DOCSLIB.ORG

On the Constant-Depth Complexity of K-Clique

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Constant-Depth Complexity of K-Clique 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 !(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 !(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 ¯rst-order logic, bounded variable hierarchy on the constant d in the exponent of n, thus breaking the mold of the traditional size-depth tradeo®. 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 : f0; 1g 2 ¡! f0; 1g 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 f1; : : : ; ng and independent edge probabilities n¡® bounded fan-in.) Early results of Ajtai [1] and Furst, Saxe 1 and Sipser [12] proved that the parity problem (given an n- where ® · 2t¡1 . Let A be a uniform random k-element subset of f1; : : : ; ng (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 2= 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 Hºastad [15] among others, who eventually es- ¯nite model theory (going back at least to Immerman in tablished that exponential-size constant-depth circuits are 1982). The m-variable fragment of ¯rst-order logic, denoted required. by FOm, consists of the ¯rst-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 (speci¯ed by a bitstring of length ), does variable hierarchy FO ½ FO ½ ¢ ¢ ¢ ½ FO ½ ¢ ¢ ¢ is strict 2 in terms of expressive power on ¯nite 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 ¯rst-order logic on ¯nite 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 signi¯- 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 tradeo®s. Lynch [23] in 1986 Mathematical Logic|Model theory established the ¯rst size-depth tradeo® for the k-clique prob- p 1:5 lem by proving a lower bound of n­( 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 !(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 Hºastad [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 tradeo®s 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 p 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 ¡® number of input bits). These lower bounds thus do not rule 0 < ® · 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 a®airs by proving a lower bound of !(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 lowerp 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 !(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 signi¯cantly 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 tradeo®s, 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 Hºastad's original switching lemma. Rather, our technique (¯nite 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 ¯rst-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 ¯rst-order logic, denoted by f;G m (see x3.1). Thsi (A) is a subset of A with certain nice prop- FO , consists of the ¯rst-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 quanti¯ers, 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- 8x8y9z 9y (x » y) ^ (y » z) ^ 9x (z » x) ^ (x » y) : mately, our approach leads to an e®ective new way of using the switching lemma in conjunction with a union bound over More generally, 3 variables su±ce 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 in¯nitary counterparts L1;!) are important has been studied in various models of computation besides objects of study in both ¯nite and classical model theory constant-depth circuits. In the setting of monotone circuits, (see [14, 18]). The chain of m-variable fragments FO1 ½ it is known that the k-clique problem (for speci¯c increasing FO2 ½ ¢ ¢ ¢ ½ FOm ½ ¢ ¢ ¢ is known as the bounded (or ¯nite) k = k(n)) requires exponential size [3, 13, 26]. A super- variable hierarchy. On ¯nite 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].
Load more

Recommended publications
  • On Uniformity Within NC
    On Uniformity Within NC David A Mix Barrington Neil Immerman HowardStraubing University of Massachusetts University of Massachusetts Boston Col lege Journal of Computer and System Science Abstract In order to study circuit complexity classes within NC in a uniform setting we need a uniformity condition which is more restrictive than those in common use Twosuch conditions stricter than NC uniformity RuCo have app eared in recent research Immermans families of circuits dened by rstorder formulas ImaImb and a unifor mity corresp onding to Buss deterministic logtime reductions Bu We show that these two notions are equivalent leading to a natural notion of uniformity for lowlevel circuit complexity classes Weshow that recent results on the structure of NC Ba still hold true in this very uniform setting Finallyweinvestigate a parallel notion of uniformity still more restrictive based on the regular languages Here we givecharacterizations of sub classes of the regular languages based on their logical expressibility extending recentwork of Straubing Therien and Thomas STT A preliminary version of this work app eared as BIS Intro duction Circuit Complexity Computer scientists have long tried to classify problems dened as Bo olean predicates or functions by the size or depth of Bo olean circuits needed to solve them This eort has Former name David A Barrington Supp orted by NSF grant CCR Mailing address Dept of Computer and Information Science U of Mass Amherst MA USA Supp orted by NSF grants DCR and CCR Mailing address Dept of
    [Show full text]
  • Week 1: an Overview of Circuit Complexity 1 Welcome 2
    Topics in Circuit Complexity (CS354, Fall’11) Week 1: An Overview of Circuit Complexity Lecture Notes for 9/27 and 9/29 Ryan Williams 1 Welcome The area of circuit complexity has a long history, starting in the 1940’s. It is full of open problems and frontiers that seem insurmountable, yet the literature on circuit complexity is fairly large. There is much that we do know, although it is scattered across several textbooks and academic papers. I think now is a good time to look again at circuit complexity with fresh eyes, and try to see what can be done. 2 Preliminaries An n-bit Boolean function has domain f0; 1gn and co-domain f0; 1g. At a high level, the basic question asked in circuit complexity is: given a collection of “simple functions” and a target Boolean function f, how efficiently can f be computed (on all inputs) using the simple functions? Of course, efficiency can be measured in many ways. The most natural measure is that of the “size” of computation: how many copies of these simple functions are necessary to compute f? Let B be a set of Boolean functions, which we call a basis set. The fan-in of a function g 2 B is the number of inputs that g takes. (Typical choices are fan-in 2, or unbounded fan-in, meaning that g can take any number of inputs.) We define a circuit C with n inputs and size s over a basis B, as follows. C consists of a directed acyclic graph (DAG) of s + n + 2 nodes, with n sources and one sink (the sth node in some fixed topological order on the nodes).
    [Show full text]
  • A Short History of Computational Complexity
    The Computational Complexity Column by Lance FORTNOW NEC Laboratories America 4 Independence Way, Princeton, NJ 08540, USA [email protected] http://www.neci.nj.nec.com/homepages/fortnow/beatcs Every third year the Conference on Computational Complexity is held in Europe and this summer the University of Aarhus (Denmark) will host the meeting July 7-10. More details at the conference web page http://www.computationalcomplexity.org This month we present a historical view of computational complexity written by Steve Homer and myself. This is a preliminary version of a chapter to be included in an upcoming North-Holland Handbook of the History of Mathematical Logic edited by Dirk van Dalen, John Dawson and Aki Kanamori. A Short History of Computational Complexity Lance Fortnow1 Steve Homer2 NEC Research Institute Computer Science Department 4 Independence Way Boston University Princeton, NJ 08540 111 Cummington Street Boston, MA 02215 1 Introduction It all started with a machine. In 1936, Turing developed his theoretical com- putational model. He based his model on how he perceived mathematicians think. As digital computers were developed in the 40's and 50's, the Turing machine proved itself as the right theoretical model for computation. Quickly though we discovered that the basic Turing machine model fails to account for the amount of time or memory needed by a computer, a critical issue today but even more so in those early days of computing. The key idea to measure time and space as a function of the length of the input came in the early 1960's by Hartmanis and Stearns.
    [Show full text]
  • Multiparty Communication Complexity and Threshold Circuit Size of AC^0
    2009 50th Annual IEEE Symposium on Foundations of Computer Science Multiparty Communication Complexity and Threshold Circuit Size of AC0 Paul Beame∗ Dang-Trinh Huynh-Ngoc∗y Computer Science and Engineering Computer Science and Engineering University of Washington University of Washington Seattle, WA 98195-2350 Seattle, WA 98195-2350 [email protected] [email protected] Abstract— We prove an nΩ(1)=4k lower bound on the random- that the Generalized Inner Product function in ACC0 requires ized k-party communication complexity of depth 4 AC0 functions k-party NOF communication complexity Ω(n=4k) which is in the number-on-forehead (NOF) model for up to Θ(log n) polynomial in n for k up to Θ(log n). players. These are the first non-trivial lower bounds for general 0 NOF multiparty communication complexity for any AC0 function However, for AC functions much less has been known. for !(log log n) players. For non-constant k the bounds are larger For the communication complexity of the set disjointness than all previous lower bounds for any AC0 function even for function with k players (which is in AC0) there are lower simultaneous communication complexity. bounds of the form Ω(n1=(k−1)=(k−1)) in the simultaneous Our lower bounds imply the first superpolynomial lower bounds NOF [24], [5] and nΩ(1=k)=kO(k) in the one-way NOF for the simulation of AC0 by MAJ ◦ SYMM ◦ AND circuits, showing that the well-known quasipolynomial simulations of AC0 model [26]. These are sub-polynomial lower bounds for all by such circuits are qualitatively optimal, even for formulas of non-constant values of k and, at best, polylogarithmic when small constant depth.
    [Show full text]
  • Lecture 13: Circuit Complexity 1 Binary Addition
    CS 810: Introduction to Complexity Theory 3/4/2003 Lecture 13: Circuit Complexity Instructor: Jin-Yi Cai Scribe: David Koop, Martin Hock For the next few lectures, we will deal with circuit complexity. We will concentrate on small depth circuits. These capture parallel computation. Our main goal will be proving circuit lower bounds. These lower bounds show what cannot be computed by small depth circuits. To gain appreciation for these lower bound results, it is essential to first learn about what can be done by these circuits. In next two lectures, we will exhibit the computational power of these circuits. We start with one of the simplest computations: integer addition. 1 Binary Addition Given two binary numbers, a = a1a2 : : : an−1an and b = b1b2 : : : bn−1bn, we can add the two using the elementary school method { adding each column and carrying to the next. In other words, r = a + b, an an−1 : : : a1 a0 + bn bn−1 : : : b1 b0 rn+1 rn rn−1 : : : r1 r0 can be accomplished by first computing r0 = a0 ⊕ b0 (⊕ is exclusive or) and computing a carry bit, c1 = a0 ^ b0. Now, we can compute r1 = a1 ⊕ b1 ⊕ c1 and c2 = (c1 ^ (a1 _ b1)) _ (a1 ^ b1), and in general we have rk = ak ⊕ bk ⊕ ck ck = (ck−1 ^ (ak _ bk)) _ (ak ^ bk) Certainly, the above operation can be done in polynomial time. The main question is, can we do it in parallel faster? The computation expressed above is sequential. Before computing rk, one needs to compute all the previous output bits.
    [Show full text]
  • Parameterized Circuit Complexity and the W Hierarchy
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science ELSEVIER Theoretical Computer Science 191 (1998) 97-115 Parameterized circuit complexity and the W hierarchy Rodney G. Downey a, Michael R. Fellows b,*, Kenneth W. Regan’ a Department of Mathematics, Victoria University, P.O. Box 600, Wellington, New Zealand b Department of Computer Science, University of Victoria, Victoria, BC, Canada V8W 3P6 ‘Department of Computer Science, State University of New York at Buffalo, 226 Bell Hall, Buffalo, NY 14260-2000 USA Received June 1995; revised October 1996 Communicated by M. Crochemore Abstract A parameterized problem (L,k) belongs to W[t] if there exists k’ computed from k such that (L, k) reduces to the weight-k’ satisfiability problem for weft-t circuits. We relate the fundamental question of whether the w[t] hierarchy is proper to parameterized problems for constant-depth circuits. We define classes G[t] as the analogues of AC0 depth-t for parameterized problems, and N[t] by weight-k’ existential quantification on G[t], by analogy with NP = 3 . P. We prove that for each t, W[t] equals the closure under fixed-parameter reductions of N[t]. Then we prove, using Sipser’s results on the AC0 depth-t hierarchy, that both the G[t] and the N[t] hierarchies are proper. If this separation holds up under parameterized reductions, then the kF’[t] hierarchy is proper. We also investigate the hierarchy H[t] defined by alternating quantification over G[t].
    [Show full text]
  • Time, Hardware, and Uniformity
    This is page Printer Opaque this Time Hardware and Uniformity David Mix Barrington Neil Immerman ABSTRACT We describ e three orthogonal complexity measures parallel time amount of hardware and degree of nonuniformity which together parametrize most complexity classes Weshow that the descriptive com plexity framework neatly captures these measures using three parameters quantier depth number of variables and typ e of numeric predicates re sp ectively A fairly simple picture arises in which the basic questions in complexity theory solved and unsolved can b e understo o d as ques tions ab out tradeos among these three dimensions Intro duction An initial presentation of complexity theory usually makes the implicit as sumption that problems and hence complexity classes are linearly ordered by diculty In the Chomsky Hierarchyeach new typ e of automaton can decide more languages and the Time Hierarchy Theorem tells us that adding more time allowsaTuring machine to decide more languages In deed the word complexity is often used eg in the study of algorithms to mean worstcase Turing machine running time under which problems are linearly ordered Those of us who study structural complexityknow that the situation is actually more complicated For one thing if wewant to mo del parallel com putation we need to distinguish b etween algorithms whichtakethesame amountofwork ie sequential time wecarehowmany pro cesses are op erating in parallel and howmuch parallel time is taken These two dimen sions of complexity are identiable in all the usual mo
    [Show full text]
  • Circuit Complexity Contents 1 Announcements 2 Relation to Other
    CS221: Computational Complexity Prof. Salil Vadhan Lecture 19: Circuit Complexity 11/6 Scribe: Chris Crick Contents 1 Announcements ² Problem Set 3, problem 2: You may use ², the regular expression for the empty string. ² PS3 due tomorrow. ² Be aware that you do not necessarily have to solve all of the problems on the problem set — the point is to get you to think about them. ² Reading: Papadimitriou x4.3, x11.4 2 Relation to other complexity measures Definition: The circuit complexity or circuit size of a function f : f0; 1gn ! f0; 1g is the number of gates in the smallest boolean circuit that computes f. Circuit complexity depends our choice of a universal basis, but only up to a constant. We can therefore use any such basis to represent our circuit. Because it’s familiar and convenient, then, we will usually use AND, OR and NOT, though sometimes the set of all boolean functions of two inputs and one output is more natural. Example: To provide a sense for how one programs with circuits, we will examine the thresh- 1 if at least two x s are 1 old function Th (x ; ¢ ¢ ¢ ; x ) = f i g. Figure 1 shows a simple circuit to 2 1 n 0 otherwise compute this function. In this circuit, li = 1 if at least one of (x1; : : : ; xi) = 1. ri becomes 1 once two xis equal 1. li is just xi _ li¡1, and ri is ri¡1 _ (li¡1 ^ xi). The output node is rn. Conventions for counting circuit gates vary a little – for example, some people count the inputs in the total, while others don’t.
    [Show full text]
  • Circuit Complexity and Computational Complexity
    Circuit Complexity and Computational Complexity Lecture Notes for Math 267AB University of California, San Diego January{June 1992 Instructor: Sam Buss Notes by: Frank Baeuerle Sam Buss Bob Carragher Matt Clegg Goran Gogi¶c Elias Koutsoupias John Lillis Dave Robinson Jinghui Yang 1 These lecture notes were written for a topics course in the Mathematics Department at the University of California, San Diego during the winter and spring quarters of 1992. Each student wrote lecture notes for between two and ¯ve one and one half hour lectures. I'd like to thank the students for a superb job of writing up lecture notes. January 9-14. Bob Carragher's notes on Introduction to circuit complexity. Theorems of Shannon and Lupanov giving upper and lower bounds of circuit complexity of almost all Boolean functions. January 14-21. Matt Clegg's notes on Spira's theorem relating depth and formula size Krapchenko's lower bound on formula size over AND/OR/NOT January 23-28. Frank Baeuerle's notes on Neciporuk's theorem Simulation of time bounded TM's by circuits January 30 - February 4. John Lillis's notes on NC/AC hierarchy Circuit value problem Depth restricted circuits for space bounded Turing machines Addition is in NC1. February 6-13. Goran Gogi¶c's notes on Circuits for parallel pre¯x vector addition and for symmetric functions. February 18-20. Dave Robinson's notes on Schnorr's 2n-3 lower bound on circuit size Blum's 3n-o(n) lower bound on circuit size February 25-27. Jinghui Yang's notes on Subbotovskaya's theorem Andreev's n2:5 lower bound on formula size with AND/OR/NOT.
    [Show full text]
  • BQP and the Polynomial Hierarchy∗
    BQP and the Polynomial Hierarchy∗ Scott Aaronson† MIT ABSTRACT 1. INTRODUCTION The relationship between BQP and PH has been an open A central task of quantum computing theory is to un- problem since the earliest days of quantum computing. We derstand how BQP—Bounded-Error Quantum Polynomial- present evidence that quantum computers can solve prob- Time, the class of all problems feasible for a quantum computer— lems outside the entire polynomial hierarchy, by relating this fits in with classical complexity classes. In their original question to topics in circuit complexity, pseudorandomness, 1993 paper defining BQP, Bernstein and Vazirani [9] showed #P 1 and Fourier analysis. that BPP BQP P . Informally, this says that quan- ⊆ ⊆ First, we show that there exists an oracle relation problem tum computers are at least as fast as classical probabilistic (i.e., a problem with many valid outputs) that is solvable in computers and no more than exponentially faster (indeed, BQP, but not in PH. This also yields a non-oracle relation they can be simulated using an oracle for counting). Bern- problem that is solvable in quantum logarithmic time, but stein and Vazirani also gave evidence that BPP = BQP, 6 not in AC0. by exhibiting an oracle problem called Recursive Fourier Ω(log n) Second, we show that an oracle decision problem separat- Sampling that requires n queries on a classical com- ing BQP from PH would follow from the Generalized Linial- puter but only n queries on a quantum computer. The Nisan Conjecture, which we formulate here and which is evidence for the power of quantum computers became dra- likely of independent interest.
    [Show full text]
  • 1 Circuit Complexity
    CS 6743 Lecture 15 1 Fall 2007 1 Circuit Complexity 1.1 Definitions A Boolean circuit C on n inputs x1, . , xn is a directed acyclic graph (DAG) with n nodes of in-degree 0 (the inputs x1, . , xn), one node of out-degree 0 (the output), and every node of the graph except the input nodes is labeled by AND, OR, or NOT; it has in-degree 2 (for AND and OR), or 1 (for NOT). The Boolean circuit C computes a Boolean function f(x1, . , xn) in the obvious way: the value of the function is equal to the value of the output gate of the circuit when the input gates are assigned the values x1, . , xn. The size of a Boolean circuit C, denoted |C|, is defined to be the total number of nodes (gates) in the graph representation of C. The depth of a Boolean circuit C is defined as the length of a longest path (from an input gate to the output gate) in the graph representation of the circuit C. A Boolean formula is a Boolean circuit whose graph representation is a tree. Given a family of Boolean functions f = {fn}n≥0, where fn depends on n variables, we are interested in the sizes of smallest Boolean circuits Cn computing fn. Let s(n) be a function such that |Cn| ≤ s(n), for all n. Then we say that the Boolean function family f is computable by Boolean circuits of size s(n). If s(n) is a polynomial, then we say that f is computable by polysize circuits.
    [Show full text]
  • On AC0[⊕] and Variants of the Majority Function
    Electronic Colloquium on Computational Complexity, Report No. 133 (2019) More on AC0[⊕] and Variants of the Majority Function Nutan Limaye∗ Srikanth Srinivasan y Utkarsh Tripathi z Abstract In this paper we prove two results about AC0[⊕] circuits. 1=d2 • We show that for d(N) = o(plog N= log log N) and N ≤ s(N) ≤ 2dN there is an N explicit family of functions ffN : f0; 1g ! f0; 1gg such that 0 { fN has uniform AC formulas of depth d and size at most s; 0 " { fN does not have AC [⊕] formulas of depth d and size s , where " is a fixed absolute constant. This gives a quantitative improvement on the recent result of Limaye, Srinivasan, Sreenivasaiah, Tripathi, and Venkitesh, (STOC, 2019), which proved a similar Fixed- Ω(d) Depth Size-Hierarchy theorem but for d log log N and s exp(N 1=2 ). As in the previous result, we use the Coin Problem to prove our hierarchy theorem. Our main technical result is the construction of uniform size-optimal formulas for solving O(d) the coin problem with improved sample complexity (1/δ)d+4 (down from (1/δ)2 in the previous result). • In our second result, we show that randomness buys depth in the AC0[⊕] setting. Formally, we show that for any fixed constant d ≥ 2, there is a family of Boolean functions that has polynomial-sized randomized uniform AC0 circuits of depth d but no polynomial-sized (deterministic) AC0[⊕] circuits of depth d. Previously Viola (Computational Complexity, 2014) showed that an increase in depth (by at least 2) is essential to avoid superpolynomial blow-up while derandomizing randomized AC0 circuits.
    [Show full text]