A Delay-Averaged Logistic Model

For the equation x(t) = x(t) (1-(1/) _t-- ^t- x(u)du), > 0, > 0, > 0, conditions for the stability of a nonzero stationary solution under small perturbations are determined.     

On characterizations of recursively enumerable languages

Summary Geffert has shown that earch recursively enumerable languageL over can be expressed in the formL{h(x) ^–1 g(x)x in ⁺} ^* where is an alphabet andg, h is a pair of morphisms. Our purpose is to give a simple proof for Geffert's result and then sharpen it into the form where both of the morphisms are nonerasing. In our method we modify constructions used in a representation of recursively enumerable languages in terms of equality sets and in a characterization of simple transducers in terms of morphisms. As direct consequences, we get the undecidability of the Post correspondence problem and various representations ofL. For instance,L =(L ₀) ^* whereL ₀ is a minimal linear language and is the Dyck reductiona, A.     

Lipschitz continuity of theε-subdifferential and nonmonotonic methods with averaging ofε-subgradients for nonsmooth optimization

Inertial methods with averaging of-subgradient are developed for convex optimization problems. Convergence is proved using Lipschitz continuity of the-subdifferential.Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 70–78, March–April, 1992.     

ε-Entropy and Fractal Dimension of a State for a Gaussian Measure

Kolmogorov introduced the concept of -entropy to analyze information in classical continuous systems. The fractal dimension of a geometric set was introduced by Mandelbrot as a new criterion to analyze the geometric complexity of the set. The -entropy and the fractal dimension of a state in a general quantum system were introduced by one of the present authors (MO) in order to characterize chaotic properties of general states.In this paper, we show that -entropy of a state includes Kolmogorov's -entropy, and that the fractal dimension of a state describes fractal structure of Gaussian measures.     

Context-free text grammars

A text is a triple=(, ₁, ₂) such that is a labeling function, and ₁ and ₂ are linear orders on the domain of ; hence may be seen as a word (, ₁) together with an additional linear order ₂ on the domain of . The order ₂ is used to give to the word (, ₁) itsindividual hierarchical representation (syntactic structure) which may be a tree but it may be also more general than a tree. In this paper we introducecontext-free grammars for texts and investigate their basic properties. Since each text has its own individual structure, the role of such a grammar should be that of a definition of a pattern common to all individual texts. This leads to the notion of ashapely context-free text grammar also investigated in this paper.     

Path Integration on a Quantum Computer

We study path integration on a quantum computer that performs quantum summation. We assume that the measure of path integration is Gaussian, with the eigenvalues of its covariance operator of order j^-k with k>1. For the Wiener measure occurring in many applications we have k=2. We want to compute an -approximation to path integrals whose integrands are at least Lipschitz. We prove: Path integration on a quantum computer is tractable. Path integration on a quantum computer can be solved roughly ^-1 times faster than on a classical computer using randomization, and exponentially faster than on a classical computer with a worst case assurance. The number of quantum queries needed to solve path integration is roughly the square root of the number of function values needed on a classical computer using randomization. More precisely, the number of quantum queries is at most 4.46 ^-1. Furthermore, a lower bound is obtained for the minimal number of quantum queries which shows that this bound cannot be significantly improved. The number of qubits is polynomial in ^-1. Furthermore, for the Wiener measure the degree is 2 for Lipschitz functions, and the degree is 1 for smoother integrands. PACS: 03.67.Lx; 31.15Kb; 31.15.-p; 02.70.-c     

Separating complexity classes related to bounded alternating ω-branching programs

We develop a theory of communication within branching programs that provides exponential lower bounds on the size of branching programs that are bounded alternating. Our theory is based on the algebraic concept of -branching programs, : , a semiring homomorphism, that generalizes ordinary branching programs, -branching programs [M2] andMOD _p-branching programs [DKMW].Due to certain exponential lower and polynomial upper bounds on the size of bounded alternating -branching programs we are able to separate the corresponding complexity classesN _ba ,co-N _{ba ba} , andMOD _p - _ba ,p prime, from each other, and from that classes corresponding to oblivious linear length-bounded branching programs investigated in the past.     

Tracking Drifting Concepts By Minimizing Disagreements

In this paper we consider the problem of tracking a subset of a domain (called thetarget) which changes gradually over time. A single (unknown) probability distribution over the domain is used to generate random examples for the learning algorithm and measure the speed at which the target changes. Clearly, the more rapidly the target moves, the harder it is for the algorithm to maintain a good approximation of the target. Therefore we evaluate algorithms based on how much movement of the target can be tolerated between examples while predicting with accuracy . Furthermore, the complexity of the classH of possible targets, as measured byd, its VC-dimension, also effects the difficulty of tracking the target concept. We show that if the problem of minimizing the number of disagreements with a sample from among concepts in a classH can be approximated to within a factork, then there is a simple tracking algorithm forH which can achieve a probability of making a mistake if the target movement rate is at most a constant times ²/(k(d +k) ln 1/), whered is the Vapnik-Chervonenkis dimension ofH. Also, we show that ifH is properly PAC-learnable, then there is an efficient (randomized) algorithm that with high probability approximately minimizes disagreements to within a factor of 7d + 1, yielding an efficient tracking algorithm forH which tolerates drift rates up to a constant times ²/(d ² ln 1/). In addition, we prove complementary results for the classes of halfspaces and axisaligned hyperrectangles showing that the maximum rate of drift that any algorithm (even with unlimited computational power) can tolerate is a constant times ²/d.     

Comparative complexity of the representation of languages by probabilistic automata

We analyze the effect of the degree of isolation of a cut point on the number of states P(U, ) of a probabilistic automaton representing the language U. We give an example of a language Uⁿ consisting of words of length n such that there exist numbers < for which P(Uⁿ, )/P(Uⁿ, )0 as n.Translated from Kibernetika, No. 3, pp. 21–25, May–June, 1989.     

Conditions for the Light-Traffic Insensitivity to Loss Probability in a GI/G/m/0 Queueing System

A loss queueing system GI/G/m/0 is considered. Let a(x) be a p.d.f. of interarrival intervals. Assume that this function behaves like cx^-1 for small x. Further let B(x) be a d.f. of service time; (1/) be the mean service time. Conditions are derived for the light-traffic insensitivity of the loss probability to the form of B(x) as (/ ) 0. In particular, the condition = 1 is necessary. Estimates for the loss probability are obtained.     

On computing Boolean connectives of characteristic functions

This paper is a study of the existence of polynomial time Boolean connective functions for languages. A languageL has an AND function if there is a polynomial timef such thatf(x,y) L x L andy L. L has an OR function if there is a polynomial timeg such thatg(x,y) xL oryL. While all NP complete sets have these functions, Graph Isomorphism, which is probably not complete, is also shown to have both AND and OR functions. The results in this paper characterize the complete sets for the classes D^p and p^SAT[O(logn)] in terms of AND and OR and relate these functions to the structure of the Boolean hierarchy and the query hierarchies. Also, this paper shows that the complete sets for the levels of the Boolean hierarchy above the second level cannot have AND or OR unless the polynomial hierarchy collapses. Finally, most of the structural properties of the Boolean hierarchy and query hierarchies are shown to depend only on the existence of AND and OR functions for the NP complete sets.The first author was supported in part by NSF Research Grants DCR-8520597 and CCR-88-23053, and by an IBM Graduate Fellowship.     

Mesh Selection for a Nearly Singular Boundary Value Problem

In this paper, we investigate the numerical solution of a model equation u _xx = exp(– ) (and several slightly more general problems) when 1 using the standard central difference scheme on nonuniform grids. In particular, we are interested in the error behaviour in two limiting cases: (i) the total mesh point number N is fixed when the regularization parameter 0, and (ii) is fixed when N. Using a formal analysis, we show that a generalized version of a special piecewise uniform mesh 12 and an adaptive grid based on the equidistribution principle share some common features. And the optimal meshes give rates of convergence bounded by |log()| as 0 and N is given, which are shown to be sharp by numerical tests.     

Range Estimation Is NP-Hard for ε2 Accuracy and Feasible for ε2−δ

The basic problem of interval computations is: given a function f(x ₁,..., x _n) and n intervals [x _i, x _i], find the (interval) range yof the given function on the given intervals. It is known that even for quadratic polynomials f(x ₁,..., x _n), this problem is NP-hard. In this paper, following the advice of A. Neumaier, we analyze the complexity of asymptotic range estimation, when the bound on the width of the input intervals tends to 0. We show that for small c > 0, if we want to compute the range with an accuracy c ₂, then the problem is still NP-hard; on the other hand, for every > 0, there exists a feasible algorithm which asymptotically, estimates the range with an accuracy c ^2–.     

The expected additive weight of trees

Summary We consider a general additive weight of random trees which depends on the structure of the subtrees, on weight functions defined on the number of internal and external nodes and on the degrees of the nodes appearing in the tree and its subtrees. Choosing particular weight functions, the corresponding weight is an important parameter appearing in the analysis of sorting and searching algorithms. For a simply generated family of rooted planar trees , we shall derive a general approach to the computation of the average weight of a tree T with n nodes and m leaves for arbitrary weight functions. This general result implies exact and asymptotic expressions for many types of average weights of a tree T with n nodes if the weight functions are arbitrary polynomials in the number of nodes and leaves with coefficients depending on the node degrees.     

A geometric view of parametric linear programming

We present a new definition of optimality intervals for the parametric right-hand side linear programming (parametric RHS LP) Problem () = min{c ^t x¦Ax =b + ¯b,x 0}. We then show that an optimality interval consists either of a breakpoint or the open interval between two consecutive breakpoints of the continuous piecewise linear convex function (). As a consequence, the optimality intervals form a partition of the closed interval {; ¦()¦ < }. Based on these optimality intervals, we also introduce an algorithm for solving the parametric RHS LP problem which requires an LP solver as a subroutine. If a polynomial-time LP solver is used to implement this subroutine, we obtain a substantial improvement on the complexity of those parametric RHS LP instances which exhibit degeneracy. When the number of breakpoints of () is polynomial in terms of the size of the parametric problem, we show that the latter can be solved in polynomial time.This research was partially funded by the United States Navy-Office of Naval Research under Contract N00014-87-K-0202. Its financial support is gratefully acknowledged.     

Finding a closest visible vertex pair between two polygons

Given nonintersecting simple polygonsP andQ, two verticespP andq Q are said to be visible if does not properly intersectP orQ. We present a parallel algorithm for finding a closest pair among all visible pairs (p,q),pP andqQ. The algorithm runs in time O(logn) using O(n) processors on a CREW PRAM, wheren=¦P¦+¦Q¦. This algorithm can be implemented serially in (n) time, which gives a new optimal sequential solution for this problem.This paper appeared in preliminary form as [1]. This work was supported in part by an AT&T BellLaboratories Graduate Fellowship, the Joint Services Electronics Program (U.S. Army, U.S. Navy, U.S. Air Force) under Contract N00014-90-J-1270, and NSF Grant CCR-89-22008. This work was done while the author was with the Department of Computer Science at the University of Illinois at Urbana-Champaign.     

On the oracle complexity of factoring integers

The problem of factoring integers in polynomial time with the help of an infinitely powerful oracle who answers arbitrary questions with yes or no is considered. The goal is to minimize the number of oracle questions. LetN be a given compositen-bit integer to be factored, wheren = log₂ N. The trivial method of asking for the bits of the smallest prime factor ofN requiresn/2 questions in the worst case. A non-trivial algorithm of Rivest and Shamir requires onlyn/3 questions for the special case whereN is the product of twon/2-bit primes. In this paper, a polynomial-time oracle factoring algorithm for general integers is presented which, for any >0, asks at most n oracle questions for sufficiently largeN, thus solving an open problem posed by Rivest and Shamir. Based on a plausible conjecture related to Lenstra's conjecture on the running time of the elliptic curve factoring algorithm, it is shown that the algorithm fails with probability at mostN ^–/2 for all sufficiently largeN.     

Decision procedures for elementary sublanguages of set theory: XIII. Model graphs,reflection and decidability

Positive solutions to the decision problem for a class of quantified formulae of the first order set theoretic language based on , , =, involving particular occurrences of restricted universal quantifiers and for the unquantified formulae of , , =, {...}, , where {...} is the tuple operator and is a general choice operator, are obtained. To that end a method is developed which also provides strong reflection principles over the hereditarily finite sets. As far as finite satisfiability is concerned such results apply also to the unquantified extention of , , =, {...}, , obtained by adding the operators of binary union, intersection and difference and the relation of inclusion, provided no nested term involving is allowed.     

An Estimation Problem for Quasilinear Stochastic Partial Differential Equations

A problem of estimating a functional parameter (x) and functionals () based on observation of a solution u (t, x) of the stochastic partial differential equation is considered. The asymptotic problem setting, as the noise intensity 0, is investigated.     

A regularity test for dual bordered OS systems

Summary A dual bordered OS system is a triple (, P, S) where is a finite alphabet, S a finite subset of ^*, the set of axioms, and P a finite set of rules of the form aa × a, where a and x ^*. Using well-quasi-order theory, we show that the regularity problem for such systems is decidable. Whether such a system generates a regular language essentially only depends on the set of rules but not on the axioms.