On the k-Abelian Equivalence Relation of Finite Words

This thesis is devoted to the so-called k-abelian equivalence relation of sequences of symbols, that is, words. This equivalence relation is a generalization of the abelian equivalence of words. Two words are abelian equivalent if one is a permutation of the other. For any positive integer k, two words are called k-abelian equivalent if each word of length at most k occurs equally many times as a factor in the two words. The k-abelian equivalence defines an equivalence relation, even a congruence, of finite words. A hierarchy of equivalence classes in between the equality relation and the abelian equivalence of words is thus obtained. Most of the literature on the k-abelian equivalence deals with infinite words. In this thesis we consider several aspects of the equivalence relations, the main objective being to build a fairly comprehensive picture on the structure of the k-abelian equivalence classes themselves. The main part of the thesis deals with the structural aspects of k-abelian equivalence classes. We also consider aspects of k-abelian equivalence in infinite words. We survey known characterizations of the k-abelian equivalence of finite words from the literature and also introduce novel characterizations. For the analysis of structural properties of the equivalence relation, the main tool is the characterization by the rewriting rule called the k-switching. Using this rule it is straightforward to show that the language comprised of the lexicographically least elements of the k-abelian equivalence classes is regular. Further word-combinatorial analysis of the lexicographically least elements leads us to describe the deterministic finite automata recognizing this language. Using tools from formal language theory combined with our analysis, we give an optimal expression for the asymptotic growth rate of the number of k-abelian equivalence classes of length n over an m-letter alphabet. Explicit formulae are computed for small values of k and m, and these sequences appear in Sloane’s Online Encyclopedia of Integer Sequences. Due to the fact that the k-abelian equivalence relation is a congruence of the free monoid, we study equations over the k-abelian equivalence classes. The main result in this setting is that any system of equations of k-abelian equivalence classes is equivalent to one of its finite subsystems, i.e., the monoid defined by the k-abelian equivalence relation possesses the compactness property. Concerning infinite words, we mainly consider the (k-)abelian complexity function. We complete a classification of the asymptotic abelian complexities of pure morphic binary words. In other words, given a morphism which has an infinite binary fixed point, the limit superior asymptotic abelian complexity of the fixed point can be computed (in principle). We also give a new proof of the fact that the k-abelian complexity of a Sturmian word is n + 1 for length n 2k. In fact, we consider several aspects of the k-abelian equivalence relation in Sturmian words using a dynamical interpretation of these words. We reprove the fact that any Sturmian word contains arbitrarily large k-abelian repetitions. The methods used allow to analyze the situation in more detail, and this leads us to define the so-called k-abelian critical exponent which measures the ratio of the exponent and the length of the root of a k-abelian repetition. This notion is connected to a deep number theoretic object called the Lagrange spectrum

Polynomial Identity Testing via Evaluation of Rational Functions

We introduce a hitting set generator for Polynomial Identity Testing based on evaluations of low-degree univariate rational functions at abscissas associated with the variables. Despite the univariate nature, we establish an equivalence up to rescaling with a generator introduced by Shpilka and Volkovich, which has a similar structure but uses multivariate polynomials in the abscissas. We study the power of the generator by characterizing its vanishing ideal, i.e., the set of polynomials that it fails to hit. Capitalizing on the univariate nature, we develop a small collection of polynomials that jointly produce the vanishing ideal. As corollaries, we obtain tight bounds on the minimum degree, sparseness, and partition class size of set-multilinearity in the vanishing ideal. Inspired by an alternating algebra representation, we develop a structured deterministic membership test for the vanishing ideal. As a proof of concept, we rederive known derandomization results based on the generator by Shpilka and Volkovich and present a new application for read-once oblivious algebraic branching programs.Comment: Appeared at ITCS 202

Methods and applications of systems identification

A Schmidt filter is proposed to compute an optimal orthonormal basis for a set of noisy filter input functions. Procedures for determining the transfer function and inverse transfer function of the filter are given. The Schmidt filter is applied to the problem of determining mathematical models of discrete, stationary, linear, dynamic systems for the case where measurements may be corrupted by noise of unknown statistics. The identification problem is reconsidered for the case where noise and signal moments are specified. Procedures are given which insure unbiased, adaptive estimates of system order and parameters for this case. These theoretical propositions are applied to the modeling of speculative prices. The stock market is formulated as a discrete, linear, dynamic system and the results of several simulation studies are presented. Evidence indicates that certain segments of the market can be approximated by high-order linear systems computed from small samples and tends to refute the random walk hypothesis. Computer programs (written in PL/1) are presented which allow for efficient digital realization of the theoretical procedures discussed in the body of this work --Abstract, page iii

Proceedings of the Workshop on the lambda-Prolog Programming Language

The expressiveness of logic programs can be greatly increased over first-order Horn clauses through a stronger emphasis on logical connectives and by admitting various forms of higher-order quantification. The logic of hereditary Harrop formulas and the notion of uniform proof have been developed to provide a foundation for more expressive logic programming languages. The λ-Prolog language is actively being developed on top of these foundational considerations. The rich logical foundations of λ-Prolog provides it with declarative approaches to modular programming, hypothetical reasoning, higher-order programming, polymorphic typing, and meta-programming. These aspects of λ-Prolog have made it valuable as a higher-level language for the specification and implementation of programs in numerous areas, including natural language, automated reasoning, program transformation, and databases

Beta-Conversion, Efficiently

Type-checking in dependent type theories relies on conversion, i.e. testing given lambda-terms for equality up to beta-evaluation and alpha-renaming. Computer tools based on the lambda-calculus currently implement conversion by means of algorithms whose complexity has not been identified, and in some cases even subject to an exponential time overhead with respect to the natural cost models (number of evaluation steps and size of input lambda-terms). This dissertation shows that in the pure lambda-calculus it is possible to obtain conversion algorithms with bilinear time complexity when evaluation is carried following evaluation strategies that generalize Call-by-Value to the stronger case required by conversion

Introduction to Axiomatic Geometry

This book presents Euclidean Geometry and was designed for a one-semester course preparing junior and senior level college students to teach high school Geometry. The book could also serve as a text for a junior level Introduction to Proofs course.https://ohioopen.library.ohio.edu/opentextbooks/1000/thumbnail.jp

Object-oriented data mining

EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Essays on Macroeconomic Implications of International Capital Flow and Fiscal Uncertainty

This dissertation comprises three chapters that contribute to a broader and an ongoing discussion in the macroeconomics, international economics, and development economics literature. Specifically, the first chapter focuses on understanding how shocks to long-term U.S debt held by foreign official institutions such as foreign central banks and foreign ministries of finance affect the U.S economy. In the context of a dynamic stochastic general equilibrium(DSGE) model with imperfect asset substitution between short and long-term government bonds, I find that shocks to long-term U.S debt held by foreign official institutions have expansionary effects on the economy--they lower the long-term interest rate and increase output, consumption and inflation. This result is supported by empirical findings from a structural vector autoregression model (SVAR). The second chapter advances the study of foreign aid fungibility by showing how subtle characteristics of household behavior interact with fungible aid and institutional factors to impact aid effectiveness. Specifically, I build a simple dynamic optimizing model and show that the way consumers internalize an aid induced increase in government spending can have very contrasting impacts on aid effectiveness-- a feature absent in the extensive empirical literature. Finally, the third chapter studies how different discretionary government spending policy options impact the consequences of explosive government transfer payments. I employ a DSGE model with a fiscal limit-- a point where higher taxation is no longer a feasible financing for this study