The query complexity of order-finding

We consider the problem where π is an unknown permutation on {0,1,…,2ⁿ−1}, y₀{0,1,…,2ⁿ−1}, and the goal is to determine the minimum r>0 such that π^r(y₀)=1. Information about π is available only via queries that yield π^x(y) from any x{0,1,…,2^m−1} and y{0,1,…,2ⁿ−1} (where m is polynomial in n). The main resource under consideration is the number of these queries. We show that the number of queries necessary to solve the problem in the classical probabilistic bounded-error model is exponential in n. This contrasts sharply with the quantum bounded-error model, where a constant number of queries suffices.     

Asymptotic and oscillatory behavior of solutions of general nonlinear difference equations of second order

The asymptotic and oscillatory behavior of solutions of some general second-order nonlinear difference equations of the form

δ(a_nh(y_n+1)δy_n)+p_nδy_n+q_n+1f(y_σ(n+1))=0 nZ,

is studied. Oscillation criteria for their solutions, when “p_n” is of constant sign, are established. Results are also presented for the damped-forced equation

δ(a_nh(y_n+1)δy_n)+p_nδy_n+q_n+1f(y_σ(n+1))=e_n nZ.

Examples are inserted in the text for illustrative purposes.     

Lower Bounds for Dynamic Algebraic Problems

We consider dynamic evaluation of algebraic functions (matrix multiplication, determinant, convolution, Fourier transform, etc.) in the model of Reif and Tate; i.e., if f(x₁,…, x_n)=(y₁, …, y_m) is an algebraic problem, we consider serving online requests of the form “change input x_i to value v” or “what is the value of output y_i?” We present techniques for showing lower bounds on the worst case time complexity per operation for such problems. The first gives lower bounds in a wide range of rather powerful models (for instance, history dependent algebraic computation trees over any infinite subset of a field, the integer RAM, and the generalized real RAM model of Ben-Amram and Galil). Using this technique, we show optimal Ω(n) bounds for dynamic matrix–vector product, dynamic matrix multiplication, and dynamic discriminant and an Ω( ) lower bound for dynamic polynomial multiplication (convolution), providing a good match with Reif and Tate's O( ) upper bound. We also show linear lower bounds for dynamic determinant, matrix adjoint, and matrix inverse and an Ω( ) lower bound for the elementary symmetric functions. The second technique is the communication complexity technique of Miltersen, Nisan, Safra, and Wigderson which we apply to the setting of dynamic algebraic problems, obtaining similar lower bounds in the word RAM model. The third technique gives lower bounds in the weaker straight line program model. Using this technique, we show an Ω((log n)²/log log n) lower bound for dynamic discrete Fourier transform. Technical ingredients of our techniques are the incompressibility technique of Ben-Amram and Galil and the lower bound for depth-two superconcentrators of Radhakrishnan and Ta-Shma. The incompressibility technique is extended to arithmetic computation in arbitrary fields.     

Schnorr Randomness

Schnorr randomness is a notion of algorithmic randomness for real numbers closely related to Martin-Löf randomness. Since its initial development in the 1970s, the notion has received considerably less attention than Martin-Löf randomness. In this article, we explore the properties of Schnorr random reals, and in particular the c.e. Schnorr random reals. We show that there are c.e. reals that are Schnorr random but not Martin-Löf random, and provide a new characterization of Schnorr random real numbers in terms of prefix-free machines. We prove that unlike Martin-Löf random c.e. reals, not all Schnorr random c.e. reals are Turing complete, though all are in high Turing degrees. We use the machine characterization to define a notion of “Schnorr reducibility” which allows us to calibrate the Schnorr complexity of reals. We define the class of “Schnorr trivial” reals, which are ones whose initial segment complexity is identical with the computable reals, and demonstrate that this class has non-computable members.     

Tracking control with prescribed transient behaviour for systems of known relative degree

Tracking of a reference signal (assumed bounded with essentially bounded derivative) is considered for multi-input, multi-output linear systems satisfying the following structural assumptions: (i) arbitrary—but known—relative degree, (ii) the “high-frequency gain” matrix is sign definite—but of unknown sign, (iii) exponentially stable zero dynamics. The first control objective is tracking, by the output y, with prescribed accuracy: given λ>0 (arbitrarily small), determine a feedback strategy which ensures that, for every reference signal r, the tracking error e=y-r is ultimately bounded by λ (that is, e(t)<λ for all t sufficiently large). The second objective is guaranteed output transient performance: the evolution of the tracking error should be contained in a prescribed performance funnel (determined by a function ). Both objectives are achieved by a filter in conjunction with a feedback function of the filter states, the tracking error and a gain parameter. The latter is generated via a feedback function of the tracking error and the funnel parameter . Moreover, the feedback system is robust to nonlinear perturbations bounded by some continuous function of the output. The feedback structure essentially exploits an intrinsic high-gain property of the system/filter interconnection by ensuring that, if (t,e(t)) approaches the funnel boundary, then the gain attains values sufficiently large to preclude boundary contact.     

Approximate fixed point sequence for a finite family of asymptotically pseudocontractive mappings

Let E be a real Banach space and K be a nonempty, closed, convex, and bounded subset of E. Let T_i:K→K, i=1,2,…,N, be N uniformly L-Lipschitzian, uniformly asymptotically regular with sequences {ε_n}, and asymptotically pseudocontractive mappings with sequences , where {ε_n} and , i=1,2,…,N, satisfy certain mild conditions. Let a sequence {x_n} be generated from x₁K by

for all integers n1, where T_n=T_n(modN), {u_n} be a sequence in K, and {λ_n}, {θ_n} and {μ_n} are three real sequences in [0,1] satisfying appropriate conditions; then x_n−T_lx_n→0 as n→∞ for each l{1,2,…,N}.     

Representation of ℤ<Subscript>4</Subscript>-Linear Preparata Codes Using Vector Fields

A binary code is called ℤ₄-linear if its quaternary Gray map preimage is linear. We show that the set of all quaternary linear Preparata codes of length n = 2^m, m odd, m ≥ 3, is nothing more than the set of codes of the form with

QFT template generation for time-delay plants based on zero-inclusion test

Plant template generation is the key step in applying quantitative feedback theory (QFT) to design robust control for uncertain systems. In this paper we propose a technique for generating plant templates for a class of linear systems with an uncertain time delay and affine parameter perturbations in coefficients. The main contribution lies in presenting a necessary and sufficient condition for the zero inclusion of the value set f(T,Q)={f(τ,q): τT[τ⁻,τ⁺], qQ∏_k=0^m−1[q_k⁻,q_k⁺]}, where f(τ,q)=g(q)+h(q)e^−jτω*, g(q) and h(q) are both complex-valued affine functions of the m-dimensional real vector q, and ω^* is a fixed frequency. Based on this condition, an efficient algorithm which involves, in the worst case, evaluation of m algebraic inequalities and solution of m2^m−1 one-variable quadratic equations, is developed for testing the zero inclusion of the value set f(T,Q). This zero-inclusion test algorithm allows one to utilize a pivoting procedure to generate the outer boundary of a plant template with a prescribed accuracy or resolution. The proposed template generation technique has a linear computational complexity in resolution and is, therefore, more efficient than the parameter gridding and interval methods. A numerical example illustrating the proposed technique and its computational superiority over the interval method is included.     

Inclusion dynamics hybrid automata

Hybrid systems are dynamical systems with the ability to describe mixed discrete-continuous evolution of a wide range of systems. Consequently, at first glance, hybrid systems appear powerful but recalcitrant, neither yielding to analysis and reasoning through a purely continuous-time modeling as with systems of differential equations, nor open to inferential processes commonly used for discrete state-transition systems such as finite state automata. A convenient and popular model, called hybrid automata, was introduced to model them and has spurred much interest on its tractability as a tool for inference and model checking in a general setting. Intuitively, a hybrid automaton is simply a “finite-state” automaton with each state augmented by continuous variables, which evolve according to a set of well-defined continuous laws, each specified separately for each state. This article investigates both the notion of hybrid automaton and the model checking problem over such a structure. In particular, it relates first-order theories and analysis results on multivalued maps and reduces the bounded reachability problem for hybrid automata whose continuous laws are expressed by inclusions (x′f(x,t)) to a decidability problem for first-order formulæ over the reals. Furthermore, the paper introduces a class of hybrid automata for which the reachability problem can be decided and shows that the problem of deciding whether a hybrid automaton belongs to this class can be again decided using first-order formulæ over the reals. Despite the fact that the bisimulation quotient for this class of hybrid automata can be infinite, we show that our techniques permit effective model checking for a nontrivial fragment of CTL.     

Fully Scalable Fault-Tolerant Simulations for BSP and CGM

In this paper we consider general simulations of algorithms designed for fully operational BSP and CGM machines on machines with faulty processors. The BSP (or CGM) machine is a parallel multicomputer consisting of p processors for which a memory of n words is evenly distributed and each processor can send and receive at most h messages in a superstep. The faults are deterministic (i.e., worst-case distributions of faults are considered) and static (i.e., they do not change in the course of computation). We assume that a constant fraction of processors are faulty.  We present two fault-tolerant simulation techniques for BSP and CGM:  1. A deterministic simulation that achieves O(1) slowdown for local computations and O((log_h p)²) slowdown for communications per superstep, provided that a preprocessing is done that requires O((log_h p)²) supersteps and linear (in h) computation per processor in each superstep.  2. A randomized simulation that achieves O(1) slowdown for local computations and O(log_h p) slowdown for communications per superstep with high probability, after the same (deterministic) preprocessing as above.  Our results are fully scalable over all values of p from Θ(1) to Θ(n). Furthermore, our results imply that if pn for 0<<1 and h=Θ((n/p)^δ) for 0<δ1 (which hold in almost all practical BSP and CGM computations), algorithms can be made resilient to a constant fraction of processor faults without any asymptotic slowdown.     

The most nonelementary theory

We give a direct proof by generic reduction that testing validity of formulas in a decidable rudimentary theory Ω of finite typed sets (Henkin, Fundamenta Mathematicæ 52 (1963) 323–344) requires space and time exceeding infinitely often(1)where n denotes the length of input. This gives the highest currently known lower bound for a decidable logical theory and affirmatively settles Problem 10.13 from (Compton and Henson, Ann. Pure Applied Logic 48 (1990) 1–79):

Is there a “natural” decidable theory with a lower bound of the form exp_∞(f(n)), where f is not linearly bounded?

The highest previously known lower (and upper) bounds for “natural” decidable theories, like WS1S, S2S, are of the form exp_∞(dn), with just linearly growing stacks of twos.Originally, the lower bound (1) for Ω was settled in (12th Annual IEEE Symposium on Logic in Computer Science (LICS’97), 1997, 294–305) using the powerful uniform lower bounds method due to Compton and Henson, and probably would never be discovered otherwise. Although very concise, the original proof has certain gaps, because the method was pushed out of the limits it was originally designed and intended for, and some hidden assumptions were violated. This results in slightly weaker bounds—the stack of twos in (1) grows subexponentially, but superpolynomially, namely, as for formulas with fixed quantifier prefix, or as 2^cn/log(n) for formulas with varying prefix. The independent direct proof presented in this paper closes the gaps and settles the originally claimed lower bound (1) for the minimally typed, succinct version of Ω.     

Complexity of Computing the Local Dimension of a Semialgebraic Set

The paper describes several algorithms related to a problem of computing the local dimension of a semialgebraic set. Let a semialgebraic set V be defined by a system of k inequalities of the formf  ≥  0 with f  R [ X₁, ,X_n ], deg(f)  < d , andx   V . An algorithm is constructed for computing the dimension of the Zariski tangent space to V at x in time (kd)^O(n). Let x belong to a stratum of codimension l_xin V with respect to a smooth stratification ofV . Another algorithm computes the local dimension dim_x(V) with the complexity (k(l_x +  1)d)^O(l_x2n). Ifl  = max_x  Vl_x, and for every connected component the local dimension is the same at each point, then the algorithm computes the dimension of every connected component with complexity (k(l +  1)d)^O(l2n). If V is a real algebraic variety defined by a system of equations, then the complexity of the algorithm is less thankd^O(l2n) , and the algorithm also finds the dimension of the tangent space to V at x in time kd^O(n). Whenl is fixed, like in the case of a smooth V , the complexity bounds for computing the local dimension are (kd)^O(n)andkd^O(n) respectively. A third algorithm finds the singular locus ofV in time (kd)^O(n2).     

Stability and boundedness in difference systems with finite delay

In this paper, stability and boundedness theorems for delay difference systems with the condition

δV(n,x(n))≤-W₂(|x(n)|)+σ

are given, where Δ is the backward difference operator. Some known results are generalized.     

Recognizing Hand-Printed Digits with a Distance Quasi-Metric

A distance quasi-metric for pattern recognition is presented. The “quasi” modifier distinguishes the metric from “true” distance metrics which obey a set of standard constraints. By relaxing one of the constraints and coupling it with a fast multidimensional search technique, the metric demonstrates improved accuracy and efficiency compared to other metrics in recognizing hand-written digit samples. A high-level design of a fast optical comparator for computing the distance in O( ) is also presented.     

A necessary condition for quantitative exponential stability of linear state-space systems

For an arbitrary n×n constant matrix A the two following facts are well known:

• (1/n)Re(traceA)−max_j=1,…,nRe λ_j(A)0;

• If U is a unitary matrix, one can always find a skew-Hermitian matrix A so that U=e^A.

In this note we present the extension of these two facts to the context of linear time-varying dynamical systemsAs a by-product, this result suggests that, the notion of “slowly varying state-space systems”, commonly used in literature, is mathematically not natural to the problem of exponential stability.     

On the Construction of Classes of Suffix Trees for Square Matrices: Algorithms and Applications

We provide a uniform framework for the study of index data structures for a two-dimensional matrixTEXT[1:n, 1:n] whose entries are drawn from an ordered alphabetΣ. An index forTEXTcan be informally seen as the two-dimensional analog of the suffix tree for a string. It allows on-line searches and statistics to be performed onTEXTby representing compactly theΘ(n³) square submatrices ofTEXTin optimalO(n²) space. We identify 4ⁿ⁻¹families of indices forTEXT, each containing ∏ⁿ_i=1 (2i−1)! isomorphic data structures. We also develop techniques leading to a single algorithm that efficiently builds any index in any family inO(n² log n) time andO(n²) space. Such an algorithm improves in various respects the algorithms for the construction of the PAT tree and the Lsuffix tree. The framework and the algorithm easily generalize tod>2 dimensions. Moreover, as part of our algorithm, we provide new algorithmic tools that yield a space-efficient implementation of the “naming scheme” of R. Karpet al.(in“Proceedings, Fourth Symposium on Theory of Computing,” pp. 125–136) for strings and matrices.     

Fast periodic graph exploration with constant memory

We consider the problem of periodic exploration of all nodes in undirected graphs by using a finite state automaton called later a robot. The robot, using a constant number of states (memory bits), must be able to explore any unknown anonymous graph. The nodes in the graph are neither labelled nor coloured. However, while visiting a node v the robot can distinguish between edges incident to it. The edges are ordered and labelled by consecutive integers 1,…,d(v) called port numbers, where d(v) is the degree of v. Periodic graph exploration requires that the automaton has to visit every node infinitely many times in a periodic manner. In this paper, we are interested in minimisation of the length of the exploration period. In other words, we want to minimise the maximum number of edge traversals performed by the robot between two consecutive visits of a generic node, in the same state and entering the node by the same port. Note that the problem is unsolvable if the local port numbers are set arbitrarily, see [L. Budach, Automata and labyrinths, Math. Nachr. 86 (1978) 195–282]. In this context, we are looking for the minimum function π(n), such that, there exists an efficient deterministic algorithm for setting the local port numbers allowing the robot to explore all graphs of size n along a traversal route with the period π(n). Dobrev et al. proved in [S. Dobrev, J. Jansson, K. Sadakane, W.-K. Sung, Finding short right-hand-on-the-wall walks in graphs, in: Proc. 12th Colloquium on Structural Information and Communication Complexity, SIROCCO 2005, in: Lecture Notes in Comput. Sci., vol. 3499, Springer, Berlin, 2005, pp. 127–139] that for oblivious robots π(n)10n. Recently Ilcinkas proposed another port labelling algorithm for robots equipped with two extra memory bits, see [D. Ilcinkas, Setting port numbers for fast graph exploration, in: Proc. 13th Colloquium on Structural Information and Communication Complexity, SIROCCO 2006, in: Lecture Notes in Comput. Sci., vol. 4056, Springer, Berlin, 2006, pp. 59–69], where the exploration period π(n)4n−2. In the same paper, it is conjectured that the bound 4n−O(1) is tight even if the use of larger memory is allowed. In this paper, we disprove this conjecture presenting an efficient deterministic algorithm arranging the port numbers, such that, the robot equipped with a constant number of bits is able to complete the traversal period in π(n)<3.75n−2 steps hence decreasing the existing upper bound. This reduces the gap with the lower bound of π(n)2n−2 holding for any robot.     

Improved edge-coloring with three colors

We show an O(1.344ⁿ)=O(2^0.427n) algorithm for edge-coloring an n-vertex graph using three colors. Our algorithm uses polynomial space. This improves over the previous O(2^n/2) algorithm of Beigel and Eppstein [R. Beigel, D. Eppstein, 3-coloring in time O(1.3289n), J. Algorithms 54 (2) (2005) 168–204.]. We apply a very natural approach of generating inclusion–maximal matchings of the graph. The time complexity of our algorithm is estimated using the “measure and conquer” technique.     

Generalized dilations and the stabilization of uncertain systems via measurement feedback

We study the problem of semiglobally stabilizing uncertain nonlinear system

Full-size image

, with (A,B) in Brunowski form. We prove that if p₁(z,u,t)u and p₂(z,u,t)u are of order greater than 1 and 0, respectively, with “generalized” dilation δ_l(z,u)=(l¹⁻ⁿz₁,…,l⁻¹z_n−1,z_n,lu) and uniformly with respect to t, where z_i is the ith component of z, then we can achieve semiglobal stabilization via arbitrarily bounded linear measurement feedback.