The recursive book of recursion

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the second publication in the Perspectives in Logic series, is an almost self-contained introduction to higher recursion theory, in which the reader is only assumed to know the basics of classical recursion theory. The book is divided into four parts: hyperarithmetic sets, metarecursion, α-recursion, and E-recursion. This text is essential reading for all researchers in the field.

As humans, our many levels of language use distinguish us from the rest of the animal world. For many scholars, it is the recursive aspect of human speech that makes it truly human. But linguists cont

This book is a complete account of the predicate transformation calculus semantics of sequential programs, including repetitions, recursive procedures, computational induction and unbounded nondeterminacy. Predicate transformation semantics are the best specification method for the development of correct and well-structured computer programs. The author develops this theory to a greater depth than has been achieved before, and describes it in a way that makes it readily compatible with programming rules for partial and total correctness of repetitions and recursive procedures, supplies new rules for proving incorrectness, and a stronger rule for proving that two programs satisfy the same specifications. Finally, the semantics are extended so that non-terminating programs can be specified as well. This will be essential reading for all computer scientists working in specification and verification of programs.

1988 marked the first centenary of Recursion Theory, since Dedekind's 1888 paper on the nature of number. Now available in paperback, this book is both a comprehensive reference for the subject and a

This book is devoted to recursion in programming, the technique by which the solution to a problem is expressed partly in terms of the solution to a simpler version of the same problem. Ultimately the solution to the simplest version must be given explicitly. In functional programming, recursion has received its full due since it is quite often the only repetitive construct. However, the programming language used here is Pascal and the examples have been chosen accordingly. It makes an interesting contrast with the use of recursion in functional and logic programming. The early chapters consider simple linear recursion using examples such as finding the highest common factor of a pair of numbers, and processing linked lists. Subsequent chapters move up through binary recursion, with examples which include the Towers of Hanoi problem and symbolic differentiation, to general recursion. The book contains well over 100 examples.

This book presents and applies a framework for studying the complexity of algorithms. It is aimed at logicians, computer scientists, mathematicians and philosophers interested in the theory of computation and its foundations, and it is written at a level suitable for non-specialists. Part I provides an accessible introduction to abstract recursion theory and its connection with computability and complexity. This part is suitable for use as a textbook for an advanced undergraduate or graduate course: all the necessary elementary facts from logic, recursion theory, arithmetic and algebra are included. Part II develops and applies an extension of the homomorphism method due jointly to the author and Lou van den Dries for deriving lower complexity bounds for problems in number theory and algebra which (provably or plausibly) restrict all elementary algorithms from specified primitives. The book includes over 250 problems, from simple checks of the reader's understanding, to current open pr

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the tenth publication in the Perspectives in Logic series, Jens E. Fenstad takes an axiomatic approach to present a unified and coherent account of the many and various parts of general recursion theory. The main core of the book gives an account of the general theory of computations. The author then moves on to show how computation theories connect with and unify other parts of general recursion theory. Some mathematical maturity is required of the reader, who is assumed to have some acquaintance with recursion theory. This book is ideal for a second course in the subject.

The proceedings of the Los Angeles Caltech-UCLA 'Cabal Seminar' were originally published in the 1970s and 1980s. Ordinal Definability and Recursion Theory is the third in a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics and discussion of research developments since the publication of the original volumes. Focusing on the subjects of 'HOD and its Local Versions' (Part V) and 'Recursion Theory' (Part VI), each of the two sections is preceded by an introductory survey putting the papers into present context. These four volumes will be a necessary part of the book collection of every set theorist.

The Recursive Mind challenges the commonly held notion that language is what makes us uniquely human. In this compelling book, Michael Corballis argues that what distinguishes us in the animal kingdom

Recursive Block Coding, a new image data compression technique that has its roots in noncausal models for 1d and 2d signals, is the subject of this book. The underlying theory provides a multitude of

The Recursive Mind challenges the commonly held notion that language is what makes us uniquely human. In this compelling book, Michael Corballis argues that what distinguishes us in the animal kingdom

This book presents a thorough development of the modern theory of stochastic approximation or recursive stochastic algorithms for both constrained and unconstrained problems. This second edition is a

The fundamental ideas concerning computation and recursion naturally find their place at the interface between logic and theoretical computer science. The contributions in this book, by leaders in the field, provide a picture of current ideas and methods in the ongoing investigations into the pure mathematical foundations of computability theory. The topics range over computable functions, enumerable sets, degree structures, complexity, subrecursiveness, domains and inductive inference. A number of the articles contain introductory and background material which it is hoped will make this volume an invaluable resource.

This rigorous but brilliantly lucid book presents a self-contained treatment of modern economic dynamics. Stokey, Lucas, and Prescott develop the basic methods of recursive analysis and illustrate th

Recursion theory – now a well-established branch of pure mathematics, having grown rapidly over the last 35 years – deals with the general (abstract) theory of those operations which we conceive as being `computable' by idealized machines. The theory grew out of, and is usually still regarded, as a branch of mathematical logic. This book is a collection of advanced research/survey papers by eminent research workers in the field, based on their lectures given at the Leeds Logic Colloquium 1979. As such it provides an up-to-date view of current ideas and developments in the field of recursion theory as a whole. The individual contributions fit together naturally so as to provide an overview of all the main areas of research in the field. It will therefore be an important and invaluable source for advanced researchers and research students in mathematics and computer science (particularly in Europe, USA and USSR).

What can computers do in principle? What are their inherent theoretical limitations? These are questions to which computer scientists must address themselves. The theoretical framework which enables such questions to be answered has been developed over the last fifty years from the idea of a computable function: intuitively a function whose values can be calculated in an effective or automatic way. This book is an introduction to computability theory (or recursion theory as it is traditionally known to mathematicians). Dr Cutland begins with a mathematical characterisation of computable functions using a simple idealised computer (a register machine); after some comparison with other characterisations, he develops the mathematical theory, including a full discussion of non-computability and undecidability, and the theory of recursive and recursively enumerable sets. The later chapters provide an introduction to more advanced topics such as Gödel's incompleteness theorem, degrees of uns

This is a revised version of the 1984 book of the same name but considerably modified and enlarged to accommodate the developments in recursive estimation and time series analysis that have occurred

「無論你想在程式設計面試中獲得優勢，想創造美麗的數學藝術，還是頑固地非得理解這個概念的有趣特性，這本書都將成為你深入遞迴這個兔子洞的指南（兔子洞裡還有兔子洞）。」遞迴程式設計技術可以產生優雅的程式解決方案，然而更常見的是，它讓程式設計師感到困惑。這並不代表程式設計師可以（或應該）忽略遞迴；儘管遞迴以具有挑戰性而聞名，但它是一個重要的電腦科學主題，並且可以對程式設計本身產生敏銳的洞察力。至少，了解遞迴可以幫助你順利通過程式設計工作面試。如果你是對電腦科學感興趣的學生，遞迴是你要理解許多流行演算法所必須克服的必要障礙。如果你是程式設計訓練營的畢業生或自學而成的程式設計師，並且順利跳過了更具理論性的電腦科學主題，但在「白板程式設計面試」中肯定還是會遇到遞迴問題。如果你是一位經驗豐富的軟體工程師，以前從未接觸過遞迴演算法，你可能會發現遞迴是你知識中一塊尷尬的缺口。❑ 本書目標讀者本書適合那些對遞迴演算法感到恐懼或感興趣的人。對於新手程式設計師或電腦科學系的大一新生來說，遞迴主題就像難以理解的黑魔法一樣。大多數遞迴課程都很難理解，使得這個主題不但讓人覺得挫敗，甚至令人心生恐懼。對於這些讀者來說，這本書的直接解釋和充足的例子應該能幫助他們最終理解這個主題。閱讀本書的唯一先決條件是具備 Python 或 JavaScript 程式語言的基本程式設計經驗，因為各章的程式碼範例使用了這兩種語言。本書的程式已被精簡到核心的基本結構； 如果你知道如何呼叫和建立函數，以及理解全域變數和局部變數之間的區別，那麼你就足以應付這些程式設計範例了。本書使用 Python 和 JavaScript 範例來教授遞迴的基礎知識，揭示了遞迴的教學方法，並闡明了所有遞迴演算法的基本原理。你將學習何時使用遞迴函數（最重要的是，何時不使用它們），如何實現工作面試中經常提到的經典遞迴演算法，以及遞迴技術如何幫助解決有關樹走訪、組合學等等許許多多的問題，以及其他棘手的話題。這個以專案為主的指南包含了完整的、可運行的程式，將能幫助你學習：☑︎ 遞迴函數如何使用呼叫堆疊，這是一種在遞迴課程中幾乎從未討論過的關鍵資料結構☑︎ 「頭尾」與「信仰飛躍」技巧如何簡化遞迴函數的編寫☑︎ 如何使用遞迴為你的檔案系統編寫自訂搜尋腳本、繪製碎形藝術、生成迷宮等☑︎ 優化和記憶如何使遞迴演算法更有效率

This book explores recursive architectures in designing progressive hyperspectral imaging algorithms. In particular, it makes progressive imaging algorithms recursive by introducing the concept of Kal

Mathematical logic grew out of philosophical questions regarding the foundations of mathematics, but logic has now outgrown its philosophical roots, and has become an integral part of mathematics in general. This book is designed for students who plan to specialize in logic, as well as for those who are interested in the applications of logic to other areas of mathematics. Used as a text, it could form the basis of a beginning graduate-level course. There are three main chapters: Set Theory, Model Theory, and Recursion Theory. The Set Theory chapter describes the set-theoretic foundations of all of mathematics, based on the ZFC axioms. It also covers technical results about the Axiom of Choice, well-orderings, and the theory of uncountable cardinals. The Model Theory chapter discusses predicate logic and formal proofs, and covers the Completeness, Compactness, and Löwenheim-Skolem Theorems, elementary submodels, model completeness, and applications to algebra. This chapter also continu

Tara Selter, the heroine of On the Calculation of Volume, has involuntarily stepped off the train of time: in her world, November eighteenth repeats itself endlessly. We meet Tara on her 122nd November 18th: she no longer experiences the changes of days, weeks, months, or seasons. She finds herself in a lonely new reality without being able to explain why: how is it that she wakes every morning into the same day, knowing to the exact second when the blackbird will burst into song and when the rain will begin? Will she ever be able to share her new life with her beloved and now chronically befuddled husband? And on top of her profound isolation and confusion, Tara takes in with pain how slight a difference she makes in the world. (As she puts it: "That's how little the activities of one person matter on the eighteenth of November.")Balle is hypnotic and masterful in her remixing of the endless recursive day, creating curious little folds of time and foreshadowings: her flashbacks light

Macroeconomics offers a modern and comprehensive introduction to the tools and methods that define contemporary graduate training in the field. Written for first-year PhD students and advanced MA/MS students, the book provides a rigorous yet accessible foundation in dynamic optimization, recursive methods, competitive equilibrium, stochastic modelling, and welfare analysis. It emphasizes clarity and intuition, guiding readers through the logic of intertemporal decision-making that underlies modern macro theory. A key feature of the text is its explicit connection to data. Throughout the book, theory is used to interpret empirical patterns, illustrating how macroeconomic models help us understand real-world behavior and policy-relevant facts. Quantitative and computational tools are integrated throughout to show how models are taken to the data. The book develops frameworks with heterogeneity and frictions, including incomplete markets, borrowing constraints, labor-market search frictio

Recursion is an important problem-solving skill that is considered to be one of the most difficult topics to master by CS1/2 students. The main goal of the book is to help students assimilate its fund

Recursion is an important problem-solving skill that is considered to be one of the most difficult topics to master by CS1/2 students. The main goal of the book is to help students assimilate its fund

This book emphasizes three very important concepts: computability, as opposed to recursion or induction; classical computability, i.e., algorithmic functions on certain countable structures in the ori

Classical computable model theory is most naturally concerned with countable domains. There are, however, several methods – some old, some new – that have extended its basic concepts to uncountable structures. Unlike in the classical case, however, no single dominant approach has emerged, and different methods reveal different aspects of the computable content of uncountable mathematics. This book contains introductions to eight major approaches to computable uncountable mathematics: descriptive set theory; infinite time Turing machines; Blum-Shub-Smale computability; Sigma-definability; computability theory on admissible ordinals; E-recursion theory; local computability; and uncountable reverse mathematics. This book provides an authoritative and multifaceted introduction to this exciting new area of research that is still in its early stages. It is ideal as both an introductory text for graduate and advanced undergraduate students and a source of interesting new approaches for resear

This 2001 book presents a unified approach to the foundations of mathematics in the theory of sets, covering both conventional and finitary (constructive) mathematics. It is based on a philosophical, historical and mathematical analysis of the relation between the concepts of 'natural number' and 'set'. This leads to an investigation of the logic of quantification over the universe of sets and a discussion of its role in second order logic, as well as in the analysis of proof by induction and definition by recursion. The subject matter of the book falls on the borderline between philosophy and mathematics, and should appeal to both philosophers and mathematicians with an interest in the foundations of mathematics.

This 2001 book presents a unified approach to the foundations of mathematics in the theory of sets, covering both conventional and finitary (constructive) mathematics. It is based on a philosophical, historical and mathematical analysis of the relation between the concepts of 'natural number' and 'set'. This leads to an investigation of the logic of quantification over the universe of sets and a discussion of its role in second order logic, as well as in the analysis of proof by induction and definition by recursion. The subject matter of the book falls on the borderline between philosophy and mathematics, and should appeal to both philosophers and mathematicians with an interest in the foundations of mathematics.

Augment your knowledge of Python with this entertaining learning guide, which features 100 exercises and programming puzzles and solutions. Python Challenges will help prepare you for your next exam or a job interview, and covers numerous practical topics such as strings, data structures, recursion, arrays, and more.Each topic is addressed in its own separate chapter, starting with an introduction to the basics and followed by 10 to 15 exercises of various degrees of difficulty, helping you to improve your programming skills effectively. Detailed sample solutions, including the algorithms used for all tasks, are included to maximize your understanding of each area. Author Michael Inden also describes alternative solutions and analyzes possible pitfalls and typical errors.Three appendices round out the book: the first covers the Python command line interpreter, which is often helpful for trying out the code snippets and examples in the book, followed by an overview of Pytest for unit te

The lectures in this book are immensely Chomskyan in spirit, recursive-syntactic in nature, and tethered to a framework which takes as the null hypothesis the notion that language is an innate, pre-determined biological system--a system which by definition is multi-complex, human-specific, and analogous to a philosophy highly commensurate of Descartes' great proverbial adage which announces the calling for a 'ghost-in-the-machine'. The book begins with a gradual assessment of the kinds of complex constructs students of syntax need to work-up. Leading to the classic 'Four-Sentences'--each of which bears as a kind of post-mark its own decade of Chomskyan analysis--we trace the origins of generative grammar from the fields of child language acquisition (of the 1960s), to psycholinguistics (of the 1970s), to where we stand today within the Minimalist Program. Various spin-off proposals have been spawned by envisioned analyses which treat syntactic movement as the quintessential human proce

This book focuses on sequences as recursive relations and then transitions to periodic recursive patterns and eventually periodic recursive patterns. The aim of the author is to get students to unders

Loss Models contains a wealth of examples that highlight the real-world applications of the concepts presented, and puts the emphasis on calculations and spreadsheet implementation. With a focus on the loss process, the book reviews the essential quantitative techniques such as random variables, basic distributional quantities, and the recursive method, and discusses techniques for classifying and creating distributions. Parametric, non-parametric, and Bayesian estimation methods are thoroughly covered. In addition, the authors offer practical advice for choosing an appropriate model. This important text:•Presents a revised and updated edition of the classic guide for actuaries that aligns with newly introduced Exams STAM and LTAM•Contains a wealth of exercises taken from previous exams•Includes fresh and additional content related to the material required by the Society of Actuaries (SOA) and the Canadian Institute of Actuaries (CIA)•Offers a solutions manual available for further ins

This book establishes recursive relations concerning kinematics and dynamics of constrained robotic systems. It uses matrix modeling to determine the connectivity conditions on the relative velocities

This book discusses examples of discrete mathematics in school curricula, including in the areas of graph theory, recursion and discrete dynamical systems, combinatorics, logic, game theory, and the m

This book provides a comprehensive account of the role of recursion in language in two distinct but interconnected ways. First, David J. Lobina examines how recursion applies at different levels withi

This book provides a general theoretical background for constructing the recursive Bayesian estimation algorithms for mixture models. It collects the recursive algorithms for estimating dynamic mixtur

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the eleventh publication in the Perspectives in Logic series, Manuel Lerman presents a systematic study of the interaction between local and global degree theory. He introduces the reader to the fascinating combinatorial methods of recursion theory while simultaneously showing how to use these methods to prove global theorems about degrees. The intended reader will have already taken a graduate-level course in recursion theory, but this book will also be accessible to those with some background in mathematical logic and a feeling for computability. It will prove a key reference to enable readers to easily locate facts about degrees and it will direct them to further results.

The focus of this book is on the two major areas of risk theory: aggregate claims distributions and ruin theory. For aggregate claims distributions, detailed descriptions are given of recursive techniques that can be used in the individual and collective risk models. For the collective model, the book discusses different classes of counting distribution, and presents recursion schemes for probability functions and moments. For the individual model, the book illustrates the three most commonly applied techniques. Beyond the classical topics in ruin theory, this new edition features an expanded section covering time of ruin problems, Gerber–Shiu functions, and the application of De Vylder approximations. Suitable for a first course in insurance risk theory and extensively classroom tested, the book is accessible to readers with a solid understanding of basic probability. Numerous worked examples are included and each chapter concludes with exercises for which complete solutions are provi

Infinitary logic, the logic of languages with infinitely long conjunctions, plays an important role in model theory, recursion theory and descriptive set theory. This book is the first modern introduction to the subject in forty years, and will bring students and researchers in all areas of mathematical logic up to the threshold of modern research. The classical topics of back-and-forth systems, model existence techniques, indiscernibles and end extensions are covered before more modern topics are surveyed. Zilber's categoricity theorem for quasiminimal excellent classes is proved and an application is given to covers of multiplicative groups. Infinitary methods are also used to study uncountable models of counterexamples to Vaught's conjecture, and effective aspects of infinitary model theory are reviewed, including an introduction to Montalbán's recent work on spectra of Vaught counterexamples. Self-contained introductions to effective descriptive set theory and hyperarithmetic theor