Pigeonhole Principle Research Articles

Intelligent identification of telecommunication channels in the distributed systems for digital control of power facilities

This paper presents the results and problems of telecommunication power amplifier modeling. The research tool is the mathematical apparatus of artificial neural networks. Simulation of the device was made according to the black box principle, operating solely with input and output data of the power amplifier under study. The criterion for the quality of model estimation was the signal-to-noise ratio adopted in radio engineering. Brief results of the previous research are shown. The problems associated with the simulation of a power amplifier operation basing on neural networks were investigated. An example is given of nonlinear distortions appearing in the device under consideration. The description is given of the joint work of a neural networks group based on analog signal clustering which works according to the principle of Grossberg’s adaptive resonance theory. The application results of this type of systems are shown. Discussing the problems concerning the use of this approach to simulation of the investigated processes.

Typical forcings, NP search problems and an extension of a theorem of Riis

We define typical forcings encompassing many informal forcing arguments in bounded arithmetic and give general conditions for such forcings to produce models of the universal variant of relativized T21. We apply this result to study the relative complexity of total (type 2) NP search problems associated to finitary combinatorial principles.Complexity theory compares such problems with respect to polynomial time many-one or Turing reductions. From a logical perspective such problems are graded according to the bounded arithmetic theories that prove their totality. The logical analogue of a reduction is to prove the totality of one problem from the totality of another. The link between the two perspectives is tight for what we call universal variants of relativized bounded arithmetics. We strengthen a theorem of Buss and Johnson (2012) that infers relative bounded depth Frege proofs of totality from polynomial time Turing reducibility.As an application of our general forcing method we derive a strong form of Riis' finitization theorem (1993). We extend it by exhibiting a simple model-theoretic property that implies independence from the universal variant of relativized T21 plus the weak pigeonhole principle. More generally, we show that the universal variant of relativized T21 does not prove (the totality of the total NP search problem associated to) a strong finitary combinatorial principle from a weak one. Being weak or strong are simple model-theoretic properties based on the behavior of the principles with respect to finite structures that are only partially defined.

The weakness of the pigeonhole principle under hyperarithmetical reductions

The infinite pigeonhole principle for 2-partitions ([Formula: see text]) asserts the existence, for every set [Formula: see text], of an infinite subset of [Formula: see text] or of its complement. In this paper, we study the infinite pigeonhole principle from a computability-theoretic viewpoint. We prove in particular that [Formula: see text] admits strong cone avoidance for arithmetical and hyperarithmetical reductions. We also prove the existence, for every [Formula: see text] set, of an infinite low[Formula: see text] subset of it or its complement. This answers a question of Wang. For this, we design a new notion of forcing which generalizes the first and second-jump control of Cholak et al.

Schematic Refutations of Formula Schemata

Proof schemata are infinite sequences of proofs which are defined inductively. In this paper we present a general framework for schemata of terms, formulas and unifiers and define a resolution calculus for schemata of quantifier-free formulas. The new calculus generalizes and improves former approaches to schematic deduction. As an application of the method we present a schematic refutation formalizing a proof of a weak form of the pigeon hole principle.

PSEUDO-FINITE SETS, PSEUDO-O-MINIMALITY

AbstractWe give an example of two ordered structures $\mathcal {M},\mathcal {N}$ in the same language $\mathcal {L}$ with the same universe, the same order and admitting the same one-variable definable subsets such that $\mathcal {M}$ is a model of the common theory of o-minimal $\mathcal {L}$ -structures and $\mathcal {N}$ admits a definable, closed, bounded, and discrete subset and a definable injective self-mapping of that subset which is not surjective. This answers negatively two question by Schoutens; the first being whether there is an axiomatization of the common theory of o-minimal structures in a given language by conditions on one-variable definable sets alone. The second being whether definable completeness and type completeness imply the pigeonhole principle. It also partially answers a question by Fornasiero asking whether definable completeness of an expansion of a real closed field implies the pigeonhole principle.

ON OBDD-BASED ALGORITHMS AND PROOF SYSTEMS THAT DYNAMICALLY CHANGE THE ORDER OF VARIABLES

AbstractIn 2004 Atserias, Kolaitis, and Vardi proposed $\text {OBDD}$ -based propositional proof systems that prove unsatisfiability of a CNF formula by deduction of an identically false $\text {OBDD}$ from $\text {OBDD}$ s representing clauses of the initial formula. All $\text {OBDD}$ s in such proofs have the same order of variables. We initiate the study of $\text {OBDD}$ based proof systems that additionally contain a rule that allows changing the order in $\text {OBDD}$ s. At first we consider a proof system $\text {OBDD}(\land , \text{reordering})$ that uses the conjunction (join) rule and the rule that allows changing the order. We exponentially separate this proof system from $\text {OBDD}(\land )$ proof system that uses only the conjunction rule. We prove exponential lower bounds on the size of $\text {OBDD}(\land , \text{reordering})$ refutations of Tseitin formulas and the pigeonhole principle. The first lower bound was previously unknown even for $\text {OBDD}(\land )$ proofs and the second one extends the result of Tveretina et al. from $\text {OBDD}(\land )$ to $\text {OBDD}(\land , \text{reordering})$ .In 2001 Aguirre and Vardi proposed an approach to the propositional satisfiability problem based on $\text {OBDD}$ s and symbolic quantifier elimination (we denote algorithms based on this approach as $\text {OBDD}(\land , \exists )$ algorithms). We augment these algorithms with the operation of reordering of variables and call the new scheme $\text {OBDD}(\land , \exists , \text{reordering})$ algorithms. We notice that there exists an $\text {OBDD}(\land , \exists )$ algorithm that solves satisfiable and unsatisfiable Tseitin formulas in polynomial time (a standard example of a hard system of linear equations over $\mathbb {F}_2$ ), but we show that there are formulas representing systems of linear equations over $\mathbb {F}_2$ that are hard for $\text {OBDD}(\land , \exists , \text{reordering})$ algorithms. Our hard instances are satisfiable formulas representing systems of linear equations over $\mathbb {F}_2$ that correspond to checksum matrices of error correcting codes.

Ramsey theory without pigeonhole principle and the adversarial Ramsey principle

We develop a general framework for infinite-dimensional Ramsey theory with and without pigeonhole principle, inspired by Gowers’ Ramsey-type theorem for block sequences in Banach spaces and by its exact version proved by Rosendal. In this framework, we prove the adversarial Ramsey principle for Borel sets, a result conjectured by Rosendal that generalizes at the same time his version of Gowers’ theorem and Borel determinacy of games on integers.

О линейных приближающих формах

Обобщенная гипергеометрическая функция определяется суммой степенного ряда, коэффициентами которого являются произведения значений некоторой дробной рациональной функции. Взятые со знаком минус корни числителя и знаменателя этой рациональной функции называются параметрами соответствующей гипергеометрической функции. Для исследования арифметической природы значений гипергеометрических функций и их производных (включая производные по параметру) часто используют метод Зигеля. Соответствующее рассуждение, как правило, начинается с построения функциональной линейной приближающей формы. Если параметры гипергеометрической функции рациональны, то для построения этой формы можно применить принцип Дирихле. При этом построение возможно не только для самих гипергеометрических функций, но и для произведений их степеней. Этим объясняется общность результатов, получаемых таким методом. Если, однако, среди параметров имеются иррациональные числа, то применение принципа Дирихле невозможно, и для проведения соответствующего исследования приходится привлекатьдополнительные соображения.Одним из способов преодоления затруднения, связанного с наличием иррациональных чисел среди параметров гипергеометрической функции является применение эффективного построения линейной приближающей формы, с которой начинается рассуждение. Первоначально эффективные конструкции построения таких приближений появились для функций специального вида (числитель рациональной функции, с помощью которой определяются коэффициенты гипергеометрической функции должен был равняться единице). Изучение свойств этих приближений показало, что они могут оказаться полезными и в случае рациональных параметров: получаемые с помощью эффективных методов количе ственные результаты оказались точнее их аналогов, полученных методом Зигеля. В дальнейшем методы эффективного построения линейной приближающей формы обобщались в различных направлениях.В данной работе предлагается новая эффективная конструкция линейной приближающей формы для случая, когда для гипергеометрических функций рассматриваются также и производные по параметру. Эта конструкция используется для уточнения оценки снизу меры линейной независимости значений соответствующих функций.

Footprints of quantum pigeons

This paper proposes a quantum setup violating the pigeonhole principle. It describes how the counterfactual reasoning of the paradox may be operationally grounded in the analysis of the tiny footprints left in the environment by the pigeons. Identifying the drawbacks of recent experiments of the quantum pigeonhole effect, the authors argue that a definitive experimental violation of the pigeonhole principle is still needed and propose an implementation using modern quantum computing hardware.

Definable combinatorics with dense linear orders

We compute the model-theoretic Grothendieck ring, $$K_0({\mathcal {Q}})$$ , of a dense linear order (DLO) with or without end points, $${\mathcal {Q}}=(Q,<)$$ , as a structure of the signature $$\{<\}$$ , and show that it is a quotient of the polynomial ring over $${\mathbb {Z}}$$ generated by $${\mathbb {N}}_+\times (Q\sqcup \{-\infty \})$$ by an ideal that encodes multiplicative relations of pairs of generators. This ring can be embedded in the polynomial ring over $${\mathbb {Q}}$$ generated by $$Q\sqcup \{-\infty \}$$ . As a corollary we obtain that a DLO satisfies the pigeon hole principle for definable subsets and definable bijections between them—a property that is too strong for many structures.

О линейной независимости значений некоторых гипергеометрических функций над мнимым квадратичным полем

Основная трудность, с которой приходится иметь дело при исследовании арифметической природы значений обобщенных гипергеометрических функций с иррациональными параметрами, состоит в том, что общий наименьший знаменатель нескольких первых коэффициентов соответствующих степенных рядов растет слишком быстро с увеличением числа этих коэффициентов. Последнее обстоятельство делает невозможным использование известного в теории трансцендентных чисел метода Зигеля для проведения упомянутого исследования. Применение названного метода предполагает использование принципа Дирихле для построения функциональной линейной приближающей формы. Это построение является первым этапом длинного и сложного рассуждения, приводящего в конечном итоге к получению требуемого арифметического результата. Попытка применить принцип Дирихле в случае функций с иррациональными параметрами наталкивается на непреодолимые трудности из-за упомянутого выше слишком быстрого роста общего наименьшего знаменателя коэффициентов соответствующих рядов Тейлора. Вследствие этого в случае функций с иррациональными параметрами обычно применяют эффективное построение линейной приближающей формы (или совокупности таких форм при использовании совместных приближений). Коэффициенты построенной формы являются многочленами с алгебраическими коэффициентами. Для общего наименьшего знаменателя этих коэффициентов требуется затем получить приемлемую оценку сверху его абсолютной величины. Известные оценки такого рода нуждаются в некоторых случаях в уточнении. Это уточнение осуществляется с применением теории делимости в квадратичных полях; дополнительно используются сведения о распределении простых чисел в арифметической прогрессии. В настоящей работе рассматривается один из вариантов эффективного построения совместных приближений для гипергеометрической функции общего вида и ее производных. Общий наименьший знаменатель коэффициентов многочленов, входящих в эти приближения, оценивается затем с помощью уточненного варианта соответствующей леммы. Все это позволяет получить новый результат об арифметической природе значений указанной функции в малой по абсолютной величине ненулевой точке мнимого квадратичного поля.

О значениях гипергеометрической функции с параметром из квадратичного поля

Для исследования арифметических свойств значений обобщенных гипергеометрических функций с рациональными параметрами обычно применяют метод Зигеля. Этим методом были получены наиболее общие результаты, относящиеся к упомянутым свойствам.Основной недостаток метода Зигеля состоит в невозможности его применения к гипергеометрическим функциям с иррациональными параметрами. В этой ситуации исследование обычно основывается на эффективной конструкции функциональной приближающей формы (в методе Зигеля существование такой формы доказывается с помощью принципа Дирихле). Построение и исследование приближающей формы является первым шагом в сложном рассуждении, которое ведет к получению арифметического результата.Используя эффективный метод, мы сталкиваемся по крайней мере с двумя проблемами, которые в значительной степени сужают область его применимости. Во-первых, неизвестна более или менее общая конструкция приближающей формы для произведений гипергеометрических функций. Используя метод Зигеля, мы не имеем дела с такой проблемой. По этой причине приходится рассматривать лишь вопросы линейной независимости над тем или иным алгебраическим полем. Выбор этого поля является второй проблемой.Подавляющее большинство опубликованных результатов, относящихся к рассматриваемому кругу задач, имеет дело с мнимым квадратичным полем (или с полем рациональных чисел). Лишь в отдельных случаях удается провести соответствующее исследование для какого-либо другого алгебраического поля.Мы рассматриваем здесь случай вещественного квадратичного поля. С помощью специального технического приема мы устанавливаем линейную независимость значений некоторой гипергеометрической функции с иррациональным параметром над таким полем.

KNOWLEDGE REPRESENTATION IN THE RECOMMENDATION SYSTEM BASED ON THE WHITE BOX PRINCIPLE

The subject matter of the article is the construction of a rating list of goods and services in recommendation systems. The goal is to develop a knowledge representation model in recommendation systems using the white box principle. Such a model contains knowledge about the possible sequences of choosing restrictions on the properties of goods and services according to user preferences. The recommender system should provide a reasonable choice of the object according to the requirements of the user, subject to restrictions on the key properties of this object. Tasks: to structure the process of applying the principles of black and white boxes when building recommendations based on the use of a knowledge base; develop knowledge representations according to the principle of the white box in order to combine a static description of the properties of objects and a description of possible sequences for clarifying the requirements for the properties of these objects. The principles used are: the principle of the white box, which provides an account of requirements that take into account the properties of the constituent elements of the object selected by the consumer. The following results are obtained. The key features of the processes of forming recommendations on the principles of black and white boxes based on the use of the knowledge base are highlighted. The knowledge representation is developed according to the principle of the white box, which allows taking into account both the properties of objects and the process of their selection by the consumer. Conclusions. The scientific novelty of the results is as follows. A model for representing knowledge in a recommendation system according to the principle of a white box is proposed. The declarative aspect of this representation of knowledge is implemented in the form of predicates that define restrictions on the values of the properties of objects offered to the consumer. The procedural aspect is implemented in the form of adapted temporal dependencies that specify the sequence of refinement of time limits. This model combines the advantages of approaches based on the similarity of customer requirements and the similarity of product characteristics, which makes it possible to adjust recommendations online for a cold start situation. With this adjustment, it is advisable to use the similarity of the user selection processes for given requirements for goods.

On Values Approximation of Some Special Type Hypergeometric Functions with Irrational Parameters

In this paper we investigate arithmetic nature of the values of generalized hypergeometric functions and their derivatives. To solve the problem one often makes use of Siegel’s method. The first step in corresponding reasoning is, using the pigeonhole principle, to construct a functional linear approximating form, which has high order of zero at the origin of the coordinates.A hypergeometric function is defined as a sum of a power series whose coefficients are the products of the values of some rational function. Taken with the opposite sign, the zeroes of a numerator and a denominator of this rational function are called parameters of the corresponding generalized hypergeometric function. If the parameters are irrational it is impossible, as a rule, to employ Siegel’s method. In this case one applies the method based on the effective construction of the linear approximating form.Additional difficulties arise in case the rational function numerator involved in the formation of the coefficients of the hypergeometric function under consideration is different from the identical unit. In this situation even the availability of the effective construction of approximating form does not enable achieving an arithmetic result yet. In this paper we consider just such a case. To overcome difficulties arisen here we consider the values of the corresponding hypergeometric function and its derivatives at small points only and impose additional restrictions on parameters of the function.

Pigeons do not jump high

The infinite pigeonhole principle for 2-partitions asserts the existence, for every set A, of an infinite subset of A or of its complement. In this paper, we develop a new notion of forcing enabling a fine analysis of the computability-theoretic features of the pigeonhole principle. We deduce various consequences, such as the existence, for every set A, of an infinite subset of it or its complement of non-high degree. We also prove that every Δ03 set has an infinite low3 solution and give a simpler proof of Liu's theorem that every set has an infinite subset in it or its complement of non-PA degree.

On Linear Independence of Some Functions

To study an arithmetic nature of the values of hyper-geometric functions (and their derivatives including those with respect to parameter), it is common practice to use one either Siegel's method or the method based on the effective construction of the linear approximating form. The main distinction between these methods consists in the mode of construction of the first approximating form. Applying Siegel's method allows us to construct such a form by means of a pigeonhole principle that makes it possible to establish, in certain cases, the algebraic independence of the values of corresponding functions. The Siegel's method can be usually applied just while considering hyper-geometric functions with rational parameters. The effective method has a certain advantage here, for in some cases this method enables us to carry out corresponding investigation also for the functions with irrational parameters. Another advantage of the effective method is that it provides obtaining of high- accuracy quantitative results. By quantitative results one usually implies the estimates of the moduli of the linear forms in the values of corresponding functions. The effective method has made it possible to obtain, in some cases, estimates being precise regarding the height of such forms with calculation of the corresponding constants. A drawback of the effective method is the narrow domain of its applications. The effective construction of the linear approximating form, which initiates reasoning, is always a challenge. So far, an effective construction of the approximating form for the product of powers of hyper-geometric functions (with the rare exceptions) failed.In both aforementioned methods one proves previously linear independence (in Siegel's method, as a rule, algebraic independence,) of the functions under consideration. Such a proof is often considered as an independent result.In this paper, by means of the method especially elaborated for this purpose we establish linear independence of some hyper-geometric functions and their derivatives (including those with respect to parameter) over the field of rational fractions. Subsequently, it will be possible to apply this result to investigate arithmetic properties of the values of such functions. Herewith we mean the application of the effective method to achieve the sufficiently accurate quantitative result.

Random resolution refutations

We study the random resolution refutation system defined in Buss et al. (J Symb Logic 79(2):496–525, 2014). This attempts to capture the notion of a resolution refutation that may make mistakes but is correct most of the time. By proving the equivalence of several different definitions, we show that this concept is robust. On the other hand, if $${{\bf P} \neq {\bf NP}}$$ , then random resolution cannot be polynomially simulated by any proof system in which correctness of proofs is checkable in polynomial time. We prove several upper and lower bounds on the width and size of random resolution refutations of explicit and random unsatisfiable CNF formulas. Our main result is a separation between polylogarithmic width random resolution and quasipolynomial size resolution, which solves the problem stated in Buss et al. (2014). We also prove exponential size lower bounds on random resolution refutations of the pigeonhole principle CNFs, and of a family of CNFs which have polynomial size refutations in constant-depth Frege.

Reductions in PPP

We show several reductions between problems in the complexity class PPP related to the pigeonhole principle, and harboring several intriguing problems relevant to Cryptography. We define a problem related to Minkowski's theorem and another related to Dirichlet's theorem, and we show them to belong to this class. We also show that Minkowski is very expressive, in the sense that all other non-generic problems in PPP considered here can be reduced to it. We conjecture that Minkowski is PPP-complete.

Generalizing the Pigeonhole Principle for Similarity Search in Hamming Space

A distance search in Hamming space finds binary vectors whose Hamming distances are no more than a threshold from a query vector. It is a fundamental problem in many applications, such as image retrieval, near-duplicate Web page detection, and scientific databases. State-of-the-art approaches to Hamming distance search are mainly based on the pigeonhole principle to generate a set of candidates and then verify them. We observe that the constraint by the pigeonhole principle is not always tight and may bring about unnecessary candidates. We also observe that the distribution in real data is often skewed, but most existing solutions adopt a simple equi-width partitioning and allocate the same threshold to all the parts, hence failing to exploit the data skewness to optimize query processing. In this paper, we propose a new form of the pigeonhole principle which allows variable partitioning and threshold allocation. Based on the new principle, we develop a tight constraint of candidates and devise cost-aware methods for partitioning and threshold allocation to optimize query processing. In addition, we extend our methods to answer Hamming distance join queries. We also discuss the application of the pigeonhole principle in set similarity search, a problem that can be converted to Hamming distance search equivalently. Our evaluation on datasets with various data distributions shows the robustness of our solution and its superior query processing performance to the state-of-the-art methods.

The Hilton--Zhao Conjecture is True for Graphs with Maximum Degree 4

A simple graph $G$ is overfull if ${|E(G)|}>\Delta\lfloor|V(G)|/2\rfloor$. By the pigeonhole principle, every overfull graph $G$ has $\chi'(G)>\Delta$. The core of a graph, denoted $G_\Delta$, is t...