Learning of event-recording automata

In regular inference, a regular language is inferred from answers to a finite set of membership queries, each of which asks whether the language contains a certain word. One of the most well-known regular inference algorithms is the L^∗ algorithm due to Dana Angluin. However, there are almost no extensions of these algorithms to the setting of timed systems. We extend Angluin’s algorithm for on-line learning of regular languages to the setting of timed systems. Since timed automata can freely use an arbitrary number of clocks, we restrict our attention to systems that can be described by deterministic event-recording automata (DERAs). We present three algorithms, , and , for inference of DERAs. In and , we further restrict event-recording automata to be event-deterministic in the sense that each state has at most one outgoing transition per action; learning such an automaton becomes significantly more tractable. The algorithm builds on , by attempts to construct a smaller (in number of locations) automaton. Finally, is a learning algorithm for a full class of deterministic event-recording automata, which infers a so called simple DERA, which is similar in spirit to the region graph.     

Pentagons: A weakly relational abstract domain for the efficient validation of array accesses

We introduce Pentagons (), a weakly relational numerical abstract domain useful for the validation of array accesses in byte-code and intermediate languages (IL). This abstract domain captures properties of the form of . It is more precise than the well known Interval domain, but it is less precise than the Octagon domain.The goal of is to be a lightweight numerical domain useful for adaptive static analysis, where is used to quickly prove the safety of most array accesses, restricting the use of more precise (but also more expensive) domains to only a small fraction of the code.We implemented the abstract domain in , a generic abstract interpreter for.NET assemblies. Using it, we were able to validate 83% of array accesses in the core runtime library in a little bit more than 3 minutes.     

The Kolmogorov Expression Complexity of Logics

The purpose of the paper is to propose a completely new notion of complexity of logics in finite-model theory. It is the Kolmogorov variant of the Vardi'sexpression complexity. We define it by considering the value of the Kolmogorov complexityC(L[]) of the infinite stringL[] of all truth values of sentences ofLin . The higher is this value, the more expressive is the logicLin . If is a class of finite models, then the value ofC(L[]) over all ∈ is a measure of expressive power ofLin . Unboundedness ofC(L[])−C(L′[]) for ∈ implies nonexistence of a recursive interpretation ofLinL′. A version of this statement with complexities modulo oracles implies the nonexistence of any interpretation ofLinL′. Thus the valuesC(L[]) modulo oracles constitute an invariant of the expressive power of logics over finite models, depending on their real (absolute) expressive power, and not on the syntax. We investigate our notion for fragments of the infinitary logic ^ω_∞ω: least fixed point logic (LFP) and partial fixed point logic (PFP). We prove a precise characterization of 0–1 laws for these logics in terms of a certain boundedness condition placed onC(L[]). We get an extension of the notion of a 0–1 law by imposing an upper bound on the value ofC(L[]) growing not too fast with cardinality of , which still implies inexpressibility results similar to those implied by 0–1 laws. We also discuss classes in whichC(PFP^k[]) is very high. It appears that then PFP or its simple extension can define all the PSPACE subsets of .     

Acyclic chromatic index of planar graphs with triangles

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by , is the least number of colors in an acyclic edge coloring of G. Let G be a planar graph with maximum degree Δ(G). In this paper, we show that , if G contains no 4-cycle; , if G contains no intersecting triangles; and if G contains no adjacent triangles.     

On generalized middle-level problem

Let be the subgraph of the hypercube Q_n induced by levels between k and n-k, where n?2k+1 is odd. The well-known middle-level conjecture asserts that is Hamiltonian for all k?1. We study this problem in for fixed k. It is known that and are Hamiltonian for all odd n?3. In this paper we prove that also is Hamiltonian for all odd n?5, and we conjecture that is Hamiltonian for every k?0 and every odd n?2k+1.     

Estimation of Kendall’s tau from censored data

This paper considers the nonparametric estimation of Kendall’s tau for bivariate censored data. Under censoring, there have been some papers discussing the nonparametric estimation of Kendall’s tau, such as Wang and Wells (2000), Oakes (2008) and Lakhal et al. (2009). In this article, we consider an alternative approach to estimate Kendall’s tau. The main idea is to replace a censored event-time by a proper imputation. Thus, it induces three estimators, say , , and . We also apply the bootstrap method to estimate the variance of , and and to construct the corresponding confidence interval. Furthermore, we analyze two data sets by the suggested approach, and compare these practical estimators of Kendall’s tau in simulation studies.     

Multi-degree reduction of Bézier curves using reparameterization

L₂-norms are often used in the multi-degree reduction problem of Bézier curves or surfaces. Conventional methods on curve cases are to minimize , where and are the given curve and the approximation curve, respectively. A much better solution is to minimize , where is the closest point to point , that produces a similar effect as that of the Hausdorff distance. This paper uses a piecewise linear function L(t) instead of t to approximate the function φ(t) for a constrained multi-degree reduction of Bézier curves. Numerical examples show that this new reparameterization-based method has a much better approximation effect under Hausdorff distance than those of previous methods.     

The size and depth of layered Boolean circuits

We consider the relationship between size and depth for layered Boolean circuits and synchronous circuits. We show that every layered Boolean circuit of size s can be simulated by a layered Boolean circuit of depth . For synchronous circuits of size s, we obtain simulations of depth . The best known result so far was by Paterson and Valiant (1976) [17], and Dymond and Tompa (1985) [6], which holds for general Boolean circuits and states that , where C(f) and D(f) are the minimum size and depth, respectively, of Boolean circuits computing f. The proof of our main result uses an adaptive strategy based on the two-person pebble game introduced by Dymond and Tompa (1985) [6]. Improving any of our results by polylog factors would immediately improve the bounds for general circuits.     

Invariance properties in the root sensitivity of time-delay systems with double imaginary roots

If is an eigenvalue of a time-delay system for the delay τ₀ then is also an eigenvalue for the delays τ_k?τ₀+k2π/ω, for any k∈Z. We investigate the sensitivity, periodicity and invariance properties of the root for the case that is a double eigenvalue for some τ_k. It turns out that under natural conditions (the condition that the root exhibits the completely regular splitting property if the delay is perturbed), the presence of a double imaginary root for some delay τ₀ implies that is a simple root for the other delays τ_k, k≠0. Moreover, we show how to characterize the root locus around . The entire local root locus picture can be completely determined from the square root splitting of the double root. We separate the general picture into two cases depending on the sign of a single scalar constant; the imaginary part of the first coefficient in the square root expansion of the double eigenvalue.     

Sensitivity versus block sensitivity of Boolean functions

Determining the maximal separation between sensitivity and block sensitivity of Boolean functions is of interest for computational complexity theory. We construct a sequence of Boolean functions with . The best known separation previously was due to Rubinstein. We also report results of computer search for functions with at most 13 variables.     

On the average cost of order-preserving encryption based on hypergeometric distribution

Order-preserving encryption (OPE) is a deterministic encryption scheme whose encryption function preserves numerical ordering of the plaintexts. The first provably-secure OPE scheme was constructed by Boldyreva, Chenette, Lee, and O?Neill. The BCLO scheme is based on a sampling algorithm for the hypergeometric distribution and is known to call the sampling algorithm at most times on average where M is the size of the plaintext-space. We show that the BCLO scheme actually calls the sampling algorithm less than times on average.     

How strong is Nisan?s pseudo-random generator?

We study the resilience of the classical pseudo-random generator (PRG) of Nisan (1992) [6] against space-bounded machines that make multiple passes over the input. Nisan?s PRG is known to fool log-space machines that read the input once. We ask what are the limits of this PRG regarding log-space machines that make multiple passes over the input. We show that for every constant k Nisan?s PRG fools log-space machines that make passes over the input, using a seed of length , for some k^′>k. We complement this result by showing that in general Nisan?s PRG cannot fool log-space machines that make nO(1) passes even for a seed of length . The observations made in this note outline a more general approach in understanding the difficulty of derandomizing BPNC¹.     

An approximation algorithm dependent on edge-coloring number for minimum maximal matching problem

We consider the minimum maximal matching problem, which is NP-hard (Yannakakis and Gavril (1980) [18]). Given an unweighted simple graph G=(V,E), the problem seeks to find a maximal matching of minimum cardinality. It was unknown whether there exists a non-trivial approximation algorithm whose approximation ratio is less than 2 for any simple graph. Recently, Z. Gotthilf et al. (2008) [5] presented a -approximation algorithm, where c is an arbitrary constant.In this paper, we present a -approximation algorithm based on an LP relaxation, where χ^′(G) is the edge-coloring number of G. Our algorithm is the first non-trivial approximation algorithm whose approximation ratio is independent of |V|. Moreover, it is known that the minimum maximal matching problem is equivalent to the edge dominating set problem. Therefore, the edge dominating set problem is also -approximable. From edge-coloring theory, the approximation ratio of our algorithm is , where Δ(G) represents the maximum degree of G. In our algorithm, an LP formulation for the edge dominating set problem is used. Fujito and Nagamochi (2002) [4] showed the integrality gap of the LP formulation for bipartite graphs is at least . Moreover, χ^′(G) is Δ(G) for bipartite graphs. Thus, as far as an approximation algorithm for the minimum maximal matching problem uses the LP formulation, we believe our result is the best possible.     

Convex invariant sets for discrete-time Lur’e systems

In this paper, a method to estimate the domain of attraction of a class of discrete-time Lur’e systems is presented. A new notion of invariance, denoted -invariance, is introduced. An algorithm to determinate the largest -invariant set for this class of systems is proposed. Moreover, it is proven that the -invariant sets provided by this algorithm are polyhedral convex sets and constitute an estimation of the domain of attraction of the non-linear system. It is shown that any contractive set for the Lur’e system is contained in the -invariant set obtained applying the results of this paper. Two illustrative examples are given.     

Kendall distributions and level sets in bivariate exchangeable survival models

For a given bivariate survival function , we study the relations between the set of the level curves of and the Kendall distribution. Then we characterize the survival models simultaneously admitting a specified Kendall distribution and a specified set of level curves. Attention will be restricted to exchangeable survival models.     

From Youla–Kucera to Identification, Adaptive and Nonlinear Control

Some 20 years ago, formulae were presented for the set of all linear time-invariant controllers stabilizing a linear time-invariant plant. This paper traces the development of many ideas from these formulae, covering linear and control, identification, adaptive control and nonlinear systems.     

Computing the nearest polynomial with a zero in a given domain by using piecewise rational functions

For a real univariate polynomial f and a closed complex domain D whose boundary C is a simple curve parameterized by a univariate piecewise rational function, a rigorous method is given for finding a real univariate polynomial such that has a zero in D and is minimal. First, it is proved that the minimum distance between f and polynomials having a zero at α∈C is a piecewise rational function of the real and imaginary parts of α. Thus, on C, the minimum distance is a piecewise rational function of a parameter obtained through the parameterization of C. Therefore, can be constructed by using the property that has a zero on C and computing the minimum distance on C. We analyze the asymptotic bit complexity of the method and show that it is of polynomial order in the size of the input.     

Almost tight upper bound for finding Fourier coefficients of bounded pseudo-Boolean functions

A k-bounded pseudo-Boolean function is a real-valued function on n{0,1} that can be expressed as a sum of functions depending on at most k input bits. The k-bounded functions play an important role in a number of areas including molecular biology, biophysics, and evolutionary computation. We consider the problem of finding the Fourier coefficients of k-bounded functions, or equivalently, finding the coefficients of multilinear polynomials on n{−1,1} of degree k or less. Given a k-bounded function f with m non-zero Fourier coefficients for constant k, we present a randomized algorithm to find the Fourier coefficients of f with high probability in function evaluations. The best known upper bound was , where λ(n,m) is between and n depending on m. Our bound improves the previous bound by a factor of . It is almost tight with respect to the lower bound . In the process, we also consider the problem of finding k-bounded hypergraphs with a certain type of queries under an oracle with one-sided error. The problem is of self interest and we give an optimal algorithm for the problem.     

Regularization parameter estimation for large-scale Tikhonov regularization using a priori information

Solutions of numerically ill-posed least squares problems for A∈R^m×n by Tikhonov regularization are considered. For D∈R^p×n, the Tikhonov regularized least squares functional is given by where matrix W is a weighting matrix and is given. Given a priori estimates on the covariance structure of errors in the measurement data , the weighting matrix may be taken as which is the inverse covariance matrix of the mean 0 normally distributed measurement errors in . If in addition is an estimate of the mean value of , and σ is a suitable statistically-chosen value, J evaluated at its minimizer approximately follows a χ² distribution with degrees of freedom. Using the generalized singular value decomposition of the matrix pair , σ can then be found such that the resulting J follows this χ² distribution. But the use of an algorithm which explicitly relies on the direct solution of the problem obtained using the generalized singular value decomposition is not practical for large-scale problems. Instead an approach using the Golub-Kahan iterative bidiagonalization of the regularized problem is presented. The original algorithm is extended for cases in which is not available, but instead a set of measurement data provides an estimate of the mean value of . The sensitivity of the Newton algorithm to the number of steps used in the Golub-Kahan iterative bidiagonalization, and the relation between the size of the projected subproblem and σ are discussed. Experiments presented contrast the efficiency and robustness with other standard methods for finding the regularization parameter for a set of test problems and for the restoration of a relatively large real seismic signal. An application for image deblurring also validates the approach for large-scale problems. It is concluded that the presented approach is robust for both small and large-scale discretely ill-posed least squares problems.