DOCSLIB.ORG

Math 310 Class Notes 5: the Recursion Theorem for N

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math 310 Class Notes 5: the Recursion Theorem for N Math 310 Class Notes 5: The Recursion Theorem for N In this set of notes we will tighten a loose end that has been left dangling. Namely, in Class Notes 3 we used the word "assign" when we defined the notion of + for N (see Definition 2 of the notes). The word "assign" actually means we have a function that does the assignment. To be precise, we have the following theorem. Theorem 1. (The recursion theorem) Let c be a fixed natural number and let f : N ! N be a function. Then there is a unique function g : N ! N such that g(1) = c and g(x0) = f(g(x)) for all x 2 N. Before proving this theorem, let us see a few applications. Application 1. Consider the function f : N ! N given by f : x 7! x0: Fix a number m 2 N. Then by the recursion theorem there is a function g such that g(1) = m0 and g(x0) = f(g(x)) for all x 2 N. Let us now define m + n := g(n) for all n 2 N. Theorem 2. Fix m 2 N. We have m + n0 = (m + n)0 for all n 2 N. Proof. m + n0 = g(n0) = f(g(n)) = f(m + n) = (m + n)0. Slightly harder to verify is the following. Theorem 3. Fix m 2 N. We have m0 + n = (m + n)0 for all n 2 N. Proof. m + n = g(n) for all n 2 N, where g(1) = m0 and g(n0) = f(g(n)). Likewise, m0 + n = h(n) for all n 2 N, where h(1) = (m0)0 and h(n0) = f(h(n)). Let S be the set of all n 2 N such that m0 + n = (m + n)0. We first show 1 2 S. Now m0 + 1 = h(1) = (m0)0 while m + 1 = g(1) = m0, so that m0 + 1 = (m0)0 = (m + 1)0. Thus 1 2 S. Next suppose n 2 S. We establish that n0 2 S. That n 2 S means m0 + n = (m + n)0; which is the induction hypothesis. Therefore, by the preceding theorem and the induction hypothesis we have m0 + n0 = (m0 + n)0 = ((m + n)0)0 = (m + n0)0: Therefore, n0 2 N and so S = N. That is m0 + n = (m + n)0 for all n 2 N. 1 2 This theorem is what we have seen in class. The advantage of this approach via the recursion formula is that we can take care of m + n for all m and n at one stroke. Application 2. Fix m 2 N. Consider the function f : N ! N given by f : x 7! x + m: Then by the recursion theorem there is a function g : N ! N such that g(1) = m and g(x0) = f(g(x)). We define m · n := g(n) for all n 2 N. Exercise. Prove that m · n0 = m · n + m and m0 · n = m · n + n for all n 2 N. Application 3. Fix a 2 N. Consider the function f : N ! N given by f : x 7! ax: Then by the recursion theorem there is a function g : N ! N such that g(1) = a and g(x0) = f(g(x)). We define an := g(n): Exerxise. By the mathematical induction, prove that am+n = am · an and (am)n = amn. Now let us look at the proof of the recursion theorem. Proof. Existence of g. By definition, a function g : N ! N is a special sort of subset of N × N. Recall that g(1) = c means (1; c) 2 g and g(x0) = f(g(x)) means that if we let y = g(x), i.e., let (x; y) 2 g, then (x0; f(y)) 2 g as this just says g(x0) = f(g(x)) = f(y): Motivated by this, let us begin to construct the function g by consid- ering the set S, where S := fA ⊂ N × N : (1; c) 2 A and (x; y) 2 A implies (x0; f(y)) 2 Ag: S is not empty since N × N 2 S. Consider the set g := \A2S A; which consists of elements common to all A 2 S. Now (1; c) 2 g since it is in all A 2 S. Moreover, if (x; y) 2 g then (x; y) 2 A for all A 2 S. But then (x0; f(y)) 2 A for all A 2 S so that 3 (x0; f(y)) 2 g. We conclude therefore that g is in fact an element of S; it is the smallest element of S. We now prove that g is a function from N to N. First we show that the domain of g is N. Let T be the subset of n 2 N such that there is a z 2 N for which (n; z) 2 g. Now 1 2 T since (1; c) 2 g. Suppose n 2 T . Then (n; z) 2 g for some z 2 N. This gives (n0; f(z)) 2 g. Hence n0 2 T . It follows by mathematical induction that T = N. That is, the domain of g is N. Secondly, we show that each domain element of g is mapped to a unique target element. To this end, consider the set U := fx 2 N : (x; y); (x; z) 2 g imply y = zg: We show 1 2 U. Suppose this is not the case. That is, suppose we have (1; z), in addition (1; c) 2 g, where z =6 c. Then the set g nf(1; z)g would be a member of S (WHY?). But this contradicts the fact that g is already the smallest element of S. Thus, 1 2 U. We show that if x 2 U then x0 2 U. Suppose this is not the case. Then there is an m 2 U so that m0 2= U. Since m 2 U, there is a unique n 2 N with (m; n) 2 g. As a consequence (m0; f(n)) 2 g. Since m0 2= U, there is an (m0; z) 2 g, in addition to (m0; f(n)) 2 g, where z =6 f(n). But then g n f(m0; z)g is an element of S that is even smaller than g (WHY?), which is absurd. The mathematical induction then says that U = N. In conclusion, g is a function. Uniqueness of g. Suppose we have two functions g1 and g2 satisfying g1(1) = c = g2(1) 0 0 and g1(x ) = f(g1(x)) and g2(x ) = f(g2(x)) for all x 2 N. Let us consider the set V := fn 2 N : g1(n) = g2(n)g: 1 2 V because g1(1) = g2(1) = c. If n 2 V , then g1(n) = g2(n), so that 0 0 g1(n ) = f(g1(n)) = f(g2(n)) = g2(n ); which implies n0 2 V . Hence V = N by the mathematical induction. That is, g1 = g2. .
Load more

Recommended publications
  • 6.5 the Recursion Theorem
    6.5. THE RECURSION THEOREM 417 6.5 The Recursion Theorem The recursion Theorem, due to Kleene, is a fundamental result in recursion theory. Theorem 6.5.1 (Recursion Theorem, Version 1 )Letϕ0,ϕ1,... be any ac- ceptable indexing of the partial recursive functions. For every total recursive function f, there is some n such that ϕn = ϕf(n). The recursion Theorem can be strengthened as follows. Theorem 6.5.2 (Recursion Theorem, Version 2 )Letϕ0,ϕ1,... be any ac- ceptable indexing of the partial recursive functions. There is a total recursive function h such that for all x ∈ N,ifϕx is total, then ϕϕx(h(x)) = ϕh(x). 418 CHAPTER 6. ELEMENTARY RECURSIVE FUNCTION THEORY A third version of the recursion Theorem is given below. Theorem 6.5.3 (Recursion Theorem, Version 3 ) For all n ≥ 1, there is a total recursive function h of n +1 arguments, such that for all x ∈ N,ifϕx is a total recursive function of n +1arguments, then ϕϕx(h(x,x1,...,xn),x1,...,xn) = ϕh(x,x1,...,xn), for all x1,...,xn ∈ N. As a first application of the recursion theorem, we can show that there is an index n such that ϕn is the constant function with output n. Loosely speaking, ϕn prints its own name. Let f be the recursive function such that f(x, y)=x for all x, y ∈ N. 6.5. THE RECURSION THEOREM 419 By the s-m-n Theorem, there is a recursive function g such that ϕg(x)(y)=f(x, y)=x for all x, y ∈ N.
    [Show full text]
  • April 22 7.1 Recursion Theorem
    CSE 431 Theory of Computation Spring 2014 Lecture 7: April 22 Lecturer: James R. Lee Scribe: Eric Lei Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. They may be distributed outside this class only with the permission of the Instructor. An interesting question about Turing machines is whether they can reproduce themselves. A Turing machine cannot be be defined in terms of itself, but can it still somehow print its own source code? The answer to this question is yes, as we will see in the recursion theorem. Afterward we will see some applications of this result. 7.1 Recursion Theorem Our end goal is to have a Turing machine that prints its own source code and operates on it. Last lecture we proved the existence of a Turing machine called SELF that ignores its input and prints its source code. We construct a similar proof for the recursion theorem. We will also need the following lemma proved last lecture. ∗ ∗ Lemma 7.1 There exists a computable function q :Σ ! Σ such that q(w) = hPwi, where Pw is a Turing machine that prints w and hats. Theorem 7.2 (Recursion theorem) Let T be a Turing machine that computes a function t :Σ∗ × Σ∗ ! Σ∗. There exists a Turing machine R that computes a function r :Σ∗ ! Σ∗, where for every w, r(w) = t(hRi; w): The theorem says that for an arbitrary computable function t, there is a Turing machine R that computes t on hRi and some input. Proof: We construct a Turing Machine R in three parts, A, B, and T , where T is given by the statement of the theorem.
    [Show full text]
  • Intuitionism on Gödel's First Incompleteness Theorem
    The Yale Undergraduate Research Journal Volume 2 Issue 1 Spring 2021 Article 31 2021 Incomplete? Or Indefinite? Intuitionism on Gödel’s First Incompleteness Theorem Quinn Crawford Yale University Follow this and additional works at: https://elischolar.library.yale.edu/yurj Part of the Mathematics Commons, and the Philosophy Commons Recommended Citation Crawford, Quinn (2021) "Incomplete? Or Indefinite? Intuitionism on Gödel’s First Incompleteness Theorem," The Yale Undergraduate Research Journal: Vol. 2 : Iss. 1 , Article 31. Available at: https://elischolar.library.yale.edu/yurj/vol2/iss1/31 This Article is brought to you for free and open access by EliScholar – A Digital Platform for Scholarly Publishing at Yale. It has been accepted for inclusion in The Yale Undergraduate Research Journal by an authorized editor of EliScholar – A Digital Platform for Scholarly Publishing at Yale. For more information, please contact [email protected]. Incomplete? Or Indefinite? Intuitionism on Gödel’s First Incompleteness Theorem Cover Page Footnote Written for Professor Sun-Joo Shin’s course PHIL 437: Philosophy of Mathematics. This article is available in The Yale Undergraduate Research Journal: https://elischolar.library.yale.edu/yurj/vol2/iss1/ 31 Crawford: Intuitionism on Gödel’s First Incompleteness Theorem Crawford | Philosophy Incomplete? Or Indefinite? Intuitionism on Gödel’s First Incompleteness Theorem By Quinn Crawford1 1Department of Philosophy, Yale University ABSTRACT This paper analyzes two natural-looking arguments that seek to leverage Gödel’s first incompleteness theo- rem for and against intuitionism, concluding in both cases that the argument is unsound because it equivo- cates on the meaning of “proof,” which differs between formalism and intuitionism.
    [Show full text]
  • Chapter 1 Logic and Set Theory
    Chapter 1 Logic and Set Theory To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. – Ian Stewart Does God play dice? The mathematics of chaos In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. That is, a proof is a logical argument, not an empir- ical one. One must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture. Mathematical logic is the framework upon which rigorous proofs are built. It is the study of the principles and criteria of valid inference and demonstrations. Logicians have analyzed set theory in great details, formulating a collection of axioms that affords a broad enough and strong enough foundation to mathematical reasoning. The standard form of axiomatic set theory is denoted ZFC and it consists of the Zermelo-Fraenkel (ZF) axioms combined with the axiom of choice (C). Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can lead to contradictions. Avoiding such contradictions was one of the original motivations for the axiomatization of set theory. 1 2 CHAPTER 1. LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic.
    [Show full text]
  • The Logic of Recursive Equations Author(S): A
    The Logic of Recursive Equations Author(s): A. J. C. Hurkens, Monica McArthur, Yiannis N. Moschovakis, Lawrence S. Moss, Glen T. Whitney Source: The Journal of Symbolic Logic, Vol. 63, No. 2 (Jun., 1998), pp. 451-478 Published by: Association for Symbolic Logic Stable URL: http://www.jstor.org/stable/2586843 . Accessed: 19/09/2011 22:53 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Symbolic Logic. http://www.jstor.org THE JOURNAL OF SYMBOLIC LOGIC Volume 63, Number 2, June 1998 THE LOGIC OF RECURSIVE EQUATIONS A. J. C. HURKENS, MONICA McARTHUR, YIANNIS N. MOSCHOVAKIS, LAWRENCE S. MOSS, AND GLEN T. WHITNEY Abstract. We study logical systems for reasoning about equations involving recursive definitions. In particular, we are interested in "propositional" fragments of the functional language of recursion FLR [18, 17], i.e., without the value passing or abstraction allowed in FLR. The 'pure," propositional fragment FLRo turns out to coincide with the iteration theories of [1]. Our main focus here concerns the sharp contrast between the simple class of valid identities and the very complex consequence relation over several natural classes of models.
    [Show full text]
  • Theorem Proving in Classical Logic
    MEng Individual Project Imperial College London Department of Computing Theorem Proving in Classical Logic Supervisor: Dr. Steffen van Bakel Author: David Davies Second Marker: Dr. Nicolas Wu June 16, 2021 Abstract It is well known that functional programming and logic are deeply intertwined. This has led to many systems capable of expressing both propositional and first order logic, that also operate as well-typed programs. What currently ties popular theorem provers together is their basis in intuitionistic logic, where one cannot prove the law of the excluded middle, ‘A A’ – that any proposition is either true or false. In classical logic this notion is provable, and the_: corresponding programs turn out to be those with control operators. In this report, we explore and expand upon the research about calculi that correspond with classical logic; and the problems that occur for those relating to first order logic. To see how these calculi behave in practice, we develop and implement functional languages for propositional and first order logic, expressing classical calculi in the setting of a theorem prover, much like Agda and Coq. In the first order language, users are able to define inductive data and record types; importantly, they are able to write computable programs that have a correspondence with classical propositions. Acknowledgements I would like to thank Steffen van Bakel, my supervisor, for his support throughout this project and helping find a topic of study based on my interests, for which I am incredibly grateful. His insight and advice have been invaluable. I would also like to thank my second marker, Nicolas Wu, for introducing me to the world of dependent types, and suggesting useful resources that have aided me greatly during this report.
    [Show full text]
  • Fractal Curves and Complexity
    Perception & Psychophysics 1987, 42 (4), 365-370 Fractal curves and complexity JAMES E. CUTI'ING and JEFFREY J. GARVIN Cornell University, Ithaca, New York Fractal curves were generated on square initiators and rated in terms of complexity by eight viewers. The stimuli differed in fractional dimension, recursion, and number of segments in their generators. Across six stimulus sets, recursion accounted for most of the variance in complexity judgments, but among stimuli with the most recursive depth, fractal dimension was a respect­ able predictor. Six variables from previous psychophysical literature known to effect complexity judgments were compared with these fractal variables: symmetry, moments of spatial distribu­ tion, angular variance, number of sides, P2/A, and Leeuwenberg codes. The latter three provided reliable predictive value and were highly correlated with recursive depth, fractal dimension, and number of segments in the generator, respectively. Thus, the measures from the previous litera­ ture and those of fractal parameters provide equal predictive value in judgments of these stimuli. Fractals are mathematicalobjectsthat have recently cap­ determine the fractional dimension by dividing the loga­ tured the imaginations of artists, computer graphics en­ rithm of the number of unit lengths in the generator by gineers, and psychologists. Synthesized and popularized the logarithm of the number of unit lengths across the ini­ by Mandelbrot (1977, 1983), with ever-widening appeal tiator. Since there are five segments in this generator and (e.g., Peitgen & Richter, 1986), fractals have many curi­ three unit lengths across the initiator, the fractionaldimen­ ous and fascinating properties. Consider four. sion is log(5)/log(3), or about 1.47.
    [Show full text]
  • I Want to Start My Story in Germany, in 1877, with a Mathematician Named Georg Cantor
    LISTENING - Ron Eglash is talking about his project http://www.ted.com/talks/ 1) Do you understand these words? iteration mission scale altar mound recursion crinkle 2) Listen and answer the questions. 1. What did Georg Cantor discover? What were the consequences for him? 2. What did von Koch do? 3. What did Benoit Mandelbrot realize? 4. Why should we look at our hand? 5. What did Ron get a scholarship for? 6. In what situation did Ron use the phrase “I am a mathematician and I would like to stand on your roof.” ? 7. What is special about the royal palace? 8. What do the rings in a village in southern Zambia represent? 3)Listen for the second time and decide whether the statements are true or false. 1. Cantor realized that he had a set whose number of elements was equal to infinity. 2. When he was released from a hospital, he lost his faith in God. 3. We can use whatever seed shape we like to start with. 4. Mathematicians of the 19th century did not understand the concept of iteration and infinity. 5. Ron mentions lungs, kidney, ferns, and acacia trees to demonstrate fractals in nature. 6. The chief was very surprised when Ron wanted to see his village. 7. Termites do not create conscious fractals when building their mounds. 8. The tiny village inside the larger village is for very old people. I want to start my story in Germany, in 1877, with a mathematician named Georg Cantor. And Cantor decided he was going to take a line and erase the middle third of the line, and take those two resulting lines and bring them back into the same process, a recursive process.
    [Show full text]
  • On the Mathematical Foundations of Learning
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 39, Number 1, Pages 1{49 S 0273-0979(01)00923-5 Article electronically published on October 5, 2001 ON THE MATHEMATICAL FOUNDATIONS OF LEARNING FELIPE CUCKER AND STEVE SMALE The problem of learning is arguably at the very core of the problem of intelligence, both biological and artificial. T. Poggio and C.R. Shelton Introduction (1) A main theme of this report is the relationship of approximation to learning and the primary role of sampling (inductive inference). We try to emphasize relations of the theory of learning to the mainstream of mathematics. In particular, there are large roles for probability theory, for algorithms such as least squares,andfor tools and ideas from linear algebra and linear analysis. An advantage of doing this is that communication is facilitated and the power of core mathematics is more easily brought to bear. We illustrate what we mean by learning theory by giving some instances. (a) The understanding of language acquisition by children or the emergence of languages in early human cultures. (b) In Manufacturing Engineering, the design of a new wave of machines is an- ticipated which uses sensors to sample properties of objects before, during, and after treatment. The information gathered from these samples is to be analyzed by the machine to decide how to better deal with new input objects (see [43]). (c) Pattern recognition of objects ranging from handwritten letters of the alpha- bet to pictures of animals, to the human voice. Understanding the laws of learning plays a large role in disciplines such as (Cog- nitive) Psychology, Animal Behavior, Economic Decision Making, all branches of Engineering, Computer Science, and especially the study of human thought pro- cesses (how the brain works).
    [Show full text]
  • Plato on the Foundations of Modern Theorem Provers
    The Mathematics Enthusiast Volume 13 Number 3 Number 3 Article 8 8-2016 Plato on the foundations of Modern Theorem Provers Ines Hipolito Follow this and additional works at: https://scholarworks.umt.edu/tme Part of the Mathematics Commons Let us know how access to this document benefits ou.y Recommended Citation Hipolito, Ines (2016) "Plato on the foundations of Modern Theorem Provers," The Mathematics Enthusiast: Vol. 13 : No. 3 , Article 8. Available at: https://scholarworks.umt.edu/tme/vol13/iss3/8 This Article is brought to you for free and open access by ScholarWorks at University of Montana. It has been accepted for inclusion in The Mathematics Enthusiast by an authorized editor of ScholarWorks at University of Montana. For more information, please contact [email protected]. TME, vol. 13, no.3, p.303 Plato on the foundations of Modern Theorem Provers Inês Hipolito1 Nova University of Lisbon Abstract: Is it possible to achieve such a proof that is independent of both acts and dispositions of the human mind? Plato is one of the great contributors to the foundations of mathematics. He discussed, 2400 years ago, the importance of clear and precise definitions as fundamental entities in mathematics, independent of the human mind. In the seventh book of his masterpiece, The Republic, Plato states “arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument” (525c). In the light of this thought, I will discuss the status of mathematical entities in the twentieth first century, an era when it is already possible to demonstrate theorems and construct formal axiomatic derivations of remarkable complexity with artificial intelligent agents the modern theorem provers.
    [Show full text]
  • Lab 8: Recursion and Fractals
    Lab 8: Recursion and Fractals In this lab you’ll get practice creating fractals with recursion. You will create a class that has will draw (at least) two types of fractals. Once completed, submit your .java file via Moodle. To make grading easier, please set up your class so that both fractals are drawn automatically when the constructor is executed. Create a Sierpinski triangle Step 1: In your class’s constructor, ask the user how large a canvas s/he wants. Step 2: Write a method drawTriangle that draws a triangle on the screen. This method will take the x,y coordinates of three points as well as the color of the triangle. For now, start with Step 3: In a method createSierpinski, determine the largest triangle that can fit on the canvas (given the canvas’s dimensions supplied by the user). Step 4: Create a method findMiddlePoints. This is the recursive method. It will take three sets of x,y coordinates for the outer triangle. (The first time the method is called, it will be called with the coordinates determined by the createSierpinski method.) The base case of the method will be determined by the minimum size triangle that can be displayed. The recursive case will be to calculate the three midpoints, defined by the three inputs. Then, by using the six coordinates (3 passed in and 3 calculated), the method will recur on the three interior triangles. Once these recursive calls have finished, use drawTriangle to draw the triangle defined by the three original inputs to the method.
    [Show full text]
  • Abstract Recursion and Intrinsic Complexity
    ABSTRACT RECURSION AND INTRINSIC COMPLEXITY Yiannis N. Moschovakis Department of Mathematics University of California, Los Angeles [email protected] October 2018 iv Abstract recursion and intrinsic complexity was first published by Cambridge University Press as Volume 48 in the Lecture Notes in Logic, c Association for Symbolic Logic, 2019. The Cambridge University Press catalog entry for the work can be found at https://www.cambridge.org/us/academic/subjects/mathematics /logic-categories-and-sets /abstract-recursion-and-intrinsic-complexity. The published version can be purchased through Cambridge University Press and other standard distribution channels. This copy is made available for personal use only and must not be sold or redistributed. This final prepublication draft of ARIC was compiled on November 30, 2018, 22:50 CONTENTS Introduction ................................................... .... 1 Chapter 1. Preliminaries .......................................... 7 1A.Standardnotations................................ ............. 7 Partial functions, 9. Monotone and continuous functionals, 10. Trees, 12. Problems, 14. 1B. Continuous, call-by-value recursion . ..................... 15 The where -notation for mutual recursion, 17. Recursion rules, 17. Problems, 19. 1C.Somebasicalgorithms............................. .................... 21 The merge-sort algorithm, 21. The Euclidean algorithm, 23. The binary (Stein) algorithm, 24. Horner’s rule, 25. Problems, 25. 1D.Partialstructures............................... ......................
    [Show full text]