if it is clear from context. If is an internal node, we denote the subformulas that are rooted in its left and right children by and , respectively.
We prove the following lemma, which says that in order to upperbound it suffices to upperbound, for every internal gateΒ , the probability that and shrink to sizeΒ underΒ .
Lemma .
For every projectionΒ it holds that
We would like to use Sections 3.1 andΒ 3.1 to prove the shrinkage theorem. As a warmup, let us make the simplifying assumption that for every two functions , the events and are independent. If this was true, we could have upperboundedΒ as follows:
(Section 3.1)  
( is of depthβ)  
(simplifying assumption)  
(Section 3.1)  
(AMβGM inequality)  
The last sum counts every leafΒ ofΒ once for each internal ancestor ofΒ , so the last expression is equal to
which is the bound we wanted. However, the above calculation only works under our simplifying assumption, which is false: the events and will often be dependent. In particular, in order for the foregoing calculation to work, we need to the following inequality to hold:
This inequality holds under our simplifying assumption by Section 3.1, but may not hold in general. Nevertheless, we prove the following similar statement in Section 3.3.
Lemma .
Let be a fixing projection. Let , let , and let be a variable. Then,
Intuitively, Section 3.1 breaks the dependency between the events and by fixing inΒ the single literal to which has shrunk. We would now like to use Section 3.1 to prove the theorem. To this end, we prove an appropriate variant of Section 3.1, which allows using the projection rather than in the second function. This variant is motivated by the following βonevariable simplification rulesβ of [HΓ₯s98], which are easy to verify.
Fact (onevariable simplification rules).
Let be a function over the variables , and let . We denote by the function obtained from by settingΒ to the bitΒ . Then:

The function is equal to the function .

The function is equal to the function .
In order to use the simplification rules, we define, for every internal gateΒ ofΒ and projectionΒ , an event as follows: if is an OR gate, then is the event that there exists some literalΒ (for ) such that and . If is an AND gate, then is defined similarly, except that we replace with . We have the following lemma, which is proved in Section 3.4.
Lemma .
For every projectionΒ it holds that
We can now use the following corollary of Section 3.1 to replace our simplifying assumption.
Corollary .
For every internal gateΒ ofΒ it holds that
Proof.
Let be an internal gate ofΒ . We prove the corollary for the case where is an OR gate, and the proof for the case that is an AND gate is similar. It holds that
as required. β
The shrinkage theorem now follows using the same calculation as above, replacing Section 3.1 with Section 3.1 and the simplifying assumption with Section 3.1:
(Section 3.1)  
(Section 3.1)  
(AMβGM inequality)  
In the remainder of this section, we prove Sections 3.1, 3.1 andΒ 3.1.
Remark .
In this paper, we do not prove Section 3.1, since we do not actually need it for our proof. However, this lemma can be established using the proof of Section 3.1, with some minor changes.
3.2 Proof of Section 3.1
Let , and let be the set of projectionsΒ such that . We prove that the probability that is at most . Our proof follows closely the proof of [HΓ₯s98, Lemma 4.1].
Let be a protocol that solvesΒ and has leaves (such a protocol exists by Theorem 2.2). Let and be the sets of projectionsΒ for which is the constants andΒ , respectively. We extend the protocolΒ to take inputs fromΒ as follows: when Alice and Bob are given as inputs the projections andΒ , respectively, they construct strings from by substitutingΒ in all the variables , and invokeΒ on the inputs andΒ . Observe that andΒ are indeed legal inputs forΒ (since and ). Moreover, recall that the protocolΒ induces a partition ofΒ to combinatorial rectangles, and that we denote the rectangle of the leafΒ by (see Section 2).
Our proof strategy is the following: We associate with every projectionΒ a leaf ofΒ , denoted . We consider the two disjoint events that correspond to the event that is a single positive literal or a single negative literal, respectively, and show that for every leafΒ it holds that
(2)  
(3) 
Together, the two inequalities imply that
The desired bound on will follow by summing the latter bound over all the leavesΒ ofΒ .
We start by explaining how to associate a leaf with every projectionΒ . Let . Then, it must be the case that for some . We define the projections and , and observe that and . We now define to be the leaf to whichΒ arrives when invoked on inputs and . Observe that the output ofΒ atΒ must be a variableΒ that satisfies , and thus .
Next, fix a leafΒ . We prove that . Let be the output of the protocolΒ atΒ . Then,
Similarly, it can be proved that . Together, the two bounds imply that
for every leafΒ ofΒ . We define for projections in an analogous way, and then a similar argument shows that
It follows that
Finally, let denote the set of leaves ofΒ . It holds that
(CauchySchwarz β seeΒ Section 2)  
We conclude the proof by showing that . To this end, let
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