Computability, douglas s bridges

Graduate Texts in Mathematics 146 Editorial Board J.H Ewing F.W Gehring P.R Halmos Graduate Texts in Mathematics 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 TAKEUTI/ZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFFER Topological Vector Spaces HILTON/STAMMBACH A Course in Homological Algebra MAc LANE Categories for the Working Mathematician HUGHES!PIPER Projective Planes SERRE A Course in Arithmetic TAKEUTI/ZARING Axiometic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable 2nd ed BEALS Advanced Mathematical Analysis ANDERSON/FULLER Rings and Categories of Modules 2nd ed GOLUBITSKy/GUILEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLAIT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nd ed HUSEMOLLER Fibre Bundles 3rd ed HUMPHREYS Linear Algebraic Groups BARNES/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and Its Applications HEWIIT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKI/SAMUEL Commutative Algebra Vol I ZARISKI/SAMUEL Commutative Algebra Vol II JACOBSON Lectures in Abstract Algebra I Basic Concepts JACOBSON Lectures in Abstract Algebra II Linear Algebra JACOBSON Lectures in Abstract Algebra III Theory of Fields and Galois Theory HIRSCH Differential Topology SPITZER Principles of Random Walk 2nd ed WERMER Banach Algebras and Several Complex Variables 2nd ed KELLEY!NAMIOKA et al Linear Topological Spaces MONK Mathematical Logic GRAUERT!FRITZSCHE Several Complex Variables ARVESON An Invitation to C·-Algebras KEMENY/SNELl)KNAPP Denumerable Markov Chains 2nd ed ApOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed SERRE Linear Representations of Finite Groups GILLMAN/JERISON Rings of Continuous Functions KENDIG Elementary Algebraic Geometry LoEvE Probability Theory I 4th ed LoEVE Probability Theory II 4th ed MOISE Geometric Topology in Dimensions and continued qfter Index Douglas S Bridges Computability A Mathematical Sketchbook With 29 Illustrations Springer Science+Business Media, LLC Douglas S Bridges Department of Mathematics University of Waikato Private Bag 3105 Hamilton, New Zealand Editorial Board J H Ewing Department of Mathematics Indiana University Bloomington, IN 47405 USA F W Gehring Department of Mathematics University of Michigan Ann Arbor, MI 48109 USA P.R Halmos Department of Mathematics Santa Clara University Santa Clara, CA 95053 USA Mathematics Subject Classifications (1991): 03Dxx Library of Congress Cataloging-in-Publication Data Bridges, D.S (Douglas S.), 1945Computability : a mathematical sketchbook / Douglas S Bridges p cm - (Graduate texts in mathematics) lncludes bibliographical references and index ISBN 978-1 -4612-6925-0 ISBN 978-1 -4612-0863-1 (eBook) DOI 10.1007/978-1-4612-0863-1 Computable functions Title Il Series QA9.59.B75 1994 511.3-dc20 93-21313 Printed on acid-free paper © 1994 Springer Science+Business Media New York Originally published by Springer-Verlag New York,lnc in 1994 Softcover reprint of the hardcover 1st edition 1994 AII rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely byanyone Production managed by Hal Henglein; manufacturing supervised by Vincent Scelta Photocomposed pages prepared from the author's LaTeX file 987654321 ISBN 978-1-4612-6925-0 For Vivien, Jain, Hamish, and Catriona 'I can't believe thaU' said Alice 'Can't you?' the Queen said in a pitying tone 'Try again: draw a long breath and shut your eyes.' Alice laughed 'There's no use trying,' she said: 'One can't believe impossible things ' 'I daresay you haven't had much practice, ' said the Queen LEWIS CARROLL, Through the Looking Glass Preface My intention in writing this book is to provide mathematicians and mathematically literate computer scientists with a brief but rigorous introduction to a number of topics in the abstract theory of computation, otherwise known as computability theory or recursion theory It develops major themes in computability, such as Rice's Theorem and the Recursion Theorem, and provides a systematic account of Blum's abstract complexity theory up to his famous Speed-up Theorem A relatively unusual aspect of the book is the material on computable real numbers and functions, in Chapter Parts of this material are found in a number of books, but I know of no other at the senior/beginning graduate level that introduces elementary recursive analysis as a natural development of computability theory for functions from natural numbers to natural numbers This part of the book is definitely for mathematicians rather than computer scientists and has a prerequisite of a first course in elementary real analysis; it can be omitted, without rendering the subsequent chapters unintelligible, in a course including the more standard topics in computability theory found in Chapters 4-6 I believe, against the trend towards weighty, all-embracing treatises (vide the typical modern calculus text), that many mathematicians would like to be able to purchase books that give them insight into unfamiliar branches of the subject in a relatively short compass and without requiring a major investment of time, effort, or money Following that belief, I have had to exclude from this book many topics-such as detailed proofs of the equivalence of various mathematical models of computation, the theory of degrees of unsolvability, and polynomial and nonpolynomial complexitywhose absence will be deplored by at least some of the experts in the field I hope that my readers will be inspired to pursue their study of recursion theory in such major works as [9, 24, 28, 29] A number of excellent texts on computability theory are primarily aimed at computer scientists rather than mathematicians, and so not always maintain the level of rigour that would be expected in a modern text on, say, abstract algebra I have tried to maintain that higher level of rigour lSome of the work in this book-notably, Proposition (4.28) and the application of the Recursion Theorem preceding Exercises (5 14)-appears to be original Vlll Preface throughout, even at the risk of deflecting the interest of mathematically insecure computer scientists Ideally, all mathematics and computer science majors should be exposed to at least some of the material found in this book It horrifies me that in some universities such majors can still graduate ignorant of the theoretical limitations of the computer, as expressed, for example, by the undecidability of the halting problem (Theorem (4.2)) A short course on computability, accessible even to students below junior level, would comprise Chapters 1-3 and the material in Chapter up to Exercises (4.7) A longer course for more advanced undergraduates would also include Rice's theorem and the Recursion Theorem, from Chapter 5, and at least parts of Chapter The entire book including the difficult material on recursive analysis from Chapter 4, would be suitable for a course for bright seniors or beginning graduate students I have tried to make the book suitable for self-study To this end, it includes solutions for most of the exercises Those exercises for which no solutions are given have been marked with the asterisk (*); of varying levels of difficulty, they provide the instructor with material for homework and tests The exercises form an integral part of the book and are not just there for the student's practice; many of them develop material that is used in later proofs, which is another reason for my inclusion of solutions I\Iy interest in constructive mathematics [5] leads me to comment here on the logic of computability theory This is classical logic, the logic used by almost all mathematicians in their daily work However, the use of classical logic has some perhaps undesirable consequences Consider the following definition of a function f on the set N of natural numbers: for all n, f(n) equals if the Continuum Hypothesis is true, and equals if the Continuum Hypothesis is false Since 'most mathematicians are formalists on weekdays and Platonists on Sundays', at least on Sundays most of us would accept this as a good definition of a function f According to classical logic, f is computable because there exists an algorithm that computes it: that algorithm is either the one which, applied to any natural number n, outputs L or else the one which, applied to any natural number n, outputs O But the Continuum Hypothesis is independent of the axioms of ZFC (ZermeloFraenkel set theory plus the axiom of choice), the standard framework of mathematics, so we will never be able to tell, using ZFC alone, which of the two algorithms actually is the one that computes f It appears from this example, eccentric though it may be, that the standard theory of computation does not exactly match computational practice, 2Thc Continuum Hypothesis (CH) says that the smallest cardinal number greater than ~o, the cardinality of N, is No , the cardinality of the set of all subsets of N The work of Cohen [13] and G6del [17] shows that neither CH nor its negation can be proved within Zermelo-Fraenkel set theory plus the axiom of choice: see also [3], pages 420-428 Preface IX in which we would expect to pin down the algorithms that we use A facetious question may reinforce my point: what would happen to an employee who, in response to a request that he write software to perform a certain computation, presented his boss with two programs and the information that, although one of those programs performed the required computation, nobody could ever tell which one? With classical logic there seems to be no way to distinguish between functions that are computed by programs which we can pin down and those that are computable but for which there is no hope of our telling which of a range of programs actually performs the desired computation To handle this problem successfully, we need a different logic, one capable of distinguishing between existence in principle and existence in practice For example, with constructive (intuitionistic) logic the problem disappears,3 since f is then not properly defined: it is only properly defined if we can decide the truth or falsehood of the Continuum Hypothesis (which we cannot) and therefore which of the two possible algorithms computes f Having said this, let me stress that, despite the inability of classical logic to make certain distinctions of the type I have just dealt with, I have followed standard practice and used classical logic throughout this book Not only the logic but also most of the material that I have chosen is standard, although some of the exercises and examples are new I have drawn on a number of books, including [34] for the treatment of Turing machines in Chapter 1; [20] for the first parts of Chapters and 5; and [9, 14, 29] for parts of Chapter The origins of my book lie in courses I gave at the University of Buckingham (England), New Mexico State University (USA), and the University of Waikato (New Zealand) I am grateful to the students in those classes for the patience with which they received various slowly improving draft versions Special thanks are due to Fred Richman for many illuminating conversations about recursion theory; to Paul Halmos for his advice and encouragement; and to Cris Calude, Nick Dudley Ward, Graham French, Hazel Locke, and Steve Merrin, all of whom have read versions of the text and made many helpful corrections and suggestions As always, it is my wife and children who suffered most as the prolonged birth of this work took so much of my care and attention; I present the book to them with love and gratitude May 1993 Douglas S Bridges 3For a development of computability theory using intuitionistic logic see Chapter of [8] 4The first drafts of this book were prepared using the r3 Scientific Word Processing System The final version was produced by converting the drafts to rEX and then using Scientific Word r3 and Scientific Word are both products of TCI Software Research, Inc The diagrams were drawn with Aldus Freehand v 3.1 (©Aldus Corporation) 166 Solutions to Exercises Solutions for Chapter (6.1.1) (i) Take Ii == 'Pi for each i Then BI is automatically satisfied But if B2 holds, then {i EN: 'Pi(O) = O} is a recursive set; since this set clearly respects indices, is nonempty, and is a proper subset of N, this contradicts Rice's Theorem (ii) (6.1.2) costs' : N:3 Take ~ri(n) == for all i and n It is clear that Bl is satisfied On the other hand, the function N defined by -+ costs'('i, n, k) costs(i, n, k) I o if i if i ifi i= = = j, j and k = 0, j and k ~ 1, is computable, and costs' (i, n, k) I o if ,: = k, otherwise r' satisfies B2 Using r' as our complexity measure, we see that the cost of computing 'Pj (n) is 0; in other words, it costs nothing to decide whether or not n belongs to the recursive set S In the particular case where S is taken as the set of all prime numbers, this situation certainly does not reflect reality: it is well known that testing integers for primality is an extremely costly business Indeed, all known algorithms for primality testing have cost that grows exponentially as a function of the size, in bits, of the integer under test For further information on this topic, see Chapter of [:33] So (6.1.3) It is clear from axiom BI, applied to 'Pi and Ii, that domainb:) = domain( 'Pd On the other hand, given positive integers nand k, and using axiom B2, we can decide whether or not there exists j ::; k such that Ii (n) = i If such j exists, then, by BI, 'Pi(n) is defined, so I~(n) is defined; moreover, by comparing !('Pi(n)) with k - j we can decide whether or not ,;(n) equals k If, however, no such j exists, then it is impossible for ~fHn) to equal k Thus the function costs' : N -+ N, defined by costs' (i, n, k) is computable o ifl;(n) = k, otherwise, Solutions for Chapter (6.1.4) 167 We have t(k) G(i,n,k) = Lcosts(i,n,j) j=O Since t and costs are total computable functions, so is G (6.1.5) The existence of s is a simple consequence of the s-m-n theorem We compute G(n,i,j,k) as follows Compute first v(n) and then costs(i,v(n),j) If the latter equals 1, then 'Yi(v(n» = j, cpi(v(n» is defined, and we can compute costs(cpi(v(n»,n,k); if that equals 1, then cps(i)(n) = CPcp;ov(n)(n) is defined and we set G(n, i,j, k) = 'Ys(i) (n) On the other hand, if either costs(i,v(n),j) = or costs(i,v(n),j) = and costs(cpi(v(n»,n,k) = 0, we set G(n,i,j,k) (6.3) = O Since cI>(i,n) = mink[costs(i,n,k) = 1], the computability of cI> follows from Exercise (2.7.3) (6.5.1) Define 1': =1 + 'Yi + CPi; = then, by Exercise (6.1.3), r' 'Yb,'YL 'Y~, is a complexity measure Let F : N2 ~ N be a total computable function By the s-m-n theorem, there exists a total computable function s : N ~ N such that cps(i)(n) = F(n,'Yi(n» for each i and for all n E domainbi) Applying the Recursion Theorem, we obtain an index 1/ such that CPs (v) = CPV' Thus 'Y~ = + 'Yv + CPs(v) = + 'Yv + F(·, 'Yv(-), so 'Y~(n) > F(n,'Yv(n» for all n E domain(cpv) Despite appearances, this result does not contradict Theorem (6.4), since it does not guarantee that 'Y~(n) > F(n,'YII(n» infinitely often Indeed, it follows from Theorem (6.4) that domain (CPII) must be finite 168 Solutions to Exercises ~~, ( \ \'llB,R / , FIGURE 29 The Turing machine Tk in solution (6.7.1) (6.7.1) Let ii (n) the number of distinct cells visited by Mi during the computation of k on N There is a one-one total computable function h : N -'> N such that ~ = Mh(k) for each k Moreover, as is easily verified, the range of h is a recursive subset of N, and the partial function

F(O,O) and set i == h(k) Then 1i(O) = k + 2, so that 1';(0) = max{O, k + - N By the s-m-n theorem, there exists a total computable function s : N -> N such that 'Pa(m) = w(m, ) If 'Pm is total, then, by the foregoing, 'Ps(m) is total, and P(m,'Pa(m)(n),n) holds for each n; whence 'Pm(n):S: 'Ps(m)(n) and So if 'Ps(m)(n) :S: I'i(n) :S: F(n, 'Ps(m) (n», then n :S: i (6.16) Using Theorem (6.4), construct a total computable function F : N -> N such that I'i(n) :S: F(n,I':(n» almost everywhere We may assume that n < F(m, n) < Fern, n + 1) for all m and n According to Theorem (6.13), for each total computable function t : N + N there exists a total computable function f : N -> N such that fen) ~ ten) for all n, and such that if fen) :S: I'i(n) :S: F(n, fen»~, then n :::; i Consider any i, n such that n > i, IHn) :S: fen), and li(n) :S: F(n,I'Hn» If fen) :S: I'i(n), then fen) :S: I'i(n) :S: F(n, I'~(n» :S: F(n, fen»~, so n :S: i, a contradiction; hence I'i(n) < fen) It follows that Cf' C Cf· The reverse inequality follows from the hypothesis that (n) :S: Aii (n) for aU i and all n E domain('Pi) 1': (6.19) If 'Pi(n) is defined, then, by BI, so is I'i(n) So, using Exercise (6.1.4), we can decide, for each j < n, whether or not P(i, j, n) holds There will be at most n values of j < n for which P(i,j,n) holds, and therefore at most n corresponding values 'Pj (n) Straightfoward computations enable us to find k, from among the n + values 0,1, , n, such that k i- 'PjCn) for all j < n for which P(i,j, n) holds (Note that in view of BI, 'Pj(n) is defined for each such j.) Hence Wei, n) is defined and at most n 172 Solutions to Exercises (6.22.1) If n ::; i, then C(e, i, n) is defined to be 0, which is certainly both finite and recursive In particular, C(e, i, 0) is defined, finite, and recursive Assume that for 0::; m < n, if C(e, i, m) is defined, then it is finite and recursive If C(e, i, n) is defined, then in order to complete our inductive proof we need only deal with the case i < n Then C(e, i, m) is definedand therefore both finite and recursive for ::; m < n, and Is(e,j+l)(n) is defined whenever i ::; j < n Given j E N, we can decide whether or not i ::; j < n Moreover, using our induction hypothesis and Exercise (6.1.4), we can decide, for each j with i ::; j < n, whether or not j rt n-l UC(e,i,m) and Ij(n) < F(n, Is(e,j+l) (n)); m=O so C(e, i, n), which is obviously finite, is recursive (6.24.1) Let f be the identity function id : N - N Then f is computed by the normalised binary Turing machine M == {{O}, 0, 0, O} (cf the solution to Exercise (5.7.1)) Let v be the index of M, and define a complexity measure r == 10, II, by li(n) == ,:(n) + Iv - ii, where ,: is defined as in the proof of the Speed-up Theorem Consider any total computable function F : N2 - N such that F(m, n + 1) 2: F(m, n) for all m, n For each j E IND(f) with j I- v, and for all n E N, we have F(n, Ij(n) 2: IJ(n) Hence f + Iv - jl > IJ(n) ~ = Iv(n) is not F -speedable relative to the complexity measure r (6.24.3) Let r be any complexity measure, and take F(m, n) == n + for all m, n E N Then F-speedable functions exist, by the Speed-up Theorem Let f be anyone of them Since is a set of nonnegative integers, it has a least member; that is, there exists v E IND(f) such that IV(O) = minhi(O) : i E IND(f)} For all j E IND(f) we have F(O, Ij (0)) = Ij (0) + > IV(O) References [1] Aberth, Oliver: Computable Analysis New York: McGraw-Hill 1980 [2] Barendregt, H.P.: The Lambda Calculus: its Syntax and Semantics Amsterdam: North-Holland 1981 [3] Barwise, J (ed.): Handbook of Mathematical Logic Amsterdam: North-Holland 1977 [4] Beeson, Michael J.: Foundations of Constructive Mathematics New York-Heidelberg-Berlin: Springer-Verlag 1985 [5] Bishop, Errett, and Bridges, Douglas S.: Constructive Analysis (Grundlehren der math Wissenschaften 279) New York-HeidelbergBerlin: Springer-Verlag 1985 [6] Blum, M.: A machine-independent theory of the complexity of recursive functions J Assoc Comput Mach 14,322-336 (1967) [7] Blum, M.: On effective procedures for speeding-up algorithms ACM Symposium on Theory of Computing, 43-53 (1969) [8] Bridges, D.S., and Richman, Fred: Varieties of Constructive Mathematics (London Mathematical Society Lecture Notes 97) Cambridge: Cambridge University Press 1987 [9] Calude, C.: Theories of Computational Complexity Amsterdam: North-Holland 1988 [10] Calude, C., and Zimand, M.: Recursive Baire classification and speedable functions, Zeitschr math Logik Grundlagen Math 38, 169-178 (1992) [11] Ceitin, G.S.: Algorithmic operators in constructive complete separable metric spaces (Russian) Doklady Akad Nauk 128, 49-52 (1959) [12] Chaitin, G.: Lisp Program-size Complexity Appl Math and Comput 49, 79-93 (1992) [13] Cohen, Paul J.: Set Theory and the Continuum Hypothesis New York: Benjamin 1966 174 References [14J Cutland, N.J.: Computability, an Introduction to Recursive Function Theory Cambridge: Cambridge University Press 1980 [15J Dieudonne, J.: Foundations of Modern Analysis New York: Academic Press 1960 [16J Dowling, W.F.: Computer Viruses: diagonalization and fixed points Notices Amer Math Soc 37(7), 858-861 (1990) [17J G6del, Kurt: The Consistency of the Continuum Hypothesis (Ann Math Studies 3) Princeton: Princeton University Press 1940 [18J Halmos, P.R.: Naive Set Springer-Verlag 1974 Theory New York-Heidelberg-Berlin: [19] Kalmar, L.: An argument against the plausibility of Church's thesis In: Heyting, A (ed.), Constructivity in Mathematics (Proceedings of the Colloquium at Amsterdam, 1957) Amsterdam: North-Holland 1959 [20] Kfoury, A.J., Moll, R.N., and Arbib, M.A.: A Programming Approach to Computability New York-Heidelberg-Berlin: Springer-Verlag 1982 [21J Ko, Ker-I: Complexity Theory of Real Functions Boston-Basel-Berlin: Birkhaiiser 1991 [22] Kreisel, G., Lacombe, D., and Schoenfield, J.: Partial recursive functions and effective operations In: Heyting, A (ed.), Constructivity in Mathematics (Proceedings of the Colloquium at Amsterdam, 1957) Amsterdam: North-Holland 1959 [23J Machtey, M., and Young, P.: An Introduction to the General Theory of Algorithms Amsterdam: North-Holland 1978 [24J Odifreddi, P.: Classical Recursion Theory, Volume Amsterdam: North-Holland 1990 [25] Peter, Rozsa: Recursive Functions in Computer Theory Chichester: Ellis-Horwood 1981 [26J Pour-EI, Marian B., and Richards, Jonathan I.: Computability in Analysis and Physics New York-Heidelberg-Berlin: Springer-Verlag 1988 [27] Rayward-Smith, V.J.: A First Course in Computability Oxford: Blackwell Scientific Publications 1986 [28J Rogers, Hartley Jr.: Theory of Recursive Functions and Effective Computability New York: McGraw-Hill 1967 [29J Salomaa, Arto: Computation and Automata Cambridge: Cambridge University Press 1985 References 175 [30] Schnorr, C.P.: Does the computational speed-up concern programming? In Nivat, M (ed.): Automata, Languages and Programming Amsterdam: North-Holland 1973 [31] Todd, John: Introduction to the Constructive Theory of Functions Basel: Birkhaiiser Verlag 1963 [32] Weihrauch, Klaus: Computability Springer-Verlag 1987 New York-Heidelberg-Berlin: [33] Wilf, H.S.: Algorithms and Complexity Englewood Cliffs, New Jersey: Prentice-Hall Inc 1986 [34] Wood, Derick: Theory of Computation New York: Harper & Row 1987 If you are interested in pursuing the study of computability, try reading [28], [29], or [24] The first of these is rather dated, but certainly a classic; the other two are likely to become classics Other references for computability are [14], [20], [23], [27], [32], and [34] A comprehensive account of abstract complexity theories, including several not mentioned in this book, is given in [9] For an authoritative account of the complexity-theoretic analysis of real variables see [21] Accounts of recursive and constructive mathematics using informal intuitionistic logic are found in [4], [5], and [8] Index A acceptable programming system, 45, 83, 161 Ackermann's function, 33, 129 admissible, admissible for minimisation, 28 almost all, 95 almost everywhere, 95 alphabet, B Baire category, 115 base functions, 26, 125 blank,5 Blum's axioms, 93 Boolean functions, 21, 122 C canonical enumeration, 41 Cantor's Theorem, 50 Cartesian product, characteristic function, 38 Church's thesis, 32 Church-Markov-Turing thesis, 32, 161 classical logic, ix code, 39 completes a computation, complexity, 93 complexity class, 101, 170 complexity function, 93 complexity measure, 93 composite, composition, 125 Compression Theorem, 104 computable d-ary expansion, 59 computable analysis, 66 computable partial function, 22, 40, 52, 56-57, 87, 145 computable real number, 53 computable real number generator, 53 computable sequence, 66 computation, concatenation of sets, concatenation of strings, configuration, configuration, reaches, 37 constructive logic, ix Continuum Hypothesis, viii converges effectively, 66 converges effectively and uniformly, 67 cost function, 93 costs, 93, 166-167 countable, 35 cutoff, 29, 125 cyclic left shift, 17 D Decidability Problem, 47 decidable, 48 decision problem, 47 defined, diagonal argument, 50 div,57 domain, Double Recursion Theorem, 83 E effective enumeration, 35, 40-41 effective enumeration, diagonal, 80 effective sequence of open intervals, 67 effectively continuous, 69 effectively enumerable, 35 effectively uniformly continuous, 69 empty partial function, 21, 120 empty string, Index encoding, 39, 87 enll rneration, 35 Equivalence Problem, 47 erase, 21 erase, 122 F factorial function, 28, 125 fails to complete a computation, 10 final configuration, finite 8ubfunction, 86 G Gap Theorem, 101, 103 Gap Theorem, Uniform Version, 103 Goldbach Conjecture, 89 H Halmos, 169 halt, 11 halt state, Halting Problem, 47 halts in k steps, 37 halts in at most n steps, 37 Heine-Borel Theorem, 67 I identity function, 76 inadmissible, increasing, 66 IND,98 index set, 98 index, of a Turing machine, 41 index of