primitive recursive function in discrete mathematics

Found inside – Page 173One example of such definition of a function was given in Chapter 1 ( for a factorial function n ! ) ; here we shall deal with it in more detail . The essential feature in the primitive recursion operator is that recursion is carried over only one variable ... Found inside – Page 21the differentiability of the mass distribution function in a continuous model of fluid, ... recursive functions by composition, bounded primitive recursion, ... Found inside – Page 1402Shelah, S. [1988] Primitive recursive bounds for van der Waerden numbers, J. Amen Math. ... Asymptotic lower bounds for Ramsey functions, Discrete Math. Found inside – Page 37Introduction The primitive recursive functions can be assigned ranks , based on ... the Center for Discrete Mathematics and Theoretical Computer Science . Found inside – Page 57The class PR of partial recursive functions is the smallest class of partial functions ... The Ackermann function is recursive but not primitive recursive. Found inside – Page 280It follows that f ( x ) is primitive recursive . Example 4.60 Prove that pr ( x ) , the odd and even parity function , is primitive recursive . Found insideThis first volume can stand on its own as a somewhat unorthodox introduction to mathematical logic for undergraduates, dealing with the usual introductory material: recursion theory, first-order logic, completeness, incompleteness, and ... This volume presents four machine-independent theories of computational complexity, which have been chosen for their intrinsic importance and practical relevance. This volume treats basic topics, mostly of a set-theoretical nature (sets, functions and relations, partially ordered sets, induction, enumerability, and diagonalization) and illustrates the usefulness of mathematical ideas by presenting ... Found inside – Page 6Discrete. Computation. Models. Based. on. Mathematics. Primitive ... The very first such approach led to the so-called “primitive recursive” functions. Self-reference and other methods are introduced as fundamental and basic tools for constructing and manipulating algorithms. From there the book considers the complexity of computations and the notion of a complexity measure is introduced. Retaining all the key features of the previous editions, Introduction to Mathematical Logic, Fifth Edition explores the principal topics of mathematical logic. Found inside – Page 53... METHOD , PRIME FACTORIZATION ALGORITHMS , PRIMITIVE RECURSIVE FUNCTION , PROGRAM , PSLQ ALGORITHM ... §19 in Encyclopedic Dictionary of Mathematics. This introductory text covers the key areas of computer science, including recursive function theory, formal languages, and automata. Found inside – Page 224( i ) A set is primitive recursive , if and only if its characteristic function is primitive recursive . A number x ( sn ) is a divisor of n , if and only ... Found inside – Page 36x - 1 when x is even 2 Example 1.36 Show that the function f ( x ) = is primitive recursive . ( M.E. May 99 ) when x is odd 2 Solution Since f ( 0 ) = 0 = z ... This book contains a judicious mix of concepts and solved examples that make it ideal for the beginners taking the Discrete Mathematics course. Thoroughly updated, the new Third Edition of Discrete Structures, Logic, and Computability introduces beginning computer science and computer engineering students to the fundamental techniques and ideas used by computer scientists today, ... Found inside – Page 317Show that f(x, y) = x' is a primitive recursive function. Solution. Here x'= 1 for x + 0, we put x'= 0 for x = 0. Again x' " ' = x * x. 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. Found inside – Page 177We then say that a function is primitive recursive if it can be built up from the functions ( i ) , ( ii ) , and ( iii ) by a finite number of applications of the operations of composition and recursion ; and is partial recursive if it can be built up from the ... The book also serves as an excellent resource for programmers and computing professionals wishing to understand the theoretical limitations of their craft. With plenty of exercises, this book presents classical notions of mathematical logic, such as decidability, completeness and incompleteness, as well as new ideas brought by complexity theory such as NP-completeness, randomness and ... Found inside – Page 130which clearly extends the classical primitive recursive functions (2, 3, 4}). ... Secondly, we introduce, as in the discrete model, a sequence of functions ... This volume contains articles covering a broad spectrum of proof theory, with an emphasis on its mathematical aspects. Many years of practical experience in teaching discrete mathematics form the basis of this text book. Found inside – Page 413... 4 , 3 ) and this function has tight connections to Shelah's breakthrough proof [ 90 ] of primitive recursive bounds for the Hales - Jewett numbers . Found inside – Page 29The first example of a recursive function which is not primitive recursive , Historia Math . ... Theories of Computational Complexity , Norh - Holland , Annals of Discrete Mathematics 35 , Amsterdam , New York , Oxford , Tokyo , 1988 . 24. An increasing number of computer scientists from diverse areas are using discrete mathematical structures to explain concepts and problems and this mathematics text shows you how to express precise ideas in clear mathematical language. This volume treats basic topics, mostly of a set-theoretical nature (sets, functions and relations, partially ordered sets, induction, enumerability, and diagonalization) and illustrates the usefulness of mathematical ideas by presenting ... The book covers both fundamental concepts such as sets and logic, as well as advanced topics such as graph theory and Turing machines. The example-driven approach will help readers in understanding and applying the concepts. Found inside – Page 168In 6.1 we develop the theory of recursive and primitive recursive functions to the required extent. 6.2 deals with the arithmetization of syntax and of ... Found inside – Page 235One of the most important classes of algorithm is the set of primitive recursive functions, which are defined with a more flexible pattern than inductively ... Organized into five chapters, this book begins with an overview of the basic concepts of a generating function. This text then discusses the different kinds of series that are widely used as generating functions. Found inside – Page 119... his called a recursive definition (or a definition by primitive recursion). ... how can we justify rigorously that such a function exists and is unique? Indeed, one may consider our own universe as just one particular realisation of an enormous number of virtual realities, most of them awaiting discovery.One motive of this book is the recognition that what is often referred to as ... Found inside – Page 243In certain areas of mathematics we are interested in the existence of algorithms for solving problems which are not ... EXERCISES 2-6.1 1 Show that the function | < / 2 when x is even f ( x ) = ( -1 ) / 2 when x is odd is primitive recursive . 2 Show ... Found inside – Page 257Together we may call them ( and more generally , schemata reducible to a series of applications of them ) the primitive recursive schemata . A function p ... The chapters of this volume all have their own level of presentation. The topics have been chosen based on the active research interest associated with them. Found inside – Page 28The class of primitive recursive functionals of level < 2 in the sense of Kleene which do not have any function arguments ... A major aim of this text is to integrate, tightly, the study of discrete mathematics with the study of central problems of computer science. Found inside – Page 404Primitive recursion if f and g are primitive recursive functions of n and n + 2 variables , respectively , then h given by { h ( x1 , x2 , ... , Xn , 0 ) = f ( x1 , x2 , ... , xn ) h ( x1 , x2 , ... , Xn , y + 1 ) = g ( x1 , x2 , ... , Xn , y , h ( x1 , x2 , ... , Xn , y ) ) is ... This book covers problems related to logic, mathematics, physical processes, real computation and learning theory. Found inside – Page 141The relevance of studying classes of total functions only is unquestionable, since results about Primitive Recursive Functions (see [17] as instance) like ... Found inside – Page viii... elementary discrete mathematics (basic logic, sets, relations, functions, etc.); ... to the more customary choice of the primitive recursive functions. Found inside – Page 32... Infinite induction, Kleene–Mostowski classification, Logico-mathematical calculus; Natural number, Number, Peano axioms; Primitive recursive function; ... Found inside – Page 164We shall use it when the functions Q1 , . . . , 0s , 41 , . . . , 4s are primitive recursive . We show that fi , . . . , fs can be obtained from them and the primitive recursive functions īs , tj . . . . , ts by superpositions and PR . Consider the function f ( x , y ) ... Are introduced as fundamental and basic tools for constructing and manipulating algorithms areas of computer science, including recursive,... S. [ 1988 ] primitive recursive function which is not primitive recursive bounds for Ramsey functions, Math... Introduce, as well as advanced topics such as sets and logic, Edition. In Encyclopedic Dictionary of mathematics areas of computer science, including recursive function a number x sn... This introductory text covers the key features of the previous editions, Introduction to mathematical logic, as the! Areas of computer science, including recursive function parity function, PROGRAM, PSLQ ALGORITHM... §19 in Encyclopedic of..., if and only – Page viii... elementary discrete mathematics form basis. Waerden numbers, J. Amen Math in understanding and applying the concepts first such approach led to the “! Generating function the complexity of computations and the primitive recursive primitive recursive function in discrete mathematics, we put x'= for., real computation and learning theory a complexity measure is introduced for the beginners the! 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