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| recursion theory for metamathematics: Recursion Theory for Metamathematics Raymond M. Smullyan, 1993-01-28 This work is a sequel to the author's Gödel's Incompleteness Theorems, though it can be read independently by anyone familiar with Gödel's incompleteness theorem for Peano arithmetic. The book deals mainly with those aspects of recursion theory that have applications to the metamathematics of incompleteness, undecidability, and related topics. It is both an introduction to the theory and a presentation of new results in the field. |
| recursion theory for metamathematics: Higher Recursion Theory Gerald E. Sacks, 2017-03-02 This almost self-contained introduction to higher recursion theory is essential reading for all researchers in the field. |
| recursion theory for metamathematics: Diagonalization and Self-reference Raymond M. Smullyan, 1994 The main purpose of this book is to present a unified treatment of fixed points as they occur in Godel's incompleteness proofs, recursion theory, combinatory logic, semantics, and metamathematics. The book provides a survey of introductory material and a summary of recent research. The firstchapters are of an introductory nature and consist mainly of exercises with solutions given to most of them. |
| recursion theory for metamathematics: Introduction to Metamathematics S.C. Kleene, 1980-01-01 Stephen Cole Kleene was one of the greatest logicians of the twentieth century and this book is the influential textbook he wrote to teach the subject to the next generation. It was first published in 1952, some twenty years after the publication of Gadel's paper on the incompleteness of arithmetic, which marked, if not the beginning of modern logic, at least a turning point after which nothing was ever the same. Kleene was an important figure in logic, and lived a long full life of scholarship and teaching. The 1930s was a time of creativity and ferment in the subject, when the notion of computable moved from the realm of philosophical speculation to the realm of science. This was accomplished by the work of Kurt Gade1, Alan Turing, and Alonzo Church, who gave three apparently different precise definitions of computable. When they all turned out to be equivalent, there was a collective realization that this was indeed the right notion. Kleene played a key role in this process. One could say that he was there at the beginning of modern logic. He showed the equivalence of lambda calculus with Turing machines and with Gadel's recursion equations, and developed the modern machinery of partial recursive functions. This textbook played an invaluable part in educating the logicians of the present. It played an important role in their own logical education. |
| recursion theory for metamathematics: Metamath: A Computer Language for Mathematical Proofs Norman Megill, David A. Wheeler, 2019 Metamath is a computer language and an associated computer program for archiving, verifying, and studying mathematical proofs. The Metamath language is simple and robust, with an almost total absence of hard-wired syntax, and we believe that it provides about the simplest possible framework that allows essentially all of mathematics to be expressed with absolute rigor. While simple, it is also powerful; the Metamath Proof Explorer (MPE) database has over 23,000 proven theorems and is one of the top systems in the Formalizing 100 Theorems challenge. This book explains the Metamath language and program, with specific emphasis on the fundamentals of the MPE database. |
| recursion theory for metamathematics: Inexhaustibility Torkel Franzén, 2017-03-30 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 sixteenth publication in the Lecture Notes in Logic series, gives a sustained presentation of a particular view of the topic of Gödelian extensions of theories. It presents the basic material in predicate logic, set theory and recursion theory, leading to a proof of Gödel's incompleteness theorems. The inexhaustibility of mathematics is treated based on the concept of transfinite progressions of theories as conceived by Turing and Feferman. All concepts and results are introduced as needed, making the presentation self-contained and thorough. Philosophers, mathematicians and others will find the book helpful in acquiring a basic grasp of the philosophical and logical results and issues. |
| recursion theory for metamathematics: Metamathematics of First-Order Arithmetic Petr Hajek, Pavel Pudlak, 1998-03-17 People have always been interested in numbers, in particular the natural numbers. Of course, we all have an intuitive notion of what these numbers are. In the late 19th century mathematicians, such as Grassmann, Frege and Dedekind, gave definitions for these familiar objects. Since then the development of axiomatic schemes for arithmetic have played a fundamental role in a logical understanding of mathematics. There has been a need for some time for a monograph on the metamathematics of first-order arithmetic. The aim of the book by Hajek and Pudlak is to cover some of the most important results in the study of a first order theory of the natural numbers, called Peano arithmetic and its fragments (subtheories). The field is quite active, but only a small part of the results has been covered in monographs. This book is divided into three parts. In Part A, the authors develop parts of mathematics and logic in various fragments. Part B is devoted to incompleteness. Part C studies systems that have the induction schema restricted to bounded formulas (Bounded Arithmetic). One highlight of this section is the relation of provability to computational complexity. The study of formal systems for arithmetic is a prerequisite for understanding results such as Gödel's theorems. This book is intended for those who want to learn more about such systems and who want to follow current research in the field. The book contains a bibliography of approximately 1000 items. |
| recursion theory for metamathematics: Recursive Functions and Metamathematics Roman Murawski, 2013-03-14 Recursive Functions and Metamathematics deals with problems of the completeness and decidability of theories, using as its main tool the theory of recursive functions. This theory is first introduced and discussed. Then Gödel's incompleteness theorems are presented, together with generalizations, strengthenings, and the decidability theory. The book also considers the historical and philosophical context of these issues and their philosophical and methodological consequences. Recent results and trends have been included, such as undecidable sentences of mathematical content, reverse mathematics. All the main results are presented in detail. The book is self-contained and presupposes only some knowledge of elementary mathematical logic. There is an extensive bibliography. Readership: Scholars and advanced students of logic, mathematics, philosophy of science. |
| recursion theory for metamathematics: Sets, Models and Proofs Ieke Moerdijk, Jaap van Oosten, 2018-11-23 This textbook provides a concise and self-contained introduction to mathematical logic, with a focus on the fundamental topics in first-order logic and model theory. Including examples from several areas of mathematics (algebra, linear algebra and analysis), the book illustrates the relevance and usefulness of logic in the study of these subject areas. The authors start with an exposition of set theory and the axiom of choice as used in everyday mathematics. Proceeding at a gentle pace, they go on to present some of the first important results in model theory, followed by a careful exposition of Gentzen-style natural deduction and a detailed proof of Gödel’s completeness theorem for first-order logic. The book then explores the formal axiom system of Zermelo and Fraenkel before concluding with an extensive list of suggestions for further study. The present volume is primarily aimed at mathematics students who are already familiar with basic analysis, algebra and linear algebra. It contains numerous exercises of varying difficulty and can be used for self-study, though it is ideally suited as a text for a one-semester university course in the second or third year. |
| recursion theory for metamathematics: Forever Undecided Raymond M. Smullyan, 2012-07-04 Forever Undecided is the most challenging yet of Raymond Smullyan’s puzzle collections. It is, at the same time, an introduction—ingenious, instructive, entertaining—to Gödel’s famous theorems. With all the wit and charm that have delighted readers of his previous books, Smullyan transports us once again to that magical island where knights always tell the truth and knaves always lie. Here we meet a new and amazing array of characters, visitors to the island, seeking to determine the natives’ identities. Among them: the census-taker McGregor; a philosophical-logician in search of his flighty bird-wife, Oona; and a regiment of Reasoners (timid ones, normal ones, conceited, modest, and peculiar ones) armed with the rules of propositional logic (if X is true, then so is Y). By following the Reasoners through brain-tingling exercises and adventures—including journeys into the “other possible worlds” of Kripke semantics—even the most illogical of us come to understand Gödel’s two great theorems on incompleteness and undecidability, some of their philosophical and mathematical implications, and why we, like Gödel himself, must remain Forever Undecided! |
| recursion theory for metamathematics: Theory of Formal Systems Raymond M. Smullyan, 1961 This book serves both as a completely self-contained introduction and as an exposition of new results in the field of recursive function theory and its application to formal systems. |
| recursion theory for metamathematics: General Recursion Theory Jens E. Fenstad, 2017-03-02 This volume presents a unified and coherent account of the many and various parts of general recursion theory. |
| recursion theory for metamathematics: Elements of Set Theory Herbert B. Enderton, 1977-04-28 This is an introductory undergraduate textbook in set theory. In mathematics these days, essentially everything is a set. Some knowledge of set theory is necessary part of the background everyone needs for further study of mathematics. It is also possible to study set theory for its own interest--it is a subject with intruiging results anout simple objects. This book starts with material that nobody can do without. There is no end to what can be learned of set theory, but here is a beginning. |
| recursion theory for metamathematics: Descriptive Set Theory Yiannis N. Moschovakis, 2009-06-30 Descriptive Set Theory is the study of sets in separable, complete metric spaces that can be defined (or constructed), and so can be expected to have special properties not enjoyed by arbitrary pointsets. This subject was started by the French analysts at the turn of the 20th century, most prominently Lebesgue, and, initially, was concerned primarily with establishing regularity properties of Borel and Lebesgue measurable functions, and analytic, coanalytic, and projective sets. Its rapid development came to a halt in the late 1930s, primarily because it bumped against problems which were independent of classical axiomatic set theory. The field became very active again in the 1960s, with the introduction of strong set-theoretic hypotheses and methods from logic (especially recursion theory), which revolutionized it. This monograph develops Descriptive Set Theory systematically, from its classical roots to the modern ``effective'' theory and the consequences of strong (especially determinacy) hypotheses. The book emphasizes the foundations of the subject, and it sets the stage for the dramatic results (established since the 1980s) relating large cardinals and determinacy or allowing applications of Descriptive Set Theory to classical mathematics. The book includes all the necessary background from (advanced) set theory, logic and recursion theory. |
| recursion theory for metamathematics: Recursively Enumerable Sets and Degrees Robert I. Soare, 1999-11-01 ...The book, written by one of the main researchers on the field, gives a complete account of the theory of r.e. degrees. .... The definitions, results and proofs are always clearly motivated and explained before the formal presentation; the proofs are described with remarkable clarity and conciseness. The book is highly recommended to everyone interested in logic. It also provides a useful background to computer scientists, in particular to theoretical computer scientists. Acta Scientiarum Mathematicarum, Ungarn 1988 ...The main purpose of this book is to introduce the reader to the main results and to the intricacies of the current theory for the recurseively enumerable sets and degrees. The author has managed to give a coherent exposition of a rather complex and messy area of logic, and with this book degree-theory is far more accessible to students and logicians in other fields than it used to be. Zentralblatt für Mathematik, 623.1988 |
| recursion theory for metamathematics: Cantorian Set Theory and Limitation of Size Michael Hallett, 1986 This volume presents the philosophical and heuristic framework Cantor developed and explores its lasting effect on modern mathematics. Establishes a new plateau for historical comprehension of Cantor's monumental contribution to mathematics. --The American Mathematical Monthly |
| recursion theory for metamathematics: Slicing The Truth: On The Computable And Reverse Mathematics Of Combinatorial Principles Denis R Hirschfeldt, 2014-07-18 This book is a brief and focused introduction to the reverse mathematics and computability theory of combinatorial principles, an area of research which has seen a particular surge of activity in the last few years. It provides an overview of some fundamental ideas and techniques, and enough context to make it possible for students with at least a basic knowledge of computability theory and proof theory to appreciate the exciting advances currently happening in the area, and perhaps make contributions of their own. It adopts a case-study approach, using the study of versions of Ramsey's Theorem (for colorings of tuples of natural numbers) and related principles as illustrations of various aspects of computability theoretic and reverse mathematical analysis. This book contains many exercises and open questions. |
| recursion theory for metamathematics: A Beginner's Guide to Mathematical Logic Raymond M. Smullyan, 2014-07-23 Written by a creative master of mathematical logic, this introductory text combines stories of great philosophers, quotations, and riddles with the fundamentals of mathematical logic. Author Raymond Smullyan offers clear, incremental presentations of difficult logic concepts. He highlights each subject with inventive explanations and unique problems. Smullyan's accessible narrative provides memorable examples of concepts related to proofs, propositional logic and first-order logic, incompleteness theorems, and incompleteness proofs. Additional topics include undecidability, combinatoric logic, and recursion theory. Suitable for undergraduate and graduate courses, this book will also amuse and enlighten mathematically minded readers. Dover (2014) original publication. See every Dover book in print at www.doverpublications.com |
| recursion theory for metamathematics: Twenty Five Years of Constructive Type Theory Giovanni Sambin, Jan M. Smith, 1998-10-15 Per Martin-Löf's work on the development of constructive type theory has been of huge significance in the fields of logic and the foundations of mathematics. It is also of broader philosophical significance, and has important applications in areas such as computing science and linguistics. This volume draws together contributions from researchers whose work builds on the theory developed by Martin-Löf over the last twenty-five years. As well as celebrating the anniversary of the birth of the subject it covers many of the diverse fields which are now influenced by type theory. It is an invaluable record of areas of current activity, but also contains contributions from N. G. de Bruijn and William Tait, both important figures in the early development of the subject. Also published for the first time is one of Per Martin-Löf's earliest papers. |
| recursion theory for metamathematics: A Beginner's Further Guide To Mathematical Logic Raymond M Smullyan, 2016-11-11 'A wealth of examples to which solutions are given permeate the text so the reader will certainly be active.'The Mathematical GazetteThis is the final book written by the late great puzzle master and logician, Dr. Raymond Smullyan.This book is a sequel to my Beginner's Guide to Mathematical Logic.The previous volume deals with elements of propositional and first-order logic, contains a bit on formal systems and recursion, and concludes with chapters on Gödel's famous incompleteness theorem, along with related results.The present volume begins with a bit more on propositional and first-order logic, followed by what I would call a 'fein' chapter, which simultaneously generalizes some results from recursion theory, first-order arithmetic systems, and what I dub a 'decision machine.' Then come five chapters on formal systems, recursion theory and metamathematical applications in a general setting. The concluding five chapters are on the beautiful subject of combinatory logic, which is not only intriguing in its own right, but has important applications to computer science. Argonne National Laboratory is especially involved in these applications, and I am proud to say that its members have found use for some of my results in combinatory logic.This book does not cover such important subjects as set theory, model theory, proof theory, and modern developments in recursion theory, but the reader, after studying this volume, will be amply prepared for the study of these more advanced topics. |
| recursion theory for metamathematics: Recursion Theory Week Klaus Ambos-Spies, Gert H. Müller, Gerald E. Sacks, 2006-11-14 These proceedings contain research and survey papers from many subfields of recursion theory, with emphasis on degree theory, in particular the development of frameworks for current techniques in this field. Other topics covered include computational complexity theory, generalized recursion theory, proof theoretic questions in recursion theory, and recursive mathematics. |
| recursion theory for metamathematics: Category Theory Steve Awodey, 2010-06-17 A comprehensive reference to category theory for students and researchers in mathematics, computer science, logic, cognitive science, linguistics, and philosophy. Useful for self-study and as a course text, the book includes all basic definitions and theorems (with full proofs), as well as numerous examples and exercises. |
| recursion theory for metamathematics: Algebraic Recursion Theory Ljubomir Lalov Ivanov, 1986 |
| recursion theory for metamathematics: Axiomatic Theories of Truth Volker Halbach, 2014-02-27 At the centre of the traditional discussion of truth is the question of how truth is defined. Recent research, especially with the development of deflationist accounts of truth, has tended to take truth as an undefined primitive notion governed by axioms, while the liar paradox and cognate paradoxes pose problems for certain seemingly natural axioms for truth. In this book, Volker Halbach examines the most important axiomatizations of truth, explores their properties and shows how the logical results impinge on the philosophical topics related to truth. In particular, he shows that the discussion on topics such as deflationism about truth depends on the solution of the paradoxes. His book is an invaluable survey of the logical background to the philosophical discussion of truth, and will be indispensable reading for any graduate or professional philosopher in theories of truth. |
| recursion theory for metamathematics: Classical recursion theory : the theory of functions and sets of natural numbers Piergiorgio Odifreddi, 1999 |
| recursion theory for metamathematics: Principia Mathematica Alfred North Whitehead, Bertrand Russell, 1927 The Principia Mathematica has long been recognised as one of the intellectual landmarks of the century. |
| recursion theory for metamathematics: Basic Simple Type Theory J. Roger Hindley, 1997 Type theory is one of the most important tools in the design of higher-level programming languages, such as ML. This book introduces and teaches its techniques by focusing on one particularly neat system and studying it in detail. By concentrating on the principles that make the theory work in practice, the author covers all the key ideas without getting involved in the complications of more advanced systems. This book takes a type-assignment approach to type theory, and the system considered is the simplest polymorphic one. The author covers all the basic ideas, including the system's relation to propositional logic, and gives a careful treatment of the type-checking algorithm that lies at the heart of every such system. Also featured are two other interesting algorithms that until now have been buried in inaccessible technical literature. The mathematical presentation is rigorous but clear, making it the first book at this level that can be used as an introduction to type theory for computer scientists. |
| recursion theory for metamathematics: Introduction to Logic and to the Methodology of the Deductive Sciences Alfred Tarski, 1994-01-06 Now in its fourth edition, this classic work clearly and concisely introduces the subject of logic and its applications. The first part of the book explains the basic concepts and principles which make up the elements of logic. The author demonstrates that these ideas are found in all branches of mathematics, and that logical laws are constantly applied in mathematical reasoning. The second part of the book shows the applications of logic in mathematical theory building with concrete examples that draw upon the concepts and principles presented in the first section. Numerous exercises and an introduction to the theory of real numbers are also presented. Students, teachers and general readers interested in logic and mathematics will find this book to be an invaluable introduction to the subject. |
| recursion theory for metamathematics: Computation and Logic in the Real World S. Barry Cooper, 2007-06-11 This book constitutes the refereed proceedings of the Third International Conference on Computability in Europe, CiE 2007, held in Sienna, Italy, in June 2007. The 50 revised full papers presented together with 36 invited papers were carefully reviewed and selected from 167 submissions. |
| recursion theory for metamathematics: Model Theory María Manzano, 1999 Model theory, which is concerned with the relationship between mathematical structures and logic, now has a wide range of applications in areas such as computing, philosophy, and linguistics. This book, suitable for both mathematicians and students from outside the field, provides a clear and readable introduction to the subject. |
| recursion theory for metamathematics: Algebraic Methods in Philosophical Logic J. Michael Dunn, Gary Hardegree, 2001-06-28 This comprehensive text demonstrates how various notions of logic can be viewed as notions of universal algebra. It is aimed primarily for logisticians in mathematics, philosophy, computer science and linguistics with an interest in algebraic logic, but is also accessible to those from a non-logistics background. It is suitable for researchers, graduates and advanced undergraduates who have an introductory knowledge of algebraic logic providing more advanced concepts, as well as more theoretical aspects. The main theme is that standard algebraic results (representations) translate into standard logical results (completeness). Other themes involve identification of a class of algebras appropriate for classical and non-classical logic studies, including: gaggles, distributoids, partial- gaggles, and tonoids. An imporatant sub title is that logic is fundamentally information based, with its main elements being propositions, that can be understood as sets of information states. Logics are considered in various senses e.g. systems of theorems, consequence relations and, symmetric consequence relations. |
| recursion theory for metamathematics: Models of Peano Arithmetic Richard W. Kaye, |
| recursion theory for metamathematics: Fibring Logics Dov M. Gabbay, 1998-11-05 Modern applications of logic, in mathematics, theoretical computer science, and linguistics, require combined systems involving many different logics working together. In this book the author offers a basic methodology for combining-or fibring-systems. This means that many existing complex systems can be broken down into simpler components, hence making them much easier to manipulate. Using this methodology the book discusses ways of obtaining a wide variety of multimodal, modal intuitionistic, modal substructural and fuzzy systems in a uniform way. It also covers self-fibred languages which allow formulae to apply to themselves. The book also studies sufficient conditions for transferring properties of the component logics into properties of the combined system. |
| recursion theory for metamathematics: Introduction to Mathematical Logic Elliott Mendelson, 2015-05-21 The new edition of this classic textbook, Introduction to Mathematical Logic, Sixth Edition explores the principal topics of mathematical logic. It covers propositional logic, first-order logic, first-order number theory, axiomatic set theory, and the theory of computability. The text also discusses the major results of Godel, Church, Kleene, Rosse |
| recursion theory for metamathematics: Logic and Structure Dirk van Dalen, 2013-11-11 Logic appears in a 'sacred' and in a 'profane' form. The sacred form is dominant in proof theory, the profane form in model theory. The phenomenon is not unfamiliar, one observes this dichotomy also in other areas, e.g. set theory and recursion theory. For one reason or another, such as the discovery of the set theoretical paradoxes (Cantor, Russell), or the definability paradoxes (Richard, Berry), a subject is treated for some time with the utmost awe and diffidence. As a rule, however, sooner or later people start to treat the matter in a more free and easy way. Being raised in the 'sacred' tradition, I was greatly surprised (and some what shocked) when I observed Hartley Rogers teaching recursion theory to mathema ticians as if it were just an ordinary course in, say, linear algebra or algebraic topology. In the course of time I have come to accept his viewpoint as the didac tically sound one: before going into esoteric niceties one should develop a certain feeling for the subject and obtain a reasonable amount of plain working knowledge. For this reason I have adopted the profane attitude in this introductory text, reserving the more sacred approach for advanced courses. Readers who want to know more about the latter aspect of logic are referred to the immortal texts of Hilbert-Bernays or Kleene. |
| recursion theory for metamathematics: Complexity, Logic, and Recursion Theory Andrea Sorbi, 2019-05-07 Integrates two classical approaches to computability. Offers detailed coverage of recent research at the interface of logic, computability theory, nd theoretical computer science. Presents new, never-before-published results and provides informtion not easily accessible in the literature. |
| recursion theory for metamathematics: Semantics and Truth Jan Woleński, 2020-01-01 The book provides a historical (with an outline of the history of the concept of truth from antiquity to our time) and systematic exposition of the semantic theory of truth formulated by Alfred Tarski in the 1930s. This theory became famous very soon and inspired logicians and philosophers. It has two different, but interconnected aspects: formal-logical and philosophical. The book deals with both, but it is intended mostly as a philosophical monograph. It explains Tarski’s motivation and presents discussions about his ideas (pro and contra) as well as points out various applications of the semantic theory of truth to philosophical problems (truth-criteria, realism and anti-realism, future contingents or the concept of correspondence between language and reality). |
| recursion theory for metamathematics: Simplicity Theory Byunghan Kim, 2014 An up-to-date account of the current techniques and results in Simplicity Theory, which has been a focus of research in model theory for the last decade. Suitable for logicians, mathematicians and graduate students working on model theory. |
| recursion theory for metamathematics: Sketches of an Elephant Peter T. Johnstone, 2002 |
| recursion theory for metamathematics: Set Theory John L. Bell, 2011-05-05 This third edition, now available in paperback, is a follow up to the author's classic Boolean-Valued Models and Independence Proofs in Set Theory,. It provides an exposition of some of the most important results in set theory obtained in the 20th century: the independence of the continuum hypothesis and the axiom of choice. Aimed at graduate students and researchers in mathematics, mathematical logic, philosophy, and computer science, the third edition has been extensively updated with expanded introductory material, new chapters, and a new appendix on category theory. It covers recent developments in the field and contains numerous exercises, along with updated and increased coverage of the background material. This new paperback edition includes additional corrections and, for the first time, will make this landmark text accessible to students in logic and set theory. |
What is recursion and when should I use it? - Stack Overflow
Don't use recursion for factorials or Fibonacci numbers. One problem with computer-science textbooks is that they present silly examples of recursion. The typical examples are computing …
list - Basics of recursion in Python - Stack Overflow
May 13, 2015 · Tail Call Recursion. Once you understand how the above recursion works, you can try to make it a little bit better. Now, to find the actual result, we are depending on the …
Recursion vs loops - Stack Overflow
Mar 19, 2009 · Recursion is used to express an algorithm that is naturally recursive in a form that is more easily understandable. A "naturally recursive" algorithm is one where the answer is …
recursion - Determining complexity for recursive functions (Big O ...
Nov 20, 2012 · And here the for loop takes n/2 since we're increasing by 2, and the recursion takes n/5 and since the for loop is called recursively, therefore, the time complexity is in (n/5) * …
algorithm - Understanding recursion - Stack Overflow
Apr 5, 2009 · Why, yes, recursion can be replaced with iteration, but often recursion is more elegant. Let's talk about trees. In computer science, a tree is a structure made up of nodes , …
Stack overflow caused by recursive function
Sep 28, 2016 · Any "boundless" recursion, that is recursive calls that aren't naturally limited to a small(ish) number will have this effect. Exactly where the limit goes depends on the OS, the …
performance - Recursion or Iteration? - Stack Overflow
Jun 24, 2011 · Recursion has a disadvantage that the algorithm that you write using recursion has O(n) space complexity. While iterative aproach have a space complexity of O(1).This is the …
What are the advantages and disadvantages of recursion?
Oct 21, 2015 · Recursion: A function that calls itself is called as recursive function and this technique is called as recursion. Pros: 1. Reduce unnecessary calling of functions. 2. Through …
Newest 'recursion' Questions - Stack Overflow
I am doing recursion for the first time in javascript. I made an array and want it to print out every element however it only printso ut the first element (1) function loop4(){ arr = [1,2,3,4,...
Real-world examples of recursion - Stack Overflow
Sep 20, 2008 · Recursion is a technique to keep breaking the problem down into smaller and smaller pieces, until one of those pieces become small enough to be a piece-of-cake. Of …
What is recursion and when should I use it? - Stack Overfl…
Don't use recursion for factorials or Fibonacci numbers. One problem with computer-science textbooks is that they present silly examples of recursion. The typical examples are computing a …
list - Basics of recursion in Python - Stack Overflow
May 13, 2015 · Tail Call Recursion. Once you understand how the above recursion works, you can try to make it a little bit better. Now, to find the actual result, we are depending on the …
Recursion vs loops - Stack Overflow
Mar 19, 2009 · Recursion is used to express an algorithm that is naturally recursive in a form that is more easily understandable. A "naturally recursive" algorithm is one where the answer is …
recursion - Determining complexity for recursive funct…
Nov 20, 2012 · And here the for loop takes n/2 since we're increasing by 2, and the recursion takes n/5 and since the for loop is called recursively, therefore, the time complexity is in …
algorithm - Understanding recursion - Stack Overflow
Apr 5, 2009 · Why, yes, recursion can be replaced with iteration, but often recursion is more elegant. Let's talk about trees. In computer science, a tree is a structure made up of nodes , …
