Theory and Applications of Computability – serie

Intuitively, a sequence such as 101010101010101010… does not seem random, whereas 101101011101010100…, obtained using coin tosses, does. How can we reconcile this intuition with the fact that both are statistically equally likely? What does it mean to say that an individual mathematical object such as a real number is random, or to say that one real is more random than another? And what is the relationship between randomness and computational power. The theory of algorithmic randomness uses tools from computability theory and algorithmic information theory to address questions such as these. Much of this theory can be seen as exploring the relationships between three fundamental concepts: relative computability, as measured by notions such as Turing reducibility; information content, as measured by notions such as Kolmogorov complexity; and randomness of individual objects, as first successfully defined by Martin-Löf. Although algorithmic randomness has been studied for several decades, a dramatic upsurge of interest in the area, starting in the late 1990s, has led to significant advances. This is the first comprehensive treatment of this important field, designed to be both a reference tool for experts and a guide for newcomers. It surveys a broad section of work in the area, and presents most of its major results and techniques in depth. Its organization is designed to guide the reader through this large body of work, providing context for its many concepts and theorems, discussing their significance, and highlighting their interactions. It includes a discussion of effective dimension, which allows us to assign concepts like Hausdorff dimension to individual reals, and a focused but detailed introduction to computability theory. It will be of interest to researchers and students in computability theory, algorithmic information theory, and theoretical computer science.

Intuitively, a sequence such as 101010101010101010… does not seem random, whereas 101101011101010100…, obtained using coin tosses, does. How can we reconcile this intuition with the fact that both are statistically equally likely? What does it mean to say that an individual mathematical object such as a real number is random, or to say that one real is more random than another? And what is the relationship between randomness and computational power. The theory of algorithmic randomness uses tools from computability theory and algorithmic information theory to address questions such as these. Much of this theory can be seen as exploring the relationships between three fundamental concepts: relative computability, as measured by notions such as Turing reducibility; information content, as measured by notions such as Kolmogorov complexity; and randomness of individual objects, as first successfully defined by Martin-Löf. Although algorithmic randomness has been studied for several decades, a dramatic upsurge of interest in the area, starting in the late 1990s, has led to significant advances. This is the first comprehensive treatment of this important field, designed to be both a reference tool for experts and a guide for newcomers. It surveys a broad section of work in the area, and presents most of its major results and techniques in depth. Its organization is designed to guide the reader through this large body of work, providing context for its many concepts and theorems, discussing their significance, and highlighting their interactions. It includes a discussion of effective dimension, which allows us to assign concepts like Hausdorff dimension to individual reals, and a focused but detailed introduction to computability theory. It will be of interest to researchers and students in computability theory, algorithmic information theory, and theoretical computer science.

Computable analysis is the modern theory of computability and complexity in analysis that arose out of Turing's seminal work in the 1930s. This was motivated by questions such as: which real numbers and real number functions are computable, and which mathematical tasks in analysis can be solved by algorithmic means?Nowadays this theory has many different facets that embrace topics from computability theory, algorithmic randomness, computational complexity, dynamical systems, fractals, and analog computers, up to logic, descriptive set theory, constructivism, and reverse mathematics. In recent decades computable analysis has invaded many branches of analysis, and researchers have studied computability and complexity questions arising from real and complex analysis, functional analysis, and the theory of differential equations, up to (geometric) measure theory and topology.This handbook represents the first coherent cross-section through most active research topics on the more theoretical side of the field. It contains 11 chapters grouped into parts on computability in analysis; complexity, dynamics, and randomness; and constructivity, logic, and descriptive complexity. All chapters are written by leading experts working at the cutting edge of the respective topic. Researchers and graduate students in the areas of theoretical computer science and mathematical logic will find systematic introductions into many branches of computable analysis, and a wealth of information and references that will help them to navigate the modern research literature in this field.

Computable analysis is the modern theory of computability and complexity in analysis that arose out of Turing's seminal work in the 1930s. This was motivated by questions such as: which real numbers and real number functions are computable, and which mathematical tasks in analysis can be solved by algorithmic means?Nowadays this theory has many different facets that embrace topics from computability theory, algorithmic randomness, computational complexity, dynamical systems, fractals, and analog computers, up to logic, descriptive set theory, constructivism, and reverse mathematics. In recent decades computable analysis has invaded many branches of analysis, and researchers have studied computability and complexity questions arising from real and complex analysis, functional analysis, and the theory of differential equations, up to (geometric) measure theory and topology.This handbook represents the first coherent cross-section through most active research topics on the more theoretical side of the field. It contains 11 chapters grouped into parts on computability in analysis; complexity, dynamics, and randomness; and constructivity, logic, and descriptive complexity. All chapters are written by leading experts working at the cutting edge of the respective topic. Researchers and graduate students in the areas of theoretical computer science and mathematical logic will find systematic introductions into many branches of computable analysis, and a wealth of information and references that will help them to navigate the modern research literature in this field.

Reverse mathematics studies the complexity of proving mathematical theorems and solving mathematical problems. Typical questions include: Can we prove this result without first proving that one? Can a computer solve this problem? A highly active part of mathematical logic and computability theory, the subject offers beautiful results as well as significant foundational insights.This text provides a modern treatment of reverse mathematics that combines computability theoretic reductions and proofs in formal arithmetic to measure the complexity of theorems and problems from all areas of mathematics. It includes detailed introductions to techniques from computable mathematics, Weihrauch style analysis, and other parts of computability that have become integral to research in the field. Topics and features:Provides a complete introduction to reverse mathematics, including necessary background from computability theory, second order arithmetic, forcing, induction, and model constructionOffers a comprehensive treatment of the reverse mathematics of combinatorics, including Ramsey's theorem, Hindman's theorem, and many other resultsProvides central results and methods from the past two decades, appearing in book form for the first time and including preservation techniques and applications of probabilistic argumentsIncludes a large number of exercises of varying levels of difficulty, supplementing each chapterThe text will be accessible to students with a standard first year course in mathematical logic. It will also be a useful reference for researchers in reverse mathematics, computability theory, proof theory, and related areas.Damir D. Dzhafarov is an Associate Professor of Mathematics at the University of Connecticut, CT, USA. Carl Mummert is a Professor of Computer and Information Technology at Marshall University, WV, USA.

Reverse mathematics studies the complexity of proving mathematical theorems and solving mathematical problems. Typical questions include: Can we prove this result without first proving that one? Can a computer solve this problem? A highly active part of mathematical logic and computability theory, the subject offers beautiful results as well as significant foundational insights.This text provides a modern treatment of reverse mathematics that combines computability theoretic reductions and proofs in formal arithmetic to measure the complexity of theorems and problems from all areas of mathematics. It includes detailed introductions to techniques from computable mathematics, Weihrauch style analysis, and other parts of computability that have become integral to research in the field. Topics and features:Provides a complete introduction to reverse mathematics, including necessary background from computability theory, second order arithmetic, forcing, induction, and model constructionOffers a comprehensive treatment of the reverse mathematics of combinatorics, including Ramsey's theorem, Hindman's theorem, and many other resultsProvides central results and methods from the past two decades, appearing in book form for the first time and including preservation techniques and applications of probabilistic argumentsIncludes a large number of exercises of varying levels of difficulty, supplementing each chapterThe text will be accessible to students with a standard first year course in mathematical logic. It will also be a useful reference for researchers in reverse mathematics, computability theory, proof theory, and related areas.Damir D. Dzhafarov is an Associate Professor of Mathematics at the University of Connecticut, CT, USA. Carl Mummert is a Professor of Computer and Information Technology at Marshall University, WV, USA.

It is not always clear what computer programs mean in the various languages in which they can be written, yet a picture can be worth 1000 words, a diagram 1000 instructions.In this unique textbook/reference, programs are drawn as string diagrams in the language of categories, which display a universal syntax of mathematics (Computer scientists use them to analyze the program semantics; programmers to display the syntax of computations). Here, the string-diagrammatic depictions of computations are construed as programs in a single-instruction programming language.  Such programs as diagrams show how functions are packed in boxes and tied by strings. Readers familiar with categories will learn about the foundations of computability; readers familiar with computability gain access to category theory. Additionally, readers familiar with both are offered many opportunities to improve the approach.Topics and features:Delivers a ‘crash’ diagram-based course in theory of computationUses single-instruction diagrammatic programming languageOffers a practical introduction into categories and string diagrams as computational toolsReveals how computability is programmability, rather than an ‘ether’ permeating computers Provides a categorical model of intensional computation is unique up to isomorphismServes as a stepping stone into research of computable categoriesIn addition to its early chapters introducing computability for beginners, this flexible textbook/resource also contains both middle chapters that expand for suitability to a graduate course as well as final chapters opening up new research. Dusko Pavlovic is a professor at the Department of Information and Computer Sciences at the University of Hawaii at Manoa, and by courtesy at the Department of Mathematics and the College of Engineering. He completed this book as an Excellence Professor at Radboud University in Nijmegen, The Netherlands.

It is not always clear what computer programs mean in the various languages in which they can be written, yet a picture can be worth 1000 words, a diagram 1000 instructions.In this unique textbook/reference, programs are drawn as string diagrams in the language of categories, which display a universal syntax of mathematics (Computer scientists use them to analyze the program semantics; programmers to display the syntax of computations). Here, the string-diagrammatic depictions of computations are construed as programs in a single-instruction programming language.  Such programs as diagrams show how functions are packed in boxes and tied by strings. Readers familiar with categories will learn about the foundations of computability; readers familiar with computability gain access to category theory. Additionally, readers familiar with both are offered many opportunities to improve the approach.Topics and features:Delivers a ‘crash’ diagram-based course in theory of computationUses single-instruction diagrammatic programming languageOffers a practical introduction into categories and string diagrams as computational toolsReveals how computability is programmability, rather than an ‘ether’ permeating computers Provides a categorical model of intensional computation is unique up to isomorphismServes as a stepping stone into research of computable categoriesIn addition to its early chapters introducing computability for beginners, this flexible textbook/resource also contains both middle chapters that expand for suitability to a graduate course as well as final chapters opening up new research. Dusko Pavlovic is a professor at the Department of Information and Computer Sciences at the University of Hawaii at Manoa, and by courtesy at the Department of Mathematics and the College of Engineering. He completed this book as an Excellence Professor at Radboud University in Nijmegen, The Netherlands.

The subject of this monograph are restarting automata. The definition of these automata is motivated by the linguistic technique of analysis by reduction. This technique, which can be used to analyze sentences in natural languages with a rather free word-order like Czech (or Latin or German), consists of a sequence of step-by-step simplifications of a given sentence. Each of these simplifications is realized by a single reduction operation, which consists of either the deletion of one or several words from that sentence or the replacement of a (possibly discontinuous) substring of that sentence by a shorter substring. It is required that each application of such a reduction operation must preserve the syntactical correctness of the sentence. Accordingly, a restarting automaton consists of a finite-state control, a flexible tape that initially contains the input, and a read-write window of a fixed finite size that works on that tape. The first type of restarting automaton was presented at the international conference FCT in 1995. This type was required to restart as soon as it executes a rewrite operation, that is, the window jumps back to the left end of the tape and the finite-state control is reset to the initial state. Moreover, each rewrite operation simply deletes one or more letters from the contents of the read-write window. Subsequently, many different variants of the restarting automaton have been defined and studied. In particular, proper length-reducing rewrite operations have replaced the original delete steps, additional non-input letters, called auxiliary letters, have been added to the alphabet, and the original combined rewrite/restart operation has been split into a rewrite operation and a separate restart operation. Thus, the restarting automaton is no longer just a particular type of automaton, but it has evolved into a whole family of various types of automata that are specified through several parameters. The objective of the current monograph is to collect the many results that have been obtained on the various types of restarting automata in one place and to present them in a uniform and systematic way. In particular, the influence of the various parameters on the expressive capacity of the resulting types of restarting automata is studied in detail. Other topics include the descriptional complexity and inductive inference of certain types of restarting automata, cooperating distributed and parallel communicating systems of restarting automata, restarting automata with output, weighted restarting automata, and restarting automata for picture languages and tree languages. This monograph may serve as a book of reference for researchers working in formal language and automata theory, as a guide to the literature on restarting automata, and as a text book for an advanced undergraduate or graduate course in formal language and automata theory.

The subject of this monograph are restarting automata. The definition of these automata is motivated by the linguistic technique of analysis by reduction. This technique, which can be used to analyze sentences in natural languages with a rather free word-order like Czech (or Latin or German), consists of a sequence of step-by-step simplifications of a given sentence. Each of these simplifications is realized by a single reduction operation, which consists of either the deletion of one or several words from that sentence or the replacement of a (possibly discontinuous) substring of that sentence by a shorter substring. It is required that each application of such a reduction operation must preserve the syntactical correctness of the sentence. Accordingly, a restarting automaton consists of a finite-state control, a flexible tape that initially contains the input, and a read-write window of a fixed finite size that works on that tape. The first type of restarting automaton was presented at the international conference FCT in 1995. This type was required to restart as soon as it executes a rewrite operation, that is, the window jumps back to the left end of the tape and the finite-state control is reset to the initial state. Moreover, each rewrite operation simply deletes one or more letters from the contents of the read-write window. Subsequently, many different variants of the restarting automaton have been defined and studied. In particular, proper length-reducing rewrite operations have replaced the original delete steps, additional non-input letters, called auxiliary letters, have been added to the alphabet, and the original combined rewrite/restart operation has been split into a rewrite operation and a separate restart operation. Thus, the restarting automaton is no longer just a particular type of automaton, but it has evolved into a whole family of various types of automata that are specified through several parameters. The objective of the current monograph is to collect the many results that have been obtained on the various types of restarting automata in one place and to present them in a uniform and systematic way. In particular, the influence of the various parameters on the expressive capacity of the resulting types of restarting automata is studied in detail. Other topics include the descriptional complexity and inductive inference of certain types of restarting automata, cooperating distributed and parallel communicating systems of restarting automata, restarting automata with output, weighted restarting automata, and restarting automata for picture languages and tree languages. This monograph may serve as a book of reference for researchers working in formal language and automata theory, as a guide to the literature on restarting automata, and as a text book for an advanced undergraduate or graduate course in formal language and automata theory.

The subject of this monograph are restarting automata. The definition of these automata is motivated by the linguistic technique of analysis by reduction. This technique, which can be used to analyze sentences in natural languages with a rather free word-order like Czech (or Latin or German), consists of a sequence of step-by-step simplifications of a given sentence. Each of these simplifications is realized by a single reduction operation, which consists of either the deletion of one or several words from that sentence or the replacement of a (possibly discontinuous) substring of that sentence by a shorter substring. It is required that each application of such a reduction operation must preserve the syntactical correctness of the sentence. Accordingly, a restarting automaton consists of a finite-state control, a flexible tape that initially contains the input, and a read-write window of a fixed finite size that works on that tape. The first type of restarting automaton was presented at the international conference FCT in 1995. This type was required to restart as soon as it executes a rewrite operation, that is, the window jumps back to the left end of the tape and the finite-state control is reset to the initial state. Moreover, each rewrite operation simply deletes one or more letters from the contents of the read-write window. Subsequently, many different variants of the restarting automaton have been defined and studied. In particular, proper length-reducing rewrite operations have replaced the original delete steps, additional non-input letters, called auxiliary letters, have been added to the alphabet, and the original combined rewrite/restart operation has been split into a rewrite operation and a separate restart operation. Thus, the restarting automaton is no longer just a particular type of automaton, but it has evolved into a whole family of various types of automata that are specified through several parameters. The objective of the current monograph is to collect the many results that have been obtained on the various types of restarting automata in one place and to present them in a uniform and systematic way. In particular, the influence of the various parameters on the expressive capacity of the resulting types of restarting automata is studied in detail. Other topics include the descriptional complexity and inductive inference of certain types of restarting automata, cooperating distributed and parallel communicating systems of restarting automata, restarting automata with output, weighted restarting automata, and restarting automata for picture languages and tree languages. This monograph may serve as a book of reference for researchers working in formal language and automata theory, as a guide to the literature on restarting automata, and as a text book for an advanced undergraduate or graduate course in formal language and automata theory.

The subject of this monograph are restarting automata. The definition of these automata is motivated by the linguistic technique of analysis by reduction. This technique, which can be used to analyze sentences in natural languages with a rather free word-order like Czech (or Latin or German), consists of a sequence of step-by-step simplifications of a given sentence. Each of these simplifications is realized by a single reduction operation, which consists of either the deletion of one or several words from that sentence or the replacement of a (possibly discontinuous) substring of that sentence by a shorter substring. It is required that each application of such a reduction operation must preserve the syntactical correctness of the sentence. Accordingly, a restarting automaton consists of a finite-state control, a flexible tape that initially contains the input, and a read-write window of a fixed finite size that works on that tape. The first type of restarting automaton was presented at the international conference FCT in 1995. This type was required to restart as soon as it executes a rewrite operation, that is, the window jumps back to the left end of the tape and the finite-state control is reset to the initial state. Moreover, each rewrite operation simply deletes one or more letters from the contents of the read-write window. Subsequently, many different variants of the restarting automaton have been defined and studied. In particular, proper length-reducing rewrite operations have replaced the original delete steps, additional non-input letters, called auxiliary letters, have been added to the alphabet, and the original combined rewrite/restart operation has been split into a rewrite operation and a separate restart operation. Thus, the restarting automaton is no longer just a particular type of automaton, but it has evolved into a whole family of various types of automata that are specified through several parameters. The objective of the current monograph is to collect the many results that have been obtained on the various types of restarting automata in one place and to present them in a uniform and systematic way. In particular, the influence of the various parameters on the expressive capacity of the resulting types of restarting automata is studied in detail. Other topics include the descriptional complexity and inductive inference of certain types of restarting automata, cooperating distributed and parallel communicating systems of restarting automata, restarting automata with output, weighted restarting automata, and restarting automata for picture languages and tree languages. This monograph may serve as a book of reference for researchers working in formal language and automata theory, as a guide to the literature on restarting automata, and as a text book for an advanced undergraduate or graduate course in formal language and automata theory.

This is the first book which gives a unified theory for countable and uncountable computable structures.The work treats computable linear orderings, graphs, groups and Boolean algebras unified with computable metric and Banach spaces, profinite groups, and the like. Further, it provides the first account of these that exploits effective versions of dualities, such as Stone and Pontryagin dualities. The themes are effective classification and enumeration.Topics and features:·       Delivers a self-contained, gentle introduction to priority arguments, directly applying them in algebraic contexts·       Includes extensive exercises that both cement and amplify the materials·       Provides complete introduction to the basics of computable analysis, particularly in the context of computable structures·       Offers the first monograph treatment of computable Polish groups, effective profinite groups via Stone duality, and effective abelian groups via Pontryagin duality·       Presents the first book treatment of Friedberg enumerations of structuresThis unique volume is aimed at graduate students and researchers in computability theory, as well as mathematicians seeking to understand the algorithmic content of structure theory. Being self-contained, it provides ample opportunity for self-study.

Choquet capacities, which provide the weighting mechanism for the Choquet and other fuzzy integrals, model synergistic and antagonistic interactions between variables by assigning value to all subsets rather than individual inputs.  While the flexibility of capacities (also referred to as fuzzy measures and cooperative games) comes at the expense of an exponentially increasing number of parameters, the ability to explain their behavior using various value and interaction indices makes them appealing for applications requiring transparency and interpretability. As well as a number of useful indices that in some way capture the extent to which positive and negative interactions occur, significant progress has been made in addressing the scalability issues that arise in applications.  This book provides a detailed overview of the background concepts relating to capacities and their role in fuzzy integration and aggregation, then presents specialised chapters on most recent results in learning, random sampling and optimization that involve Choquet capacities.Topics and features:·       Fundamentals of Choquet capacities (fuzzy measures) and their use in modeling importance and interaction between variables·       Definitions, properties and mappings between alternative representations that allow capacities and fuzzy integrals to be interpreted and applied in different settings·       Various simplification assumptions, from k-additive, p-symmetric and l-measures to more recent concepts such as k-interactive and hierarchical frameworks·       Capacity learning formulations that allow the diverse types to be elicited from datasets or according to user-specified requirements·       Recent findings related to random sampling and optimisation with Choquet integral objectivesThis book includes illustrative examples and guidance for implementation, including an appendix detailing functions found in the pyfmtools software library. It aims to be useful for practitioners and researchers in decision and data-driven fields, or those who wish to apply these emerging tools to new problems. The authors are all affiliated with the School of Information Technology at Deakin University, Australia. Gleb Beliakov is a professor, Simon James is an Associate Professor, and Jian-Zhang Wu is a Research Fellow.

This book questions the relevance of computation to the physical universe. Our theories deliver computational descriptions, but the gaps and discontinuities in our grasp suggest a need for continued discourse between researchers from different disciplines, and this book is unique in its focus on the mathematical theory of incomputability and its relevance for the real world. The core of the book consists of thirteen chapters in five parts on extended models of computation; the search for natural examples of incomputable objects; mind, matter, and computation; the nature of information, complexity, and randomness; and the mathematics of emergence and morphogenesis.This book will be of interest to researchers in the areas of theoretical computer science, mathematical logic, and philosophy.

This book questions the relevance of computation to the physical universe. Our theories deliver computational descriptions, but the gaps and discontinuities in our grasp suggest a need for continued discourse between researchers from different disciplines, and this book is unique in its focus on the mathematical theory of incomputability and its relevance for the real world. The core of the book consists of thirteen chapters in five parts on extended models of computation; the search for natural examples of incomputable objects; mind, matter, and computation; the nature of information, complexity, and randomness; and the mathematics of emergence and morphogenesis.This book will be of interest to researchers in the areas of theoretical computer science, mathematical logic, and philosophy.

The theory presented in this book is developed constructively, is based on a few axioms encapsulating the notion of objects (points and sets) being apart, and encompasses both point-set topology and the theory of uniform spaces. While the classical-logic-based theory of proximity spaces provides some guidance for the theory of apartness, the notion of nearness/proximity does not embody enough algorithmic information for a deep constructive development. The use of constructive (intuitionistic) logic in this book requires much more technical ingenuity than one finds in classical proximity theory -- algorithmic information does not come cheaply -- but it often reveals distinctions that are rendered invisible by classical logic.In the first chapter the authors outline informal constructive logic and set theory, and, briefly, the basic notions and notations for metric and topological spaces. In the second they introduce axioms for a point-set apartness and then explore some of the consequences of those axioms. In particular, they examine a natural topology associated with an apartness space, and relations between various types of continuity of mappings. In the third chapter the authors extend the notion of point-set (pre-)apartness axiomatically to one of (pre-)apartness between subsets of an inhabited set. They then provide axioms for a quasiuniform space, perhaps the most important type of set-set apartness space. Quasiuniform spaces play a major role in the remainder of the chapter, which covers such topics as the connection between uniform and strong continuity (arguably the most technically difficult part of the book), apartness and convergence in function spaces, types of completeness, and neat compactness. Each chapter has a Notes section, in which are found comments on the definitions, results, and proofs, as well as occasional pointers to future work. The book ends with a Postlude that refers to other constructive approaches to topology, with emphasis on the relation between apartness spaces and formal topology.Largely an exposition of the authors' own research, this is the first book dealing with the apartness approach to constructive topology, and is a valuable addition to the literature on constructive mathematics and on topology in computer science. It is aimed at graduate students and advanced researchers in theoretical computer science, mathematics, and logic who are interested in constructive/algorithmic aspects of topology.

The theory presented in this book is developed constructively, is based on a few axioms encapsulating the notion of objects (points and sets) being apart, and encompasses both point-set topology and the theory of uniform spaces. While the classical-logic-based theory of proximity spaces provides some guidance for the theory of apartness, the notion of nearness/proximity does not embody enough algorithmic information for a deep constructive development. The use of constructive (intuitionistic) logic in this book requires much more technical ingenuity than one finds in classical proximity theory -- algorithmic information does not come cheaply -- but it often reveals distinctions that are rendered invisible by classical logic.In the first chapter the authors outline informal constructive logic and set theory, and, briefly, the basic notions and notations for metric and topological spaces. In the second they introduce axioms for a point-set apartness and then explore some of the consequences of those axioms. In particular, they examine a natural topology associated with an apartness space, and relations between various types of continuity of mappings. In the third chapter the authors extend the notion of point-set (pre-)apartness axiomatically to one of (pre-)apartness between subsets of an inhabited set. They then provide axioms for a quasiuniform space, perhaps the most important type of set-set apartness space. Quasiuniform spaces play a major role in the remainder of the chapter, which covers such topics as the connection between uniform and strong continuity (arguably the most technically difficult part of the book), apartness and convergence in function spaces, types of completeness, and neat compactness. Each chapter has a Notes section, in which are found comments on the definitions, results, and proofs, as well as occasional pointers to future work. The book ends with a Postlude that refers to other constructive approaches to topology, with emphasis on the relation between apartness spaces and formal topology.Largely an exposition of the authors' own research, this is the first book dealing with the apartness approach to constructive topology, and is a valuable addition to the literature on constructive mathematics and on topology in computer science. It is aimed at graduate students and advanced researchers in theoretical computer science, mathematics, and logic who are interested in constructive/algorithmic aspects of topology.

Turing's famous 1936 paper introduced a formal definition of a computing machine, a Turing machine. This model led to both the development of actual computers and to computability theory, the study of what machines can and cannot compute. This book presents classical computability theory from Turing and Post to current results and methods, and their use in studying the information content of algebraic structures, models, and their relation to Peano arithmetic. The author presents the subject as an art to be practiced, and an art in the aesthetic sense of inherent beauty which all mathematicians recognize in their subject. Part I gives a thorough development of the foundations of computability, from the definition of Turing machines up to finite injury priority arguments. Key topics include relative computability, and computably enumerable sets, those which can be effectively listed but not necessarily effectively decided, such as the theorems of Peano arithmetic. Part IIincludes the study of computably open and closed sets of reals and basis and nonbasis theorems for effectively closed sets. Part III covers minimal Turing degrees. Part IV is an introduction to games and their use in proving theorems. Finally, Part V offers a short history of computability theory.The author has honed the content over decades according to feedback from students, lecturers, and researchers around the world. Most chapters include exercises, and the material is carefully structured according to importance and difficulty. The book is suitable for advanced undergraduate and graduate students in computer science and mathematics and researchers engaged with computability and mathematical logic.

This book offers a self-contained exposition of the theory of computability in a higher-order context, where 'computable operations' may themselves be passed as arguments to other computable operations. The subject originated in the 1950s with the work of Kleene, Kreisel and others, and has since expanded in many different directions under the influence of workers from both mathematical logic and computer science. The ideas of higher-order computability have proved valuable both for elucidating the constructive content of logical systems, and for investigating the expressive power of various higher-order programming languages. In contrast to the well-known situation for first-order functions, it turns out that at higher types there are several different notions of computability competing for our attention, and each of these has given rise to its own strand of research. In this book, the authors offer an integrated treatment that draws together many of these strands within a unifying framework, revealing not only the range of possible computability concepts but the relationships between them. The book will serve as an ideal introduction to the field for beginning graduate students, as well as a reference for advanced researchers

This book offers a self-contained exposition of the theory of computability in a higher-order context, where 'computable operations' may themselves be passed as arguments to other computable operations. The subject originated in the 1950s with the work of Kleene, Kreisel and others, and has since expanded in many different directions under the influence of workers from both mathematical logic and computer science. The ideas of higher-order computability have proved valuable both for elucidating the constructive content of logical systems, and for investigating the expressive power of various higher-order programming languages. In contrast to the well-known situation for first-order functions, it turns out that at higher types there are several different notions of computability competing for our attention, and each of these has given rise to its own strand of research. In this book, the authors offer an integrated treatment that draws together many of these strands within a unifying framework, revealing not only the range of possible computability concepts but the relationships between them. The book will serve as an ideal introduction to the field for beginning graduate students, as well as a reference for advanced researchers