On the Constant-Depth Complexity of k-Clique
∗ Benjamin Rossman Massachusetts Institute of Technology 32 Vassar St., Cambridge, MA 02139 [email protected]
ABSTRACT General Terms We prove a lower bound of ω(nk/4) on the size of constant- Theory depth circuits solving the k-clique problem on n-vertex graphs (for every constant k). This improves a lower bound 2 Keywords of ω(nk/89d ) due to Beame where d is the circuit depth. k-clique, constant-depth circuits, circuit complexity, AC0, Our lower bound has the advantage that it does not depend first-order logic, bounded variable hierarchy on the constant d in the exponent of n, thus breaking the mold of the traditional size-depth tradeoff. Our k-clique lower bound derives from a stronger result of 1. INTRODUCTION (n) Constant-depth circuits have been one of the most fruit- independent interest. Suppose fn : {0, 1} 2 −→ {0, 1} is a sequence of functions computed by constant-depth circuits ful settings for complexity lower bounds. (Here we refer of size O(nt). Let G be an Erd˝os-R´enyi random graph with to logical circuits comprised of ¬, ∧ and ∨ gates with un- vertex set {1, . . . , n} and independent edge probabilities n−α bounded fan-in.) Early results of Ajtai [1] and Furst, Saxe 1 and Sipser [12] proved that the parity problem (given an n- where α ≤ 2t−1 . Let A be a uniform random k-element subset of {1, . . . , n} (where k is any constant independent of bit string, are there an even number of 1’s?) does not have polynomial-size constant-depth circuits. This result, often n) and let KA denote the clique supported on A. We prove stated as PARITY ∈/ AC0, was subsequently sharpened by that fn(G) = fn(G ∪ KA) asymptotically almost surely. These results resolve a long-standing open question in Yao [28] and H˚astad [15] among others, who eventually es- finite model theory (going back at least to Immerman in tablished that exponential-size constant-depth circuits are 1982). The m-variable fragment of first-order logic, denoted required. by FOm, consists of the first-order sentences which involve In this paper we consider the k-clique problem for con- stant values of k. The k-clique problem asks, given a graph at most m variables. Our results imply that the bounded n 1 2 m G on n vertices (specified by a bitstring of length ), does variable hierarchy FO ⊂ FO ⊂ · · · ⊂ FO ⊂ · · · is strict 2 in terms of expressive power on finite ordered graphs. It G contain a k-clique? Unlike the parity problem, the k- was previously unknown that FO3 is less expressive than clique problem has polynomial-size constant-depth circuits (for every constant k). In particular, there is an obvious full first-order logic on finite ordered graphs. n depth-2 circuit consisting of k ∧-gates at the bottom level (deciding each potential k-clique) feeding into a single ∨-gate on top. A natural question is whether there exist signifi- Categories and Subject Descriptors cantly smaller constant-depth circuits for the k-clique prob- o(k) F.2.2 [Theory of Computation]: Nonnumerical Algo- lem (say, of size n ). 0 rithms and Problems—Computations on discrete structures; Lower bounds on the complexity of problems within AC F.4.1 [Mathematical Logic and Formal Languages]: traditionally involve size-depth tradeoffs. Lynch [23] in 1986 Mathematical Logic—Model theory established the first size-depth tradeoff for the k-clique prob- √ 1.5 lem by proving a lower bound of nΩ( k/d ) on the size of depth-d circuits solving the k-clique problem on n-vertex ∗ 2 Supported by a National Defense Science and Engineering graphs. This was improved to ω(nk/89d ) by Beame [7] in Graduate Fellowship. 1990 (in the context of CRCW PRAM’s). These results were based on developments in the switching lemma technology originally introduced by H˚astad [15] for studying the par- ity problem. For his lower bound, Beame proved powerful switching lemmas in the setting of random graphs (see [8]). Permission to make digital or hard copies of all or part of this work for Size-depth tradeoffs of this sort, however, are unsatisfac- personal or classroom use is granted without fee provided that copies are tory inasmuch as they degrade in the exponent of n as d is not made or distributed for profit or commercial advantage and that copies taken to be a larger constant. (This seems to be an unavoid- bear this notice and the full citation on the first page. To copy otherwise, to able consequence of using switching lemmas in the usual republish, to post on servers or to redistribute to lists, requires prior specific √ way.) For depth d = k, the lower bounds of Lynch and permission and/or a fee. n Copyright 200X ACM X-XXXXX-XX-X/XX/XX ...$5.00. Beame no longer beat the trivial lower bound of 2 (the 1 −α number of input bits). These lower bounds thus do not rule 0 < α ≤ 2t−1 , let G = ER(n, n ) be an Erd˝os-R´enyi ran- out the possibility that the 100-clique problem has depth-10 dom graph and let A be a uniform random set of k vertices of 2 circuits of size O(n ). G. Then fn(G) = fn(G ∪ KA) asymptotically almost surely. In this paper we create a more satisfactory state of affairs by proving a lower bound of ω(nk/4) on the size of depth-d This result directly implies our k-clique lower bound. circuits for the k-clique problem for all constants k and d. (In fact, our lower√ bound holds even for slightly increasing Theorem 1.2. For every constant k, the k-clique problem d = d(n) = o( log n) and perhaps moderately increasing on n-vertex graphs requires constant-depth circuits of size k = k(n) as well.) By contrast, our lower bound implies ω(nk/4). that 100-clique does not have size O(n25) circuits of any constant depth. Thus it may now be said that, in some In fact, Theorem 1.1 implies the even stronger assertion reasonable sense, no constant-depth circuits for the k-clique that for all ` > k, no constant-depth circuits of size O(nk/4) problem significantly beat the naive circuits of depth 2 and distinguish between the class of graphs which contain an size O(nk). `-clique and the class of graphs which contain no k-clique. Unlike traditional size-depth tradeoffs, our approach does not involve developing sharper switching lemmas for the 1.2 Corollaries in Logic problem at hand. In fact, we require nothing stronger than Our k-clique lower bound has some nice corollaries in logic H˚astad’s original switching lemma. Rather, our technique (finite model theory [18, 22]), which answer long-standing breaks from past approaches in the innovative way that the open questions, via the well-known descriptive complexity switching lemma is used. A key technical notion we intro- characterization of first-order logic in terms of the circuit f,G 0 duce is the s-bounded clique-sensitive core Thsi (A) of a set class AC [6, 11, 21]. A of vertices in a graph G with respect to a graph-function f The m-variable fragment of first-order logic, denoted by f,G m (see §3.1). Thsi (A) is a subset of A with certain nice prop- FO , consists of the first-order sentences which involve at erties. In particular, if we add to G the clique supported most m variables. (Sentences of FOm may contain more f,G than m quantifiers, as variables may be reused.) For exam- on Thsi (A), then the value of f on the resulting graph is no longer sensitive to the further addition of any subclique ple, the following sentence (in the language of simple graphs of A up to size s. This new technical notion (which puts a with a symmetric binary relation ∼ denoting adjacency) ex- twist on the familiar concept of sensitive inputs) allows for presses “diameter is at most 4” using only 3 variables: a novel inductive argument on circuits (Lemma 3.6). Ulti- ∀x∀y∃z ∃y (x ∼ y) ∧ (y ∼ z) ∧ ∃x (z ∼ x) ∧ (x ∼ y) . mately, our approach leads to an effective new way of using the switching lemma in conjunction with a union bound over More generally, 3 variables suffice to express “diameter is at the nodes in a circuit. most d” for every constant d. Bounded variable fragments m m We remark that the complexity of the k-clique problem FO (and their infinitary counterparts L∞,ω) are important has been studied in various models of computation besides objects of study in both finite and classical model theory constant-depth circuits. In the setting of monotone circuits, (see [14, 18]). The chain of m-variable fragments FO1 ⊂ it is known that the k-clique problem (for specific increasing FO2 ⊂ · · · ⊂ FOm ⊂ · · · is known as the bounded (or finite) k = k(n)) requires exponential size [3, 13, 26]. A super- variable hierarchy. On finite graphs without a linear order, polynomial lower bound was even proved for circuits with a this hierarchy is strict in terms of expressive power for the bounded number of negation gates [5]. Lower bounds have simplest of reasons: the sentence “there exist at least m also been investigated in the context of branching programs vertices” is expressible in FOm but not in FOm−1. and decision trees [27]. Here, however, we are concerned with first-order logic on Of course, the greatest hope is eventually for a lower finite ordered graphs (where sentences may speak of order < bound of nΩ(k) on the size of circuits of arbitrary depth as well as adjacency ∼ among elements). Exploiting order, for the k-clique problem. Such a result would imply NP * we are now able to express “there exist at least m vertices” P/poly and hence P 6= NP. for every m using only 3 variables (similar to the diameter example above). Perhaps, then, every first-order sentence is Organization of the Paper. equivalent on finite ordered graphs to a sentence with only 3 In the rest of this section, we state our main results and variables? It far from obvious how one might express “there discuss some interesting corollaries in logic. §2 fixes some exists a 4-clique” using only 3 variables. On the other hand, notation and covers the preliminaries on graphs, random it is unclear how to prove that 3 variable do not suffice for the graphs and circuits. In §3 we present the technique behind task. (Ehrenfeucht-Fra¨ıss´egames, the standard method for our lower bound and prove weaker versions of our main theo- arguing inexpressibility in fragments of first-order logic, are rems (modulo some technical lemmas). In §4 we prove those greatly complicated by the presence of a linear order.) In- technical lemmas (concerning the sensitivity of randomly re- deed, it was (until now) an open problem whether, in terms stricted AC0-computable functions). In §5 we prove our of expressive power on finite ordered graphs, the bounded main theorems using the technique developed in §3. We variable hierarchy is strict, or whether perhaps it collapses state some conclusions and raise some open questions in §6. down to FO3. (It is long known that FO2 is weaker than FO3.) This intriguing question was raised as early as 1982 1.1 Main Results by Immerman [16]. A survey article by Dawar [10] devoted to this very question was published in 2005. (Concerning (n) Theorem 1.1. Suppose fn : {0, 1} 2 −→ {0, 1} is a se- some variations on this question: Poizat [25] showed that quence of functions computed by constant-depth circuits of 3 variables suffice to express all first-order properties of fi- size O(nt) where t > 1/2. For any constants k ∈ N and nite colored linear orders, while Dawar [10] showed that the hierarchy of m-variable existential sentences is expressively |EG| = ∅. For example, if Kk is the complete graph on 2 strict on finite ordered graphs.) k vertices, then thres(Kk) = k−1 . Our k-clique lower bound finally resolves the status of the The class of graphs with vertex set [n] is denoted by Gn. bounded variable hierarchy on finite ordered graphs. The (n) We identify Gn with the set {0, 1} 2 , particularly when bridge between constant-depth circuits and bounded vari- viewing graphs as inputs to circuits. For a set A ⊆ [n], able fragments of first-order logic is the following well-known we denote by KA the clique supported on A(i.e., the com- result from descriptive complexity theory (discovered by var- A plete graph with VK = A and EK = ). For graphs ious authors [6, 11] and recently refined [21]). A A 2 G ∈ Gn, we will often consider the union graph G ∪ KA (i.e., G plus a clique on A). Proposition 1.3. For every m-variable first-order sen- tence Φ in the language of ordered graphs, there exist m n Random Graphs. constant-depth circuits Cn, n ∈ N, of size O(n ) on 2 A random graph is a probability distribution on Gn (or a inputs such that Cn evaluates Φ on ordered graphs of size n. sequence of such distributions for each n). For q ∈ [0, 1], Theorem 1.2 and Proposition 1.3 directly imply: we denote by ER(n, q) the Erd˝os-R´enyi random graph with vertex set [n] in which each potential edge in [n] is indepen- bk/4c 2 Corollary 1.4. No sentence in FO expresses the dently included with probability q. We write G ∈ ER(n, q) existence of a k-clique on finite ordered graphs. to express that G is a random graph with distribution ER(n, q). We are mainly interested in Erd˝os-R´enyi random (In fact, Corollary 1.4 holds not just for the class of finite graphs with edge probability q = n−α for constant values of ordered graphs, but for classes of finite graphs with arbi- α > 0 (see [4] for background). trary numerical predicates; see the full paper.) Since one m Given two graphs G and H, the number of induced sub- can express the existenceV of an m-clique in FO , namely by graphs of G isomorphic to H is denoted by sub(G, H). For a the sentence ∃x1 ... ∃xm i
Corollary 1.7. The bounded variable hierarchy is strict Circuits. n in terms of expressive power on finite ordered graphs. A circuit on n inputs (resp. 2 inputs) is an acyclic di- rected graph C in which sources, called input nodes, are la- beled by elements of [n] (resp. [n] ) and non-source nodes, 2. PRELIMINARIES 2 called gates, are labeled by elements of {¬, ∧, ∨}. Gates la- beled ¬ are required to have fan-in (in-degree) 1, while ∧ Basic Notation. and ∨ gates have unrestricted fan-in. A subset of nodes in C For a positive integer n, let [n] denote the set {1, . . . , n}. (by default: the sinks in C) are designated as output nodes. Letters A, B, C, D are reserved for subsets of [n]. For a The size and depth of C are respectively the number of gates set X and k ∈ N, let X = {Y ⊆ X : |Y | = k} and S k it contains and the maximum number of gates on a path X k X ≤k = j=0 j . The abbreviation “a.a.s.” stands for “as- from an input node to an output node. The value of a cir- ymptotically almost surely (as n → ∞)”. Logarithms have cuit C on an input string x ∈ {0, 1}n (resp. an input graph base 2 unless otherwise indicated. G ∈ Gn) is denoted by C(x) (resp. C(G)). If C has m output nodes, then C(x) is an element of {0, 1}m. Thus, a circuit n Graphs. on n inputs (resp. 2 inputs) and m outputs computes a n m Graphs are by default finite simple graphs. Formally, a function from {0, 1} (resp. Gn) to {0, 1} . graph G is a pair (VG,E G) where VG is a nonempty finite set VG 0 and EG is a subset of 2 . The quantity minH⊆G |VH |/|EH | AC . (where H ranges over induced subgraphs of G) is denoted by (Non-uniform) AC0 is the class of languages L ⊆ {0, 1}∗ thres(G) and called the threshold exponent of G (Lemma 2.1 decided by a sequence of circuits (one for each input size n) examples why); we set thres(G) = ∞ in the event that of constant depth and size polynomial in n. 3. OUR TECHNIQUE The next three lemmas give some additional properties of f,G f,G In this section we prove weaker versions of our main theo- operators T and Thsi . rems (with slightly inferior parameters) modulo some techni- f,G cal lemmas on random restrictions (which we tackle later on Lemma 3.3. Let T = Thsi (A) and suppose B is a set such in §4). This section illustrates our lower-bound technique in that T ⊆ B ⊆ A and |B| ≤ s. Then f(G∪KT ) = f(G∪KB ). a somewhat simplified form. We eventually prove our main theorems in §5 using the technique introduced here. Proof. Let b1, . . . , bm enumerate the set B \ T . For all i ∈ {1, . . . , m}, we have 3.1 s-Bounded Clique Sensitivity f(G ∪ KT ∪{b1,...,bi}) = f(G ∪ KT ∪{b1,...,bi−1}). We define a technical notion: the clique-sensitive core of f,G a set of vertices in a given graph with respect to a given (Otherwise, we would have bi ∈ T (B) ⊆ T .) It follows graph-function. This is closely related to the familiar notion immediately that f(G ∪ KT ) = f(G ∪ KB ). of sensitive vertices of a graph-function (as we will see in f,G §4.1). We also introduce the a more novel notion: the s- It is of course possible that |Thsi (A)| > s, in which case bounded clique-sensitive core (for a parameter s ∈ N). This the statement of Lemma 3.3 is vacuous. notion plays a central role in our technique. f,G Lemma 3.4. The set Thsi (A) is the union of all fully Definition 3.1. Let f be a graph-function with domain clique-sensitive subsets of A of size at most s. That is, S Gn (and arbitrary range) and let G ∈ Gn. For all A ⊆ [n] f,G f,G f,G f,G Thsi (A) = B ⊆ A : T (B) = B and |B| ≤ s . and s ∈ N, sets T (A) and Thsi (A) are defined by ( ) Proof. The inclusion ⊇ is fairly obvious. To prove the f,G f,G there exists B ⊆ A such that opposite inclusion, we must find, for each a ∈ T (A), a T (A) = a ∈ A : , hsi f(G ∪ KB ) 6= f(G ∪ KB\{a}) fully-clique sensitive subset B ⊆ A of size ≤ s which contains [ f,G f,G f,G a. Because a ∈ Thsi (A) we know there exists some subset Thsi (A) = T (B). B ⊆ A such that f(G ∪ KB ) 6= f(G ∪ KB\{a}) and |B| ≤ s. B⊆A : |B|≤s Take any minimal B satisfying these conditions. To show The set Tf,G(A) (resp. Tf,G(A)) is called the (resp. s- that B is fully clique-sensitive, we assume that there exists hsi b ∈ B \ Tf,G(B) and derive a contradiction. First, we note bounded) clique-sensitive core of A under f in G. We say that f(G ∪ K ) = f(G ∪ K ) by the minimality of that A is fully clique-sensitive under f in G if Tf,G(A) = A. B\{b} B\{a,b} our choice of B. However, the fact that b∈ / Tf,G(B) implies Intuitively, Tf,G(A) (resp. Tf,G(A)) is the set of vertices hsi f(G ∪ KB ) = f(G ∪ KB\{b}), a ∈ A such that the graph-function f, when evaluated on f(G ∪ K ) = f(G ∪ K ). G plus different subcliques of A (resp. of size ≤ s), is sensi- B\{a} B\{a,b} tive to the inclusion of a in the added subclique. A simple But these three equalities imply f(G∪KB ) = f(G∪KB\{a}), example serves to illustrate this definition. contradicting our choice of B. Therefore, B is fully clique- sensitive. After noting that B contains a (since otherwise we Example 3.2. Let cliquek : Gn −→ {0, 1} be the function again reach the contradiction f(G ∪ KB ) = f(G ∪ KB\{a})), defined by cliquek(G) = 1 ⇐⇒ G contains a k-clique. Let we are done. Gemp denote the empty graph in Gn (with no edges). For all A ⊆ [n] and s ∈ N, we have Lemma 3.5. ( f,G A if |A| ≥ k, 1. T (A) is nonempty if, and only if, A has a fully Tcliquek,Gemp (A) = hsi ∅ otherwise, clique-sensitive subset B of size 2 ≤ |B| ≤ s. ( f,G A if |A| ≥ k and s ≥ k, 2. |T (A)| > s if, and only if, A has fully clique- Tcliquek,Gemp (A) = hsi hsi ∅ otherwise. sensitive subsets B and C of size ≤ s such that |B ∪ C| ≥ s + 1. f,G f,G f,G Some elementary properties of operators T and Thsi 3. If |Thsi (A)| > s, then A has a fully clique-sensitive are stated below. subset D of size s + 1 ≤ |D| ≤ 2s. f,G f,G (i) Thsi (A) ⊆ T (A) ⊆ A for all sets A. Proof. Implications (⇐=) in both (1) and (2) are easy. We prove the implications (=⇒) in statements (1)–(3). Let f,G f,G (ii) T and Thsi are monotone. That is, A ⊆ B implies A1,...,Am, m ∈ N, enumerate the nonempty fully clique- f,G f,G f,G f,G sensitive subsets of A of size ≤ s. By Lemma 3.4, Tf,G(A) = T (A) ⊆ T (B) and Thsi (A) ⊆ Thsi (B). hsi A1 ∪ · · · ∪ Am. m f,G (iii) If f : Gn −→ {0, 1} where f1, . . . , fm : Gn −→ Suppose that Thsi (A) is nonempty, that is, m ≥ 1. Let {0, 1} are the individual coordinate-functions of f, S B = A1. We claim that |B| ≥ 2. Toward a contradiction, f,G m fi,G f,G then T (A) = i=1 T (A) and Thsi (A) = assume |B| < 2. Then (since B 6= ∅) B = {b} and hence S f,G f,G m fi,G T ({b}) = {b} for some b ∈ A. By definition of T , i=1 Thsi (A). there exists C ⊆ {b} such that f(G ∪ KC ) 6= f(G ∪ KC\{b}). (iv) If sets A and B are both fully clique-sensitive under f Clearly it must be that C = {b}. But then f(G ∪ K{b}) 6= in G, then so is their union A ∪ B. f(G∪K∅), which is absurd since K{b} (the clique supported • on the single vertex b) has no edges and hence G ∪ K{b} = claim to νout. Thus C(G ∪ KT (νout)) = C(G ∪ KA). But • G ∪ K∅ = G. We conclude that |B| ≥ 2, proving implication C(G ∪ KT (νout)) = C(G ∪ KT (νout)) by Lemma 3.3 (since • • (=⇒) of statement (1). T (νout) ⊆ T (νout) ⊆ A and |T (νout)| ≤ s) and T (νout) = f,G C,G Now suppose that |Thsi (A)| > s. Let ` ∈ {1, . . . , m} be Thsi (A) = ∅ by assumption. Hence G∪KT (νout) = G∪K∅ = the least index such that |A1 ∪ · · · ∪ A`| > s, noting that ` is G. We conclude that C(G) = C(G ∪ KA). Lemma well-defined since |A1 ∪· · ·∪Am| > s. Let B = A1 ∪· · ·∪A`−1 and C = A`. Clearly |B|, |C| ≤ s and |B ∪ C| ≥ s + 1. 3.3 Proofs of (Almost) Main Theorems The set B is fully clique-sensitive (by observation (iv), as it In this section we prove versions of our main results, The- is a union of fully clique-sensitive sets A1,...,A`−1), as is orems 1.1 and 1.2, with slightly inferior parameters. Theo- the set C. This proves implication (=⇒) of statement (2). rems 1.1 and 1.2 (stated in the introduction) will be proved Statement (3) follows by taking D = B ∪ C. in §5 using an argument similar to the one present here. The advantage of first proving weaker results is a moderate gain 3.2 Circuit Lemma in the simplicity of the argument (making things easier the n second time around). Let C be a single-output circuit with 2 inputs. For all nodes ν in C, we denote by Cν the single-output subcircuit • Lemma 3.8 (Main Technical Lemma). Suppose f = of C with output ν. We denote by Cν the circuit Cν in 0 (fn)n∈N is an AC -computable sequence of functions fn : which not only ν but also all of its children are designated n nβ {0, 1}(2) −→ {0, 1} for some constant β ≥ 0. Let α > 0 as outputs. Thus C and Cν compute functions Gn −→ {0, 1}, −α • fan-in(ν)+1 and k ∈ N. Then for a random graph G ∈ ER(n, n ) and while Cν computes a function Gn −→ {0, 1} . [n] a uniform random set A ∈ k , Lemma 3.6 (Circuit Lemma). Let G ∈ Gn and A ⊆ k • f,G α(2)+(β−1)k+o(1) C,G Cν ,G Pr T (A) = A = n . [n] and s ≥ 1. Suppose that Thsi (A) = ∅ and Thsi (A) ≤ s for all nodes ν in C. Then C(G) = C(G ∪ KA). We prove this lemma at the end of §4.1 after developing the technical preliminaries concerning random restrictions. Cν ,G Proof. To streamline notation, let T (ν) = Thsi (A) and C• ,G Corollary 3.9. Take f, β, α, k, G, A as in Lemma 3.8. T •(ν) = T ν (A) for all nodes ν in C. Note that hsi S f,G max α+2(β−1), α s +s(β−1) +o(1) • 1. Pr T (A) 6= ∅ = n (2) , T (ν) = T (ν) ∪ T (µ) hsi children µ of ν 2. Pr |Tf,G(A)| > s by observation (iii) in §3.1. We prove the following claim. hsi s+1 2s max α( 2 )+(β−1)(s+1), α( 2 )+2s(β−1) +o(1) Claim 3.7. For all nodes ν in C, = n . Proof. We prove statement (2); the proof of (1) is simi- Cν (G ∪ KT •(ν)) = Cν (G ∪ KA). lar. f,G Proof of Claim. The proof is by induction on the Pr |Thsi (A)| > s G∈ER(n,n−α),A∈([n]) depth of ν. The claim is trivial when ν is an input node. k " # Indeed, if ν is an input node corresponding to a potential _ [n] • f,G edge e ∈ 2 , then (since s ≥ 1) T (ν) is {e} if e ⊆ A and ∅ ≤ Pr T (B) = B G∈ER(n,n−α),A∈ [n] if e * A. It follows that Cν (the indicator function for edge ( k ) B⊆A : s<|B|≤2s e) has the same value on graphs G ∪ KT •(ν) and G ∪ KA, as (by Lemma 3.5(3)) the edge e is either present in both graphs or absent from ! X2s both graphs. k ≤ Pr Tf,G(B) = B For the induction step, let ν be an non-input node and r −α [n] r=s+1 G∈ER(n,n ),B∈( r ) assume that Cµ(G ∪ KT •(µ)) = Cµ(G ∪ KA) for all children (union bound) µ of ν. Consider now a particular child µ of ν. Since T (µ) ⊆ ! T •(ν) ⊆ A and |T •(ν)| ≤ s, Lemma 3.3 implies X2s k α r +(β−1)r+o(1) = n (2) (by Lemma 3.8) Cµ(G ∪ KT (µ)) = Cµ(G ∪ KT •(ν)). r r=s+1 • Invoking Lemma 3.3 a second time, since T (µ) ⊆ T (µ) ⊆ A max α s+1 +(β−1)(s+1), α 2s +2s(β−1) +o(1) = n ( 2 ) ( 2 ) . and |T •(µ)| ≤ s, The beauty of this corollary is that k (the size of A) does C (G ∪ K ) = C (G ∪ K • ). µ T (µ) µ T (µ) not appear in these upper bounds outside of the o(1) term. By the induction hypothesis, We now prove a weaker version of Theorem 1.1 (with t > 2/3 4 1 instead of t > 1/2 and α ≤ 9t−3 instead of α ≤ 2t−1 ). Cµ(G ∪ KT •(µ)) = Cµ(G ∪ KA). Theorem 3.10. Suppose f = (fn)n∈N is a sequence of Therefore, Cµ(G∪KT •(ν)) = Cµ(G∪KA). Because the value n (2) of Cν on any input graph is determined by the values of Cµ functions fn : {0, 1} −→ {0, 1} computed by circuits of t on the various children µ of ν, we conclude that Cν (G ∪ depth O(1) and size O(n ) for constant t > 1/2. Let G = −α KT •(ν)) = Cν (G ∪ KA). Claim ER(n, n ) be an Erd˝os-R´enyi random graph and let A be a [n] 4 uniform random set in k for any constants 0 < α ≤ 9t−3 Let νout be the output node of C (so that Cνout = C). and k ∈ N. Then fn(G) = fn(G∪KA) asymptotically almost To complete the proof of the lemma, we simply apply the surely as n → ∞. Proof. Let s = b3t/2c. By a routine calculation, Definition 4.1 (Sensitive Inputs). Let I be a set [n] s (we mainly consider I = [n] or ) and let f be a function max α − 2, α − s < 0, 2 2 I s+1 2s with domain {0, 1} (and arbitrary range). The set S(f) of max α 2 − (s + 1), α 2 − 2s < −t. sensitive inputs of f consists of all i ∈ I for which there I So (by continuity) there exists β > 0 such that exist x, y ∈ {0, 1} such that f(x) 6= f(y) and xj = yj for all j ∈ I \{i}. s+1 2s max α 2 + (β − 1)(s + 1), α 2 + 2s(β − 1) < −t. Definition 4.2 (Restriction). A restriction on I is Let C = (Cn)n∈N be a sequence of circuits computing f in depth O(1) and size O(nt). Without loss of generality, a function % : I −→ {0, 1,?}. Intuitively % fixes a certain we can assume that all nodes of C have fan-in nβ . This subset of input bits (assigning either 0 or 1) which leaving assumption only increases the depth of C by a factor of dt/βe other input bits unrestricted (by assigning ?). A restriction % thus may be applied to a function f with domain {0, 1}I , without even doubling its size. −1 • % (?) For all nodes ν in C, let Cν be (as defined at the beginning resulting in a function fd% with domain {0, 1} . The %−1(?) of §3.2) the subcircuit of C in which ν and all its children value of fd% on an element y ∈ {0, 1} is defined by • I as designated as outputs nodes. So Cν computes a function (fd%)(y) = f(x) where x ∈ {0, 1} satisfies x(i) = %(i) if nβ +1 t −1 −1 Gn −→ {0, 1} in depth O(1) and size O(n ). Therefore, i ∈ % ({0, 1}) and x(i) = y(i) if i ∈ % (?). by Corollary 3.9(2),
• We now consider a particular family of random restrictions Pr TCν ,G(A) > s (also studied in [7, 8]). hsi max α(s+1)+(β−1)(s+1), α(2s)+2s(β−1) +o(1) = n 2 2 Definition 4.3. For a finite index set I and p, q ∈ [0, 1], p,q −t−Ω(1) let R denote the random restriction % : I −→ {0, 1,?} = n . I where values %(i) are i.i.d. such that Taking a union bound over all nodes ν in C, Pr[%(i) = ?] = p, W • Cν ,G t −t−Ω(1) −Ω(1) Pr ν Thsi (A) > s = O(n ) · n = n . Pr[%(i) = 1] = (1 − p)q, In addition, by Corollary 3.9(1) we have Pr[%(i) = 0] = (1 − p)(1 − q). p,q p,q C,G max α+2, α s +s +o(1) −Ω(1) As a matter of notation, we write Rn for R in the case Pr T (A) 6= ∅ = n (2) = n . [n] hsi I = [n]. Therefore, it holds asymptotically almost surely that • TC,G(A) = ∅ and TCν ,G(A) ≤ s for all nodes ν in C. The Our first lemma concerning random restrictions is proved hsi hsi by a standard application of the H˚astad Switching Lemma. Circuit Lemma (Lemma 3.6) now gives the desired result The proof is included in the full paper, but omitted here that a.a.s. C(G) = C(G∪K ) (i.e., f (G) = f (G∪K )). A n n A since results similar to Lemma 4.4 are found elsewhere in 0 The following theorem is a weaker version of Theorem 1.2 the literature (e.g., in proofs of PARITY ∈/ AC [4, 15]). 2k/9 k/4 (with ω(n ) instead of ω(n )). n Lemma 4.4. Suppose functions fn : {0, 1} −→ {0, 1}, Theorem 3.11. For every constant k, the k-clique prob- n ∈ N, are computed by circuits of constant depth d and t lem on n-vertex graphs requires constant-depth circuits of size O(n ). Then for every (small) δ > 0 and (large) ` > 0, size ω(n2k/9). there is a constant c = c(d, t, δ, `) such that 2k/9 ` Proof. The lower bound of ω(n ) is certainly trivial Pr |S(fd%)| > c = O(1/n ). n−δ ,1/2 for k ≤ 2, so we assume that k ≥ 3. Let C = (Cn)n∈N % ∈ Rn n be a sequence of circuits on 2 inputs of depth O(1) and t 2k 2 4 size O(n ) for some constant t = 9 (> 3 ). Let α = 9t−3 In plain language, Lemma 4.4 says that with high proba- 2 2 and note that α = > = thres(Kk). For ran- bility (i.e., failing with probability as small an inverse poly- k−1.5 k −1 nomial in n as one desires), an AC0-computable function dom G ∈ ER(n, n−α) and A ∈ [n] , Theorem 3.10 tells us k will depend on only constant many bits after being hit with that asymptotically almost surely C(G) = C(G ∪ K ). By A a random restriction which fixes (i.e., assigns either 0 or 1 to) Lemma 2.1(1), G almost surely contains no k-clique. On all but a fractional power of n input bits. Besides the having the other hand, G ∪ K contains a k-clique (with proba- A PARITY ∈/ AC0 as an immediate corollary, Lemma 4.4 also bility 1). It follows that C does not define the property of implies that the average sensitivity of AC0 functions is no(1) containing a k-clique (for sufficiently large n). Therefore, (though better is known [9]). the k-clique problem does not have constant-depth circuits The next lemma extends Lemma 4.4 to biased random re- of size O(n2k/9). strictions. We remark that the proof of Lemma 4.5, below, 0 does not rely on biased versions of the H˚astad Switching 4. RANDOM RESTRICTIONS AND AC Lemma (e.g., Beame’s Inbalanced Switching Lemma from In this section we prove some technical results concerning [8]). Instead, we use a simple gadget (an ∧-gate) to boot- the sensitivity of AC0 functions when hit with random re- strap from Lemma 4.4. strictions. The endgoal of this section is a proof of the main n technical lemma from §3 (Lemma 3.8). We begin with some Lemma 4.5. Suppose functions fn : {0, 1} −→ {0, 1}, standard definitions. n ∈ N, are computed by circuits of constant depth d and size O(nt). Then for all α, δ > 0 and ` > 0, there is a We now prove the main result of this section. We then constant c = c(d, t, α, δ, `) such that derive Lemma 3.8 (the main technical lemma of §3.3) as a ` corollary. Pr |S(fd%)| > c = O(1/n ). −(α+δ) −α n ,n nβ 0 % ∈ Rn Proposition 4.8. Suppose f : Gn −→ {0, 1} is AC - computable where β ≥ 0. Let H be any graph and let 0 < Proof sketch. Let m = bα log2 nc and consider the function g : {0, 1}n×m −→ {0, 1}n which maps y ∈ α < thres(H). Then n×m n {0, 1} to the element x ∈ {0, 1} where x(i) = α|EH |+(β−1)|VH |+o(1) V Pr |V(fd%)| = |VH | = n . y(i, j). −α j∈[m] % ∈ GRn (H) Let % :[n]×[m] −→ {0, 1,?} be a random restriction under n (nm)−δ ,1/2 e β Rn×m . We lift % to a restriction % :[n] −→ {0, 1,?} We conveniently pretend that n is always an integer. To defined by be correct, every instance of nβ in the following proof should 8 be replaced with dnβ e. <>0 if ∃j ∈ [m],%(i, j) = 0, e Proof. Fix ε > 0 and let %(i) = >1 if ∀j ∈ [m],%(i, j) = 1, : ε thres(H) − α ? otherwise. δ = min , . (1) 2|EH | 2 %e is clearly a random restriction under Rp,q for some p = n [n] p(n) and q = q(n) in [0, 1]. By a straightforward calculation, We also pick a random restriction ξ : 2 −→ {0, 1,?} from log (p) ∼ −(α + δ) and log (q) ∼ −α. n−(α+δ),n−α n n the distribution R n . 0 (2) Since f and g are both AC -computable, so is their com- n n×m Let f , . . . , f β : {0, 1} −→ {0, 1} be the individual position f ◦ g : {0, 1} −→ {0, 1}. Therefore, by Lemma 1 n coordinate-functions of f. By Lemma 4.5 (noting that 4.4 there is a constant c such that Pr |S((f ◦ g)d%)| > c = δ > 0 and taking ` = |V |), there is a constant c such O(1/n`). Note that H |VH | β that Pr |S(fj dξ)| > c = O(1/n ) for all j ∈ [n ]. e β S(fd%) = i : ∃j (i, j) ∈ S((f ◦ g)d%) . Let X1 stand for the event that |V(fdξ)| ≤ 2cn . Our first order of business is to bound the probability of ¬X1. It follows that |S(fd%e)| ≤ |S((f ◦ g)d%)|. As required, this ` e β proves Pr |S(fd%)| > c = O(1/n ). Pr[¬X1] ≤ Pr |S(fdξ)| > cn (2) This argument, however, fails to prove the precise state- (since |V(fd )| ≤ 2|S(fd )|) ment of the lemma, since p and q are not exactly equal to W ξ ξ −(α+δ) −α n and n . This is a minor issue to get around. The ≤ Pr j∈[nβ ] |S(fj dξ)| > c resolution involves redoing the proof sketched here with a S (since S(fd ) = S(f d )) few additional parameters (see full paper). ξ j∈[nβ ] j ξ ≤ nβ · O(1/n|VH |) (union bound) 4.1 Random Graph-Restrictions = O(1/n(1−β)|VH |). We now shift our focus from random restrictions [n] −→ [n] {0, 1,?} to random graph-restrictions 2 −→ {0, 1,?}. Let Gξ be the graph with vertex set VGξ = [n] and edge set E = ξ−1(?). Note that G is a random graph with Definition 4.6 (Sensitive Vertices). Suppose f is Gξ ξ distribution ER(n, n−(α+δ)). a graph-function with domain Gn (and arbitrary range) and |VH |−α|EH |−ε let % : [n] −→ {0, 1,?} be a graph-restriction. The set Let X2 denote the event that sub(Gξ,H) ≥ n . 2 [n] We have α + δ < thres(H) and δ|EH | ≤ ε/2 by definition (1) S(fd%) of sensitive inputs of fd% is a subset of (i.e., 2 of δ. We now bound the probability of ¬X2. the potential edges of a graph in Gn). We call elements of |VH |−α|EH |−ε S(fd%) sensitive edges of the restricted function fd% (rather Pr[¬X2] = Pr sub(Gξ,H) < n (3) than sensitive inputs). We define the set V(fd%) ⊆ [n] by = Pr sub(G, H) < n|VH |−α|EH |−ε G∈ER(n,n−(α+δ)) V(fd%) = i ∈ [n]: ∃j ∈ [n] {i, j} ∈ S(fd%) . ε |V |−(α+δ)|E |− H H 2 We call elements of V(fd%) sensitive vertices of fd%. That ≤ Pr sub(G, H) < n −(α+δ) is, sensitive vertices are simply the endpoints of sensitive G∈ER(n,n ) edges. = exp(−nΩ(1)) (by Lemma 2.1(2)).
Notice that |V(fd%)| ≤ 2|S(fd%)| since each sensitive edge In the (very likely) event that X1 and X2 both hold, we 0 of fd% contributes two sensitive vertices (with possible over- proceed to pick a random graph H and a random graph- lap). restriction % as follows. First we choose H0 uniformly at random from among the induced subgraphs of Gξ isomorphic Definition 4.7. For every graph H and q ∈ [0, 1] and to H (noting that X2 =⇒ sub(Gξ,H) > 0). Having chosen q n ≥ |VH |, let GRn(H) denote the random graph-restriction H0, we then pick a random-graph restriction % : [n] −→ [n] 2 % : 2 −→ {0, 1,?} defined as follows. First, a one-to-one {0, 1,?} subject to function w : V −→ [n] is chosen uniformly at random. For H −1 all edges {i, j} ∈ E , the element {w(i), w(j)} ∈ [n] is • % (?) = EH0 (with probability 1), H 2 [n] • %(e) = ξ(e) (with probability 1) for all e ∈ ξ−1({0, 1}), mapped under % to ?. All remaining e ∈ 2 \ w(EH ) are then independently mapped under % to 1 with probability q • Pr[%(e) = 1] = n−α independently for all e ∈ ξ−1(?) \ and to 0 with probability 1 − q. EH0 . At this point, some observations are in order. 5. PROOFS OF THEOREMS 1.1 AND 1.2 In this section we prove our main results, Theorems 1.1 (i) Conditioned on X1 and X2 (so that % is well-defined), and 1.2. Recall that in §3.3, we proved versions of these n−α % has distribution GRn (H). theorems with weaker parameters (Theorems 3.10 and 3.11). The proofs we now present follow the same basic scheme as (ii) S(fd%) ⊆ S(fdξ) and V(fd%) ⊆ V(fdξ) by virtue of the fact that % refines ξ (i.e., ξ(e) ∈ {0, 1} =⇒ %(e) = ξ(e)). before. To pinpoint the difference this time around, recall statements (2) and (3), below, of Lemma 3.5. 2cnβ (iii) X1 =⇒ V(fdξ) contains ≤ subsets of size |VH |. (2) |VH | f,G Thsi (A) > s ⇐⇒ ∃ fully clique-sensitive B,C ⊆ A with |VH |−α|EH |−ε (iv) X2 =⇒ no matter what ξ is, there are ≥ n |B|, |C| ≤ s and |B ∪ C| ≥ s + 1 0 equally likely possibilities for the random graph H . (3) =⇒ ∃ fully clique-sensitive D ⊆ A with Combining these observations, we have s + 1 ≤ |D| ≤ 2s.
Pr |V(fd )| = |V | X , X = Pr V 0 = V(fd ) X , X In §3.3 we merely exploited statement (3) of Lemma 3.5, % H 1 2 H % 1 2 specifically in the proof of Corollary 3.9(2). This time we use (by (ii)) ≤ Pr V 0 ⊆ V(fdξ) X1, X2 H! the stronger statement (2) of Lemma 3.5 to derive superior 2cnβ bounds (without too much added complexity). (by (iii) and (iv)) ≤ /n|VH |−α|EH |−ε |V | H Definition 5.1. For a, b, c ∈ N such that a ≤ min(b, c) = nβ|VH |+o(1)/n|VH |−α|EH |−ε and b + c − a ≥ 1, let Ha,b,c denote the graph formed by gluing together a b-clique and a c-clique over a subclique of = nα|EH |+(β−1)|VH |+ε+o(1).
size a. For example, H2,3,4 is the graph . Putting this together with (2) and (3), we get b c a Clearly |VHa,b,c | = b+c−a and |EHa,b,c | = 2 + 2 − 2 . Pr |V(fd%)| = |VH | ≤ Pr |V(fd%)| = |VH | X1, X2 Also note that every induced subgraph of Ha,b,c is a graph 0 0 0 + Pr[¬X1] + Pr[¬X2] Ha0,b0,c0 for some smaller a , b , c . α|E |+(β−1)|V |+ε+o(1) = n H H Lemma 5.2. thres(H ) > (1 + √1 )/ max(b, c). a,b,c 2 (β−1)|V | Ω(1) + O(n H ) + exp(−n ) Proof. Let s = max(b, c). We assume s ≥ 2, since oth- = nα|EH |+(β−1)|VH |+ε+o(1). erwise thres(Ha,b,c) = ∞ (as Ha,b,c has no edges). We have
|VHa0,b0,c0 | The result follows, as this inequality holds for all ε > 0. thres(Ha,b,c) = min induced subgraphs |EHa0,b0,c0 | Ha0,b0,c0 ⊆Ha,b,c Finally, we conclude this section by proving our main tech- 0 0 0 b + c − a nical lemma from §3.3. ≥ min b0 c0 a0 . a0,b0,c0∈N: + − 0 0 0 2 2 2 Proof of Lemma 3.8. We assume that α < 2 , since a ≤b ,c ≤s k−1 b0+c0−a0≥1 n otherwise the lemma is trivial. Suppose f : {0, 1}(2) −→ This last quantity is clearly minimal for b0 = c0 = s. Notice nβ 0 {0, 1} is AC -computable. Consider a random graph G ∈ that a0 can be at most s − 1. Letting λ = s/a0, we have ER(n, n−α) and a random set A ∈ [n] . k 0 G [n] 2s − a Let %A : −→ {0, 1,?} be the graph-restriction which thres(Ha,b,c) ≥ min 0 2 a0∈{0,...,s−1} s a [n] A A 2 2 − 2 maps e ∈ 2 to ? if e ∈ 2 , to 1 if e ∈ EG \ 2 , and to 0 G 1 2(2 − λ) 1 1 otherwise. So in other words, %A is obtained from the char- > min = 1 + √ . [n] s 0≤λ≤1 2 − λ2 2 s acteristic function of the edge set EG ⊆ 2 by changing A G the value at every element of 2 to ?. Note that %A is a Definition 5.3. For s ∈ N, let Ws denote the set of n−α 3 random graph-restriction with distribution GRn (Kk). It triples (a, b, c) ∈ N such that a ≤ min(b, c) and max(b, c) ≤ f,G is easy to see that T (A) ⊆ V(fd G ) ⊆ A. That is, every s and b + c − a ≥ s + 1. %A a in the clique-sensitive core of A under f in G is clearly The next lemma will play the same role in proving The- a sensitive vertex of the restricted function fd G . Conse- %A orem 1.1 as our main technical lemma (Lemma 3.8) played f,G quently, T (A) = A =⇒ |V(fd%G )| = k. Therefore, by in proving the weaker Theorem 3.10. A 2 Proposition 4.8 (noting that α < = thres(Kk)), 0 k−1 Lemma 5.4. Suppose f = (fn)n∈N is an AC -computable (n) nβ f,G sequence of functions fn : {0, 1} 2 −→ {0, 1} for some Pr T (A) = A ≤ Pr |V(fd%G )| = k G,A G,A A constant β ≥ 0. Let s ∈ N and (a, b, c) ∈ Ws and 0 < α ≤ −α 1/s. Then for a random graph G ∈ ER(n, n ) and uniform = Pr |V(fd%)| = k [n] [n] n−α %∈GRn (Kk) random sets B ∈ b and C ∈ c subject to |B ∩ C| = a, α|E |+(β−1)|V |+o(1) = n Kk Kk Pr Tf,G(B) = B and Tf,G(C) = C α k +(β−1)k+o(1) b c a (2) α( + − ) + (β − 1)(b + c − a) + o(1) = n . = n 2 2 2 . Proof. Consider the graph-restriction %G : [n] −→ We are finally ready to prove our main theorems. B,C 2 {0, 1,?} which maps e ∈ [n] to ? if e ∈ B ∪ C , to 2 2 2 Proof of Theorem 1.1. Let s = d2t − 1e and note that B C 1 if e ∈ EG \ ( ∪ ), and to 0 otherwise. That is, 1 2 2 α = 2t−1 ≤ 1/s. By a straightforward calculation (included G %B,C is obtained from the characteristic function of the edge in the full paper), we have [n] setEG ⊆ 2 by changing the value at every element of s B C G max α − 2, α 2 − s < 0, 2 ∪ 2 to ?. Note that %B,C is a random graph-restriction −α b c a n max α( 2 + 2 − 2 ) − (b + c − a) < −t. with distribution GRn (Ha,b,c). (a,b,c)∈Ws Every element of Tf,G(B) ∪ Tf,G(C) must clearly be a sensitive vertex with respect to the restricted graph-function (The maximizing (a, b, c) ∈ Ws in the second inequality is the triple (s − 1, s, s).) fd%G . In particular, B,C From this point forward, the argument continues exactly f,G f,G as in the proof of Theorem 3.10 (but using Corollary 5.5 T (B) ∪ T (C) ⊆ V(fd%G ) ⊆ B ∪ C. B,C instead of Corollary 3.9). That is, we choose a small positive
By Proposition 4.8 (noting that α ≤ 1/s < thres(Ha,b,c) by β > 0 such that γ < −t where Lemma 5.2), we have γ = max α( b + c − a ) + (β − 1)(b + c − a). (a,b,c)∈W 2 2 2 Pr Tf,G(B) = B and Tf,G(C) = C s G,B,C β We next choose circuits C = (Cn)n∈N with fan-in n which t ≤ Pr |V(fd G )| = |B ∪ C| compute functions fn in depth O(1) and size O(n ) with %B,C G,B,C fan-in nβ . For each node ν ∈ C, Corollary 5.5 gives us ≤ Pr |V(fd%)| = |VH | a,b,c C• ,G %∈GRn−α (H ) ν γ+o(1) −t−Ω(1) n a,b,c Pr Thsi (A) > s = n = n . α|E | + (β − 1)|V | + o(1) = n Ha,b,c Ha,b,c So by a union bound b c a W C• ,G α( 2 + 2 − 2 ) + (β − 1)(b + c − a) + o(1) ν t −t−Ω(1) −Ω(1) = n . Pr ν Thsi (A) > s = O(n ) · n = n . The following corollary of Lemma 5.4 directly strengthens Also, by Corollary 3.9(1) we have Corollary 3.9(2). C,G max α+2, α s +s +o(1) −Ω(1) Pr T (A) 6= ∅ = n (2) = n . Corollary 5.5. Take f, β, s, α as in Lemma 5.4. Let hsi k be a constant (independent of n). For a random graph Therefore, it holds asymptotically almost surely that −α [n] • G ∈ ER(n, n ) and a uniform random set A ∈ , C,G Cν ,G k T (A) = ∅ and T (A) ≤ s for all ν ∈ C. The Cir- hsi hsi f,G γ+o(1) cuit Lemma (Lemma 3.6) now gives the desired result that Pr |Thsi (A)| > s = n a.a.s. C(G) = C(G ∪ KA) (i.e., fn(G) = fn(G ∪ KA)). b c a where γ = max α( 2 + 2 − 2 ) + (β − 1)(b + c − a). k/4 (a,b,c)∈Ws Proof of Theorem 1.2. The lower bound of ω(n ) is certainly trivial for k ≤ 3, so we assume that k ≥ 4. Let Proof. Similar to the proof of Corollary 3.9, we have n C = (Cn)n∈N be a sequence of circuits on inputs of f,G 2 t k 1 Pr |Thsi (A)| > s depth O(1) and size O(n ) for some constant t = (> ). G∈ER(n,n−α),A∈ [n] 4 2 ( k ) 1 2 2 h _ i Let α = 2t−1 and note that α = k−2 > k−1 = thres(Kk). f,G f,G = Pr T (B) = B and T (C) = C For random G ∈ ER(n, n−α) and A ∈ [n] , Theorem 1.1 G∈ER(n,n−α) k B,C∈ A : tells us that C(G) = C(G ∪ KA) asymptotically almost A∈([n]) (≤s) k |B∪C|>s surely. By Lemma 2.1(1), G a.a.s. contains no k-clique as 2 (by Lemma 3.5(2)) α > thres(Kk) = k−1 . On the other hand, G ∪ KA con- ! X tains a k-clique (with probability 1). It follows that C does k (b + c − a)! ≤ × not define the property of containing a k-clique (for suffi- b + c − a a!(b − a)!(c − a)! ciently large n). Therefore, the k-clique problem does not (a,b,c)∈Ws have constant-depth circuits of size O(nk/4). Pr Tf,G(B) = B and Tf,G(C) = C G∈ER(n,n−α) B∈ [n] ,C∈ [n] : ( b ) ( c ) 6. CONCLUSION |B∩C|=a (union bound) Superconstant Depth. O(1) The constant-depth assumption is really only used in one z ! }| { X place in this paper, namely Lemma 4.4. In fact, Lemma 4.4 k (b + c − a)! = × (and hence all other results in this paper) actually√ holds b + c − a a!(b − a)!(c − a)! for slightly increasing circuit depth d = d(n) = o( log n). (a,b,c)∈Ws To accommodate this change, the statement of Lemma 4.4 α( b + c − a ) + (β − 1)(b + c − a) + o(1) n 2 2 2 changes from “there exists a constant c = c(d, t, δ, `)” to o(1) (by Lemma 5.4) “there exists a function cd,t,δ,`(n) = n ”. The proof of Proposition 4.8 (the only place where Lemma 4.4 is used) γ+o(1) = n . goes through unchanged when c = no(1). Though we have not yet looked into the question, we be- [10] A. Dawar. How many first-order variables are needed lieve that our lower bound of ω(nk/4) should also hold for on finite ordered structures? In We Will Show Them: superconstant clique-size k = k(n), perhaps up to k ≤ log n Essays in Honour of Dov Gabbay, pages 489–520, (as is the case with the lower bounds of Lynch [23] and 2005. Beame [7] mentioned in the introduction). [11] L. Denenberg, Y. Gurevich, and S. Shelah. Definability by constant-depth polynomial-size circuits. Open Questions. Information and Control, 70(2/3):216–240, 1986. What is the maximal α(t) such that for all k ∈ N and [12] M. L. Furst, J. B. Saxe, and M. Sipser. Parity, (n) every sequence (fn)n∈N of functions fn : {0, 1} 2 −→ {0, 1} circuits, and the polynomial-time hierarchy. computed by constant-depth circuits of size O(nt), it holds Mathematical Systems Theory, 17:13–27, 1984. that fn(G) = fn(G ∪ KA) asymptotically almost surely [13] M. Goldmann and J. H˚astad. A simple lower bound where G ∈ ER(n, nα(t)−Ω(1)) and A is a random k-element for the depth of monotone circuits computing clique 1 using a communication game. Information Processing subset of [n]? We showed that α(t) ≥ 2t−1 , but it is con- 2 Letters, 41(4):221–226, 1992. ceivable that α(t) = t−1 . What is the minimal t(k) such that the k-clique problem [14] M. Grohe. Finite variable logics in descriptive complexity theory. Bulletin of Symbolic Logic, has constant-depth circuits of size nt(k)+o(1)? We showed 4(4):345–398, 1998. that t(k) ≥ k/4, but it is conceivable that t(k) = k. What is the fewest number of variables needed to define [15] J. H˚astad. Almost optimal lower bounds for small the class of finite ordered graphs which contain a k-clique depth circuits. In STOC ’86: Proceedings of the in first-order logic? We showed that bk/4c variables are Eighteenth Annual ACM Symposium on Theory of necessary, but it is conceivable that k variables are necessary. Computing, pages 6–20, 1986. Finally, can our technique be used to obtain lower bound [16] N. Immerman. Upper and lower bounds for first order for problems other than k-clique? expressibility. J. Comput. Syst. Sci., 25(1):76–98, 1982. [17] N. Immerman. Expressibility and parallel complexity. Acknowledgements. SIAM J. Comput., 18(3):625–638, 1989. I am grateful to Neil Immerman for pointing out that our [18] N. Immerman. Descriptive Complexity. Graduate non-collapse result (Corollary 1.5) implies strictness of the Texts in Computer Science. Springer-Verlag, New bounded variable hierarchy on finite ordered graphs (Corol- York, 1999. lary 1.7) and for allowing me to include his proof of Propo- [19] S. Janson. Poisson approximation for large deviations. sition 1.6 in the full version of this paper. I am also grateful Random Struct. Algorithms, 1(2):221–230, 1990. to Madhu Sudan for many helpful discussions and to Philipp [20] S. Janson, T. Luczak, and A. Rucinski. An Weis for his comments on an earlier draft of this paper. I also exponential bound for the probability of nonexistence thank the anonymous referees for their detailed comments. of a specified subgraph in a random graph. In Random Graphs ’87, pages 73–87, 1990. 7. REFERENCES [21] M. Koucky, C. Lautemann, S. Poloczek, and 1 D. Therien. Circuit lower bounds via [1] M. Ajtai. Σ1 formulae on finite structures. Annals of Ehrenfeucht-Fraisse games. In CCC ’06: Proceedings Pure and Applied Logic, 24:1–48, 1983. of the 21st Annual IEEE Conference on [2] M. Ajtai. First-order definability on finite structures. Computational Complexity, pages 190–201, 2006. Ann. Pure Appl. Logic, 45(3):211–225, 1989. [22] L. Libkin. Elements of Finite Model Theory. [3] N. Alon and R. B. Boppana. The monotone circuit Springer-Verlag, 2004. complexity of boolean functions. Combinatorica, [23] J. F. Lynch. A depth-size tradeoff for boolean circuits 7(1):1–22, 1987. with unbounded fan-in. In Structure in Complexity [4] N. Alon and J. Spencer. The Probabilistic Method. Theory Conference, pages 234–248, 1986. John Wiley, 1992. [24] M. Otto. Bounded variable logics and counting – A [5] K. Amano and A. Maruoka. A superpolynomial lower study in finite models, volume 9. Springer-Verlag, 1997. bound for a circuit computing the clique function with [25] B. Poizat. Deux ou trois choses que je sais de Ln. J. at most (1/6) log log n negation gates. SIAM J. Symb. Log., 47(3):641–658, 1982. Comput., 35(1):201–215, 2005. [26] A. A. Razborov. Lower bounds on the monotone [6] D. M. Barrington, N. Immerman, and H. Straubing. 1 complexity of some boolean functions. Doklady On uniformity within NC . Journal of Computer and Akademii Nauk SSSR, 281:798–801, 1985. English System Sciences, 41(3):274–306, 1990. translation in Soviet Math. Doklady 31 (1985), [7] P. Beame. Lower bounds for recognizing small cliques 354–357. on CRCW PRAM’s. Discrete Appl. Math., 29(1):3–20, [27] I. Wegener. On the complexity of branching programs 1990. and decision trees for clique functions. J. ACM, [8] P. Beame. A switching lemma primer. Technical 35(2):461–471, 1988. Report UW-CSE-95-07-01, Department of Computer [28] A. C.-C. Yao. Separating the polynomial-time Science and Engineering, University of Washington, hierarchy by oracles. In Proc. 26th Annual Symposium November 1994. on Foundations of Computer Science, pages 1–10, [9] R. B. Boppana. The average sensitivity of 1985. bounded-depth circuits. Inf. Process. Lett., 63(5):257–261, 1997.