On the Ehrenfeucht-Fraïssé game in theoretical computer science

We present effective criteria for first-order definability of regular tree languages. It is known that over words the absence of modulo counting (the “noncounting property”) characterizes the expressive power of first-order logic (McNaughton, Schutzenberger), whereas non-counting regular tree languages exist which are not first-order definable. We present new conditions on regular tree languages (more precisely, on tree automata) which imply nondefinability in first-order logic. One method is based on tree homomorphisms which allow to deduce nondefinability of one tree language from nondefinability of another tree language. Additionly we introduce a structural property of tree automata (the socalled Λ-∨-patterns) which also causes tree languages to be undefinable in first-order logic. Finally, it is shown that this notion does not yet give a complete characterization of first-order logic over trees. The proofs rely on the method of Ehrenfeucht-Fraisse games.

This chapter describes four interdisciplinary fields originated and defined by Ashu M. G. Solo in 2011 called political engineering, public policy engineering, computational politics, and computational public policy. Political engineering is the application of engineering, computer science, mathematics, or natural science to solving problems in politics. Computational politics is the application of computer science or mathematics to solving problems in politics. Political engineering and computational politics include, but are not limited to, principles and methods for political decision-making, analysis, modeling, optimization, forecasting, simulation, and expression. Public policy engineering is the application of engineering, computer science, mathematics, or natural science to solving problems in public policy. Computational public policy is the application of computer science or mathematics to solving problems in public policy. Public policy engineering and computational public policy include, but are not limited to, principles and methods for public policy formulation, decision-making, analysis, modeling, optimization, forecasting, and simulation. The chapter describes the scope of research and development in these fields, provides examples of research and development in these fields, and provides possible university curricula for academic programs in these fields.

The purpose of the study is to assess the application of Computer science in the fractional distillation of petroleum. The study sought to: Determine the merit of using Computer science in fractional distillation of petroleum. Evaluate the cost effectiveness to the application of Computer science in fractional distillation of petroleum to the welfare of citizens. Determine if Computer science can help in controlling the natural effects of fractional distillation of petroleum on the environment. Computer science has gained wider application in so many areas of human endeavor ranging from the homes to more scientific and industrial applications, hence its application of Computer science in fractional distillation of petroleum cannot be overemphasized. Irrespective of the benefits of the application of Computer science in fractional distillation of petroleum as a whole, it has become pertinent to ask how effective and efficient its application of Computer science in fractional distillation of petroleum has been. This research work tries to investigate how effective the application of Computer science in fractional distillation of petroleum. The project exertion covered statement of the problem, purpose of the study, research questions/hypotheses were formulated to enable the researcher find out facts about the study, highlighted in division one. Subdivision two was based on the literature review of this study. The study covers the research methodology which guided the study, the next stage was based on the presentation of data and its’ analysis as well as the discussion of findings. The final stage of the research was concerned with the conclusion as well as recommendations and suggestions for further studies

The non-linear invariance principle of Mossel, O'Donnell and Oleszkiewicz establishes that if f(x1, ..., xn) is a multilinear low-degree polynomial with low influences then the distribution of f(B1, ..., Bn) is close (in various senses) to the distribution of f(G1, ..., Gn), where Bi ∈R {-1, 1} are independent Bernoulli random variables and Gi ~ N(0, 1) are independent standard Gaussians. The invariance principle has seen many application in theoretical computer science, including the Majority is Stablest conjecture, which shows that the Goemans--Williamson algorithm for MAXCUT is optimal under the Unique Games Conjecture. More generally, MOO's invariance principle works for any two vectors of hypercontractive random variables (X1, ..., Xn), (Y1, ..., Yn) such that (i) Matching moments: Xi and Yi have matching first and second moments, (ii) Independence: the variables X1, ..., Xn are independent, as are Y1, ..., Yn. The independence condition is crucial to the proof of the theorem, yet in some cases we would like to use distributions (X1, ..., Xn) in which the individual coordinates are not independent. A common example is the uniform distribution on the slice ([EQUATION]) which consists of all vectors (x1, ..., xn) ∈ {0, 1}n with Hamming weight k. The slice shows up in theoretical computer science (hardness amplification, direct sum testing), extremal combinatorics (Erdos--Ko--Rado theorems) and coding theory (in the guise of the Johnson association scheme). Our main result is an invariance principle in which (X1, ..., Xn) is the uniform distribution on a slice ([EQUATION]) and (Y1, ..., Yn) consists either of n independent Ber(p) random variables, or of n independent N(p, p(1 - p)) random variables. As applications, we prove a version of Majority is Stablest for functions on the slice, a version of Bourgain's tail theorem, a version of the Kindler--Safra structural theorem, and a stability version of the t-intersecting Erdos--Ko--Rado theorem, combining techniques of Wilson and Friedgut. Our proof relies on a combination of ideas from analysis and probability, algebra and combinatorics. In particular, we make essential use of recent work of the first author which describes an explicit Fourier basis for the slice.

This paper reviews applications in computer science that decision theorists have addressed for years, discusses the requirements posed by these applications that place great strain on decision theory/social science methods, and explores applications in the social and decision sciences of newer decision-theoretic methods developed with computer science applications in mind. The paper deals with the relation between computer science and decision-theoretic methods of consensus, with the relation between computer science and game theory and decisions, and with “algorithmic decision theory.”

We consider a class of formula equations in first-order logic, Horn formula equations, which are defined by a syntactic restriction on the occurrences of predicate variables. Horn formula equations play an important role in many applications in computer science. We state and prove a fixed-point theorem for Horn formula equations in first-order logic with a least fixed-point operator. Our fixed-point theorem is abstract in the sense that it applies to an abstract semantics which generalises standard semantics. We describe several corollaries of this fixed-point theorem in various areas of computational logic, ranging from the logical foundations of program verification to inductive theorem proving.

In our previous paper [1], a new model of a Labeled Transition System (LTS)-type implementation was proposed. In ordinary LTSs, transitions are labeled by actions; therefore, they can be called LTSs of actions. The new model is an LTS of observations; in this model, observations and test actions (buttons) are used instead of actions. This model generalizes many testing semantics that are based on the LTS of actions but use additional observations (refusals, ready sets, etc.). Moreover, systems with priority, which are not described by the LTS of actions, are simulated uniformly. In the present paper, we develop this approach by focusing on the composition of systems. The point is that, on observation traces, one cannot define a composition with respect to which a composition of LTSs would possess the property of additivity: the set of traces of a composition of LTSs coincides with the set of all pairwise compositions of traces of LTS operands. This is explained by the fact that an observation in a composition state is not calculated based on observations in states-operands. In this paper, we propose an approach that eliminates this drawback. To this end, we label the transitions of LTSs by symbols (events) that, on the one hand, can be composed to guarantee the property of additivity, and, on the other hand, can be used to generate observations under testing: a transition by an event gives rise to an observation related to this event. This model is called an LTS of events. In this paper, we define (1) a transformation of an LTS of events into an LTS of observations to conform with the principles of our previous paper [1]; (2) a composition of LTSs of events; (3) a composition of specifications that preserves conformance: a composition of conformal implementations is conformal to a composition of specifications; and (4) a uniform simulation of LTSs of actions in terms of the LTSs of events, which allows one to consider an implementation in any interaction semantics admissible for LTSs of actions. In this case, a composition of LTSs of events obtained as a result of simulation of the original LTSs of actions is equivalent to the LTS of events obtained as a result of simulation of the composition of these LTSs of actions.

Games provide mathematical models for interaction. Numerous tasks in computer science can be formulated in game-theoretic terms. This fresh and intuitive way of thinking through complex issues reveals underlying algorithmic questions and clarifies the relationships between different domains. This collection of lectures, by specialists in the field, provides an excellent introduction to various aspects of game theory relevant for applications in computer science that concern program design, synthesis, verification, testing and design of multi-agent or distributed systems. Originally devised for a Spring School organised by the GAMES Networking Programme in 2009, these lectures have since been revised and expanded, and range from tutorials concerning fundamental notions and methods to more advanced presentations of current research topics. This volume is a valuable guide to current research on game-based methods in computer science for undergraduate and graduate students. It will also interest researchers working in mathematical logic, computer science and game theory.

Mathematics and philosophy have historically enjoyed a mutually beneficial and productive relationship, as a brief review of the work of mathematician–philosophers such as Descartes, Leibniz, Bolzano, Dedekind, Frege, Brouwer, Hilbert, Gödel, and Weyl easily confirms. In the last century, it was especially mathematical logic and research in the foundations of mathematics which, to a significant extent, have been driven by philosophical motivations and carried out by technically minded philosophers. Mathematical logic continues to play an important role in contemporary philosophy, and mathematically trained philosophers continue to contribute to the literature in logic. For instance, modal logics were first investigated by philosophers and now have important applications in computer science and mathematical linguistics.

Combinatorial Optimization is an active research area that developed from the rich interaction among many mathematical areas, including combinatorics, graph theory, geometry, optimization, probability, theoretical computer science, and many others. It combines algorithmic and complexity analysis with a mature mathematical foundation and it yields both basic research and applications in manifold areas such as, for example, communications, economics, traffic, network design, VLSI, scheduling, production, computational biology, to name just a few. Through strong inner ties to other mathematical fields it has been contributing to and benefiting from areas such as, for example, discrete and convex geometry, convex and nonlinear optimization, algebraic and topological methods, geometry of numbers, matroids and combinatorics, and mathematical programming. Moreover, with respect to applications and algorithmic complexity, Combinatorial Optimization is an essential link between mathematics, computer science and modern applications in data science, economics, and industry.

Preface

Rank bounds for design matrices with block entries and geometric applications, Discrete Analysis 2018:5, 24 pp. It is not hard to prove, when the statement is suitably formulated, that if $A$ is a random matrix, then it is not possible to reduce the rank of $A$ by changing only a few entries. However, it is remarkably hard to find an explicit example of such a matrix. It turns out that explicit examples would have major applications in theoretical computer science, and therefore the matrix-rigidity problem is a central question in the area. In order to make progress on this problem, one can ask a more basic question: are there any combinatorial conditions on a matrix that guarantee that it has a high rank? An interesting condition has recently been identified that does not solve the matrix-rigidity problem but that does have several applications, in particular to combinatorial geometry. Roughly speaking, the condition is that there should not be many non-zero entries in any row or too few non-zero entries in any column, and for any two rows the number of columns for which the entries in both rows are non-zero is small. Such matrices are known as _design matrices_, since the conditions resemble those for a design, though they are significantly looser. The proof that such matrices have high rank uses a beautiful technique called _matrix scaling_. The rough idea is to multiply rows and columns by non-zero scalars until the entries are of broadly the same magnitude, and then to show that the matrix $A^*A$ is dominated by its diagonal. From this it follows fairly easily that $A^*A$ has high rank, and therefore that $A$ does as well. The argument can be found in [a paper of Barak, Dvir, Wigderson and Yehudayoff](https://arxiv.org/pdf/1009.4375.pdf) that this paper generalizes. Such rank bounds are useful, because there are several problems in combinatorial geometry that lead naturally to design matrices. For example, the Sylvester-Gallai theorem and its generalizations concern families of points and collinearity relations between them. If the points are $v_1,\dots,v_n$, then one can write them as column vectors and put them together to form a matrix $M$. Three of the points will be collinear if there is a linear dependence amongst the columns of this matrix with three non-zero coefficients, or equivalently a vector $w$ with three non-zero entries such that $Mw=0$. Under suitable hypotheses, one can use such vectors $w$ to create a design matrix and apply rank bounds on the design matrix to deduce combinatorial consequences about the set of points $v_1,\dots,v_n$. The generalization in this paper concerns design matrices whose entries are themselves matrices, or blocks. Rank bounds are proved for such matrices, which lead to new applications in combinatorial geometry. One of these applications is to the following question. Let $T$ be a set of triples of elements of $\{1,2,\dots,n\}$ and let $V(T)$ be the variety of all sequences $(v_1,\dots,v_n)$ of points in $\mathbb C^d$ such that for every triple $\{i,j,k\}\in T$ the three points $v_i, v_j$ and $v_k$ are collinear. Given a (non-singular) sequence $V\in V(T)$, how many degrees of freedom does it have? A trivial lower bound is 8, resulting from the fact that projective transformations preserve collinearity. The authors prove a general result that has as a consequence that if every pair $\{i,j\}$ belongs to exactly one triple in $T$ and no line contains more than half the points of $V$, then the number of degrees of freedom of $V$ is at most 15. A more precise statement, as well as other interesting applications, can be found in the paper.

The \(\ell _p\) -norm regression problem is a classic problem in optimization with wide ranging applications in machine learning and theoretical computer science. The goal is to compute \(\boldsymbol {\mathit {x}}^{\star } =\arg \min _{\boldsymbol {\mathit {A}}\boldsymbol {\mathit {x}}=\boldsymbol {\mathit {b}}}\Vert \boldsymbol {\mathit {x}}\Vert _p^p\) , where \(\boldsymbol {\mathit {x}}^{\star }\in \mathbb {R}^n,\boldsymbol {\mathit {A}}\in \mathbb {R}^{d\times n},\boldsymbol {\mathit {b}}\in \mathbb {R}^d\) and \(d\le n\) . Efficient high-accuracy algorithms for the problem have been challenging both in theory and practice and the state-of-the-art algorithms require \(poly(p)\cdot n^{\frac{1}{2}-\frac{1}{p}}\) linear system solves for \(p\ge 2\) . In this article, we provide new algorithms for \(\ell _p\) -regression (and a more general formulation of the problem) that obtain a high-accuracy solution in \(O(p n^{ {(p-2)}{(3p-2)}})\) linear system solves. We further propose a new inverse maintenance procedure that speeds-up our algorithm to \(\widetilde{O}(n^{\omega })\) total runtime, where \(O(n^{\omega })\) denotes the running time for multiplying \(n \times n\) matrices. Additionally, we give the first Iteratively Reweighted Least Squares (IRLS) algorithm that is guaranteed to converge to an optimum in a few iterations. Our IRLS algorithm has shown exceptional practical performance, beating the currently available implementations in MATLAB/CVX by 10–50×.

Introduction/Purpose: This study introduced the concept of a prime ideal-based unit graph associated with a commutative ring 𝑅𝑅. In this graph, the vertices consisted of units of 𝑅𝑅 that were not contained in a chosen prime ideal 𝐼𝐼, and two such vertices were considered adjacent if their difference belonged to 𝐼𝐼. The aim was to investigate the structural, algebraic, and topological properties of this graph and examine the algebraic implications of various graph-theoretic invariants. Methods: The construction of ideal-based unit graphs was carried out using the ring ℤ𝑛𝑛, where units excluded from the chosen prime ideal formed the vertex set. The adjacency between two vertices was determined by whether their difference lay in the ideal. The analysis involved computing several topological indices including the Zagreb indices, Wiener index, Arithmetic-Geometric index, Harmonic index, Estrada index, and graph energy. Adjacency matrices and graphical visualizations were employed to understand structural complexity and connectivity. Results: It was observed that the structure of the resulting graph depended significantly on both the modulus 𝑛𝑛 and the nature of the selected ideal. Smaller ideals produced graphs with higher connectivity, while larger ideals led to sparser or disconnected graphs. The calculated indices reflected patterns in symmetry, degree distribution, and distances, revealing deeper algebraic characteristics. Conclusions: Prime ideal-based unit graphs provided a novel approach to studying the interaction between ring-theoretic and graph-theoretic concepts. The findings contributed to potential applications in mathematical chemistry, secure communications, and theoretical computer science.

In the last two decades, modal and description logics have provided a theoretical framework for important applications in many areas of computer science. For this reason, the problem of automated reasoning in modal and description logics has been thoroughly investigated. In this chapter we show how efficient Boolean reasoning techniques have been imported, used and integrated into reasoning tools for modal and description logics. To this extent, we focus on modal logics, and in particular mainly on K(m). In particular, we provide some background in modal logics; we describe a basic theoretical framework and we present and analyze the basic tableau-based and DPLL-based techniques; we describe optimizations and extensions of the DPLL-based procedures; we introduce the automata-theoretic/OBDD-based approach; finally, we present the eager approach.