Circuit Complexity of Regular Languages

We survey the current state of knowledge on the circuit complexity of regular languages and we prove that regular languages that are in AC⁰ and ACC⁰ are all computable by almost linear size circuits, extending the result of Chandra et al. (J. Comput. Syst. Sci. 30:222–234, 1985). As a consequence we obtain that in order to separate ACC⁰ from NC¹ it suffices to prove for some ε>0 an Ω(n ^1+ε ) lower bound on the size of ACC⁰ circuits computing certain NC¹-complete functions. Partially supported by grant GA ČR 201/07/P276, project No. 1M0021620808 of MŠMT ČR and Institutional Research Plan No. AV0Z10190503.     

New upper bounds on the Boolean circuit complexity of symmetric functions

In this note, we present improved upper bounds on the circuit complexity of symmetric Boolean functions. In particular, we describe circuits of size 4.5n+o(n) for any symmetric function of n variables, as well as circuits of size 3n for function.     

Communication complexity towards lower bounds on circuit depth

Karchmer, Raz, and Wigderson (1995) discuss the circuit depth complexity of n-bit Boolean functions constructed by composing up to d = log n/log log n levels of k = log n-bit Boolean functions. Any such function is in AC¹ . They conjecture that circuit depth is additive under composition, which would imply that any (bounded fan-in) circuit for this problem requires depth. This would separate AC¹ from NC¹. They recommend using the communication game characterization of circuit depth. In order to develop techniques for using communication complexity to prove circuit depth lower bounds, they suggest an intermediate communication complexity problem which they call the Universal Composition Relation. We give an almost optimal lower bound of dk—O(d ²(k log k)^1/2) for this problem. In addition, we present a proof, directly in terms of communication complexity, that there is a function on k bits requiring circuit depth. Although this fact can be easily established using a counting argument, we hope that the ideas in our proof will be incorporated more easily into subsequent arguments which use communication complexity to prove circuit depth bounds. Received: July 30, 1999.     

Which bases admit non-trivial shrinkage of formulae?

We show that the shrinkage exponent, under random restrictions, of formulae over a finite complete basis B of Boolean functions, is strictly greater than 1 if and only if all the functions in B are unate, i.e., monotone increasing or decreasing in each of their variables. As a consequence, we get non-linear lower bounds on the formula complexity of the parity function over any basis composed only of unate functions. Received: June 15, 2000.     

Computing Boolean functions by polynomials and threshold circuits

We investigate the computational power of threshold—AND circuits versus threshold—XOR circuits. In contrast to the observation that small weight threshold—AND circuits can be simulated by small weight threshold—XOR circuit, we present a function with small size unbounded weight threshold—AND circuits for which all threshold—XOR circuits have exponentially many nodes. This answers the basic question of separating subsets of the hypercube by hypersurfaces induced by sparse real polynomials. We prove our main result by a new lower bound argument for threshold circuits. Finally we show that unbounded weight threshold gates cannot simulate alternation: There are -functions which need exponential size threshold—AND circuits. Received: August 8, 1996.     

A complex-number Fourier technique for lower bounds on the Mod-m degree

We say an integer polynomial p, on Boolean inputs, weakly m-represents a Boolean function f if p is nonconstant and is zero (mod m), whenever f is zero. In this paper we prove that if a polynomial weakly m-represents the Mod _q function on n inputs, where q and m are relatively prime and m is otherwise arbitrary, then the degree of the polynomial is . This generalizes previous results of Barrington, Beigel and Rudich, and Tsai, which held only for constant or slowly growing m. In addition, the proof technique given here is quite different. We use a method (adapted from Barrington and Straubing) in which the inputs are represented as complex q-th roots of unity. In this representation it is possible to compute the Fourier transform using some elementary properties of the algebraic integers. As a corollary of the main theorem and the proof of Toda's theorem, if q, p are distinct primes, any depth-three circuit that computes the Mod _q function, and consists of an exact threshold gate at the output, Mod _p gates at the next level, and AND gates of polylog fan-in at the inputs, must be of exponential size. We also consider the question of how well circuits consisting of one exact gate over ACC(p)-type circuits (where p is an odd prime) can approximate parity. It is shown that such circuits must have exponential size in order to agree with parity for more than 1/2 + o(1) of the inputs. Received: February 21, 1996.     

Harmonic Analysis, Real Approximation, and the Communication Complexity of Boolean Functions

The two-party communication complexity of Boolean function f is known to be at least log rank (M _f ), i.e., the logarithm of the rank of the communication matrix of f [19]. Lovász and Saks [17] asked whether the communication complexity of f can be bounded from above by (log rank (M _f )) ^c , for some constant c . The question was answered affirmatively for a special class of functions f in [17], and Nisan and Wigderson proved nice results related to this problem [20], but, for arbitrary f , it remained a difficult open problem. We prove here an analogous polylogarithmic upper bound in the stronger multiparty communication model of Chandra et al. [6], which, instead of the rank of the communication matrix, depends on the L ₁ norm of function f , for arbitrary Boolean function f . Received August 24, 1996; revised October 15, 1997.     

Majority gates vs. general weighted threshold gates

In this paper we study small depth circuits that contain threshold gates (with or without weights) and parity gates. All circuits we consider are of polynomial size. We prove several results which complete the work on characterizing possible inclusions between many classes defined by small depth circuits. These results are the following:

Space Complexity Vs. Query Complexity

Combinatorial property testing deals with the following relaxation of decision problems: Given a fixed property and an input x, one wants to decide whether x satisfies the property or is “far” from satisfying it. The main focus of property testing is in identifying large families of properties that can be tested with a certain number of queries to the input. In this paper we study the relation between the space complexity of a language and its query complexity. Our main result is that for any space complexity s(n) ≤ log n there is a language with space complexity O(s(n)) and query complexity 2^Ω(s(n)). Our result has implications with respect to testing languages accepted by certain restricted machines. Alon et al. [FOCS 1999] have shown that any regular language is testable with a constant number of queries. It is well known that any language in space o(log log n) is regular, thus implying that such languages can be so tested. It was previously known that there are languages in space O(log n) that are not testable with a constant number of queries and Newman [FOCS 2000] raised the question of closing the exponential gap between these two results. A special case of our main result resolves this problem as it implies that there is a language in space O(log log n) that is not testable with a constant number of queries. It was also previously known that the class of testable properties cannot be extended to all context-free languages. We further show that one cannot even extend the family of testable languages to the class of languages accepted by single counter machines.      

Easy lower bound for a strange computational model

The paper gives a new proof of the well-known lower bound (n logn) for the algebraic complexity of the functionx ₁ ⁿ x ₂ ⁿ +...+x _n ⁿ (if the characteristic of the ground field does not dividen). As a tool, the proof uses a computational model that counts only duplicators.     

Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds

We show that derandomizing Polynomial Identity Testing is essentially equivalent to proving arithmetic circuit lower bounds for NEXP. More precisely, we prove that if one can test in polynomial time (or even nondeterministic subexponential time, infinitely often) whether a given arithmetic circuit over integers computes an identically zero polynomial, then either (i) (ii) Permanent is not computable by polynomial-size arithmetic circuits. We also prove a (partial) converse: If Permanent requires superpolynomial-size arithmetic circuits, then one can test in subexponential time whether a given arithmetic circuit of polynomially bounded degree computes an identically zero polynomial.Since Polynomial Identity Testing is a coRP problem, we obtain the following corollary: If then NEXP is not computable by polynomial-size arithmetic circuits. Thus establishing that RP = coRP or BPP = P would require proving superpolynomial lower bounds for Boolean or arithmetic circuits. We also show that any derandomization of RNC would yield new circuit lower bounds for a language in NEXP.We also prove unconditionally that NEXP^RP does not have polynomial-size Boolean or arithmetic circuits. Finally, we show that if both BPP = P and low-degree testing is in P; here low-degree testing is the problem of checking whether a given Boolean circuit computes a function that is close to some low-degree polynomial over a finite field.     

On lower bounds for the complexity of polynomials and their multiples

We prove a theorem giving arbitrarily long explicit sequences of algebraic numbers such that any nonzero polynomial f(X) satisfying has nonscalar complexity for some positive constant C independent of s. A similar result is shown for rapidly growing rational sequences. Received: April 6 1999.     

A New Quantum Lower Bound Method,with Applications to Direct Product Theorems and Time-Space Tradeoffs

We give a new version of the adversary method for proving lower bounds on quantum query algorithms. The new method is based on analyzing the eigenspace structure of the problem at hand. We use it to prove a new and optimal strong direct product theorem for 2-sided error quantum algorithms computing k independent instances of a symmetric Boolean function: if the algorithm uses significantly less than k times the number of queries needed for one instance of the function, then its success probability is exponentially small in k. We also use the polynomial method to prove a direct product theorem for 1-sided error algorithms for k threshold functions with a stronger bound on the success probability. Finally, we present a quantum algorithm for evaluating solutions to systems of linear inequalities, and use our direct product theorems to show that the time-space tradeoff of this algorithm is close to optimal. A. Ambainis supported by University of Latvia research project Y2-ZP01-100. This work conducted while at University of Waterloo, supported by NSERC, ARO, MITACS, CIFAR, CFI and IQC University Professorship. R. Špalek supported by NSF Grant CCF-0524837 and ARO Grant DAAD 19-03-1-0082. Work conducted while at CWI and the University of Amsterdam, supported by the European Commission under projects RESQ (IST-2001-37559) and QAP (IST-015848). R. de Wolf supported by a Veni grant from the Netherlands Organization for Scientific Research (NWO) and partially supported by the EU projects RESQ and QAP.     

Evaluating spectral norms for constant depth circuits with symmetric gates

The Fourier spectrum and its norms are given as explicit arithmetic expressions and evaluated, for Boolean functions computed by classes of constant depth, read-once circuits consisting of an arbitrary set of symmetric gates. Previous results of this nature estimate the spectralL ₁ norm of functions computed by certain types of decision trees [20], [7], and in some cases, give randomizedprocedures that evaluate the spectrum by clever rounding [20]. One corollary of our results provides a large class ofAC ⁰ functions whose spectralL ₁ norm is exponential, thus generalizing the single example of such a function given in [9]. This shows that almost every read-onceAC ⁰ function does not belong in the classPL ₁ of functions with polynomially bounded spectral norms.Implications of our results and technique are discussed, for estimating the spectral norms ofany function in a constant depth circuit class, using the coding theoretic concept of weight distributions. Evaluating the spectral norms for any such function reduces to estimating certain non-trivial weight distributions of simple, linear codes.     

一种最大向量平均个数的估计方法

提出一种估计n个d维向量中最大向量平均个数的方法。该方法通过分析单个向量与其他向量子集的支配关系,求出最大向量平均个数的解析式。证明解析式满足已知的递归关系,得到最大向量平均个数的近似估计。与已有方法相比,该方法可应用到估计k个其他向量支配的平均个数问题。