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6433 results in Logic, categories and Sets

Page 318 of 322

  • 3 - Other approaches to computability: Church's thesis

  • Nigel Cutland
  • Book:
    Computability
    Published online:
    28 May 2018
    Print publication:
    19 June 1980, pp 48-71

  • Summary

    Over the past fifty years there have been many proposals for a precise mathematical characterisation of the intuitive idea of effective computability. The URM approach is one of the more recent of these. In this chapter we pause in our investigation of URM-computability itself to consider two related questions.

  • How do the many different approaches to the characterisation of computability compare with each other, and in particular with URM-computability?

  • How well do these approaches (particularly the URM approach) characterise the informal idea of effective computability?

    • The first question will be discussed in §§ 1-6; the second will be taken up in § 7. The reader interested only in the technical development of the theory in this book may omit §§ 3-6; none of the development in later chapters depends on these sections.

    Other approaches to computability

    The following are some of the alternative characterisations that have been proposed:

  • (a) Gödel-Herbrand-Kleene (1936). General recursive functions defined by means of an equation calculus. (Kleene [1952], Mendelson [1964].)

  • (b) Church (1936). λ-definable functions. (Church [1936] or [1941].)

  • (c) Gödel-Kleene (1936). μrecursive functions and partial recursive functions (§ 2 of this chapter.).

  • (d) Turing (1936). Functions computable by finite machines known as Turing machines. (Turing [1936]; § 4 of this chapter.)

  • (e) Post (1943). Functions defined from canonical deduction systems. (Post [1943], Minsky [1967]; § 5 of this chapter.)

  • (f) Markov (1951). Functions given by certain algorithms over a finite alphabet. (Markov [1954], Mendelson [1964]; § 5 of this chapter.)

  • (g) Shepherdson-Sturgis (1963). URM-computable functions. (Shepherdson & Sturgis [1963].)

    • There is great diversity among these various approaches; each has its own rationale for being considered a plausible characterisation of computability. The remarkable result of investigation by many researchers is the following:

    The Fundamental result

    Each of the above proposals for a characterisation of the notion of effective computability gives rise to the same class of functions, the class that we have denoted ℘.

  • 7 - Recursive and recursively enumerable sets

  • Nigel Cutland
  • Book:
    Computability
    Published online:
    28 May 2018
    Print publication:
    19 June 1980, pp 121-142

  • Summary

    The sets mentioned in the title to this chapter are subsets of ℕ corresponding to decidable and partially decidable predicates. We discuss recursive sets briefly in § 1. The major part of this chapter is devoted to the study of recursively enumerable sets, beginning in § 2; many of the basic properties of these sets are derived directly from the results about partially decidable predicates in the previous chapter. The central new result in § 2 is the characterisation of recursively enumerable sets that gives them their name: they are sets that can be enumerated by a recursive (or computable) function.

    In §§ 3 and 4 we introduce creative sets and simple sets: these are special kinds of recursively enumerable sets that are in marked contrast to each other; they give a hint of the great variety existing within this class of sets.

    Recursive sets

    There is a close connection between unary predicates of natural numbers and subsets of ℕ: corresponding to any predicate M(x) we have the set {x : M(x) holds}, called the extent of M (which could, of course, be 0); while to a set A ⊆ ℕ there corresponds the predicate ‘x ∈ A’. The name recursive is given to sets corresponding in this way to predicates that are decidable.

    Definition

    Let A be a subset of ℕ. The characteristic function of A is the function cA given by

    Then A is said to be recursive if cA is computable, or equivalently, if ‘x ∈ A’ is a decidable predicate.

    Notes

    1. For obvious reasons, recursive sets are also called computable sets.

    2. If cA is primitive recursive, the set A is said to be primitive recursive.

    3. The idea of a recursive set can be extended in the obvious way to subsets of ℕn (n > 1), although in the text we shall (as is common practice) restrict the use of the term to subsets of ℕ.

  • 9 - Reducibility and degrees

  • Nigel Cutland
  • Book:
    Computability
    Published online:
    28 May 2018
    Print publication:
    19 June 1980, pp 157-181

  • Summary

    In earlier chapters we have used the technique of reducing one problem to another, often as means of demonstrating undecidability. We did this, for instance, in the proof of theorem 6-1.4 by showing that there is a total computable function k such that x ∈ Wx ⇔ ϕk(x) = 0, i.e. we used the function k to transform or reduce each instance of the general problem ‘x ∈ Wx’ to an instance of the general problem ‘ϕx = 0’. In this chapter we consider two ways of making the idea of reducibility precise, and for each we discuss the associated notion of degree (of difficulty) that arises.

    It is more convenient to deal with reducibility between sets rather than between problems, remembering that any problem is represented by a set of numbers. The informal idea of a set A being reducible to a set B can be expressed in various ways: for instance

  • (a) ‘Given a decision procedure for the problem’ x ∈ B’, we can construct one for ‘x ∈ A’.’

  • (b) ‘For someone who knows all about B, there is a mechanical procedure (that uses his knowledge of B) for deciding questions about A.’

  • (c) ‘Questions about A are no harder than questions about B.’

  • (d) ‘The degree of difficulty of the problem’ x ∈ A’ is no greater than that of the problem ‘x ∈ B’.’

    • It turns out that there are several non-equivalent ways of making this idea precise. The differences between these consist in the manner and extent to which information about B is allowed to be used to settle questions about A. In §§ 1-3 we shall investigate one of the simplest notions of reducibility, called many-one reducibility, which includes all of our earlier uses of the informal idea. In the final sections we shall discuss a more general notion known as Turing reducibility.

  • 8 - Arithmetic and Gödel's incompleteness theorem

  • Nigel Cutland
  • Book:
    Computability
    Published online:
    28 May 2018
    Print publication:
    19 June 1980, pp 143-156

  • Summary

    The celebrated incompleteness theorem of Gödel [1931] is one of many results about formal arithmetic that involve an interplay between computability and logic. Although full proofs in this area are beyond the scope of this book, we are able to outline some of the arguments discovered by Gödel and others. We shall highlight particularly the part played by computability theory, which in many cases can be viewed as an application of the phenomenon of creative and productive sets.

    In §§ 1 and 2 we present some results about formal arithmetic that lead up to the full Gödel incompleteness theorem in § 3. In the final section the question of undecidability in formal arithmetic, already touched upon in § 1, is taken up again. Our presentation in this chapter does not assume any knowledge of formal logic.

    Formal arithmetic

    The formalisation of arithmetic begins by specifying a formal logical language L that is adequate for making statements of ordinary arithmetic of the natural numbers. The language L has its own alphabet, which includes the symbols 0, 1, +, x, = (having the obvious meanings), and also symbols for logical notions as follows: ¬ (‘not’), ∧ (‘and’), ∨ (‘or’), → (‘implies’), ∀ (‘for all’), ∃ (‘there exists’). (In this chapter we will reserve the symbols ∀, ∃ for use in L, and write the phrase ‘for all’ and ‘there exists’ when needed in informal contexts.) In addition, L has symbols x, y, z,… for variables, and brackets (and), and there may be other symbols besides.

    The statements (or formulas) of L are defined to be the meaningful finite sequences of symbols from the alphabet of L. For instance, the statement

    ∃y(y×(1 + 1) = x)

    is the formal counterpart of the informal statement ‘x is even’.

  • 4 - Numbering computable functions

  • Nigel Cutland
  • Book:
    Computability
    Published online:
    28 May 2018
    Print publication:
    19 June 1980, pp 72-84

  • Summary

    We return now to the study of URM-computable functions. Henceforth the term computable standing alone means URM-computable, and program means URM program.

    The key fact established in this chapter is that the set of all programs is effectively denumerable1: in other words there is an effective coding of programs by the set of all natural numbers. Among other things, it follows that the class ℘ is denumerable, which implies that there are many functions that are not computable. In § 3 we discuss Cantor's diagonal method, whereby this is established.

    The numbering or coding of programs, and particularly its effectiveness, is absolutely fundamental to the development of the theory of computability. We cannot overemphasise its importance. From it we obtain codes or indices for computable functions, and this means that we are able to pursue the idea of effective operations involving such codes.

    In § 4 we prove the first of two important theorems involving codes of functions: the so-called s–m–n theorem of Kleene. (The second theorem is the main result of chapter 5.)

    Numbering programs

    We first explain the terminology that we shall use.

    Definitions

  • (a) A set X is denumerable if there is a bijection f: X → ℕ. (Note. The term countable is normally used to mean finite or denumerable; thus, for infinite sets, countable means the same as denumerable. The term countably infinite is used by some authors instead of denumerable.)

  • (b) An enumeration of a set X is a surjection g: ℕ → X; this is often represented by writing

    • X = {x0, x1, x2, …}

    where xn = g(n). This is an enumeration without repetitions if g is injective.

  • (c) Let x be a set of finite objects (for example a set of integers, or a set of instructions, or a set of programs); then X is effectively denumerable if there is a bijection f : X → ℕ such that both f and f-l are effectively computable functions.

  • 2 - Generating computable functions

  • Nigel Cutland
  • Book:
    Computability
    Published online:
    28 May 2018
    Print publication:
    19 June 1980, pp 25-47

  • Summary

    In this chapter we shall see that various methods of combining computable functions give rise to other computable functions. This will enable us to show quite rapidly that many commonly occurring functions are computable, without writing a program each time – a task that would be rather laborious and tedious.

    The basic functions

    First we note that some particularly simple functions are computable; from these basic functions (defined in lemma 1.1 below) we shall then build more complicated computable functions using the techniques developed in subsequent sections.

    Lemma

    The following basic functions are computable:

  • (a) the zero functionO(O(x) = 0 for all x);

  • (b) the successor function x + 1

  • (c) for each n ≥ 1 and 1 ≤ i ≤ n, the projection function given by

    • Proof. These functions correspond to the arithmetic instructions for the URM. Specifically, programs are as follows:

  • (a) 0: program Z(l);

  • (b) x + l: program S(l);

  • (c) program T(i, 1).

    • Joining programs together

    In each of §§ 3-5 below we need to write programs that incorporate other programs as subprograms or subroutines. In this section we deal with some technical matters so as to make the program writing of later sections as straightforward as possible.

    A simple example of program building is when we have programs P and Q, and we wish to write a program for the composite procedure: first do P, and then do Q. Our instinct is to simply write down the instructions in P followed by the instructions in Q. But there are two technical points to consider.

    Suppose that P = I1,I2,…, Is. A computation under P is completed when the ‘next instruction for the computation’ is Iv for some v > s; we then require the computation under our composite program to proceed to the first instruction of Q.

  • Prologue. Prerequisites and notation

  • Nigel Cutland
  • Book:
    Computability
    Published online:
    28 May 2018
    Print publication:
    19 June 1980, pp 1-6

  • Summary

    The only prerequisite to be able to read this book is familiarity with the basic notations of sets and functions, and the basic ideas of mathematical reasoning. Here we shall review these matters, and explain the notation and terminology that we shall use. This is mostly standard; so for the reader who prefers to move straight to chapter 1 and refer back to this prologue only as necessary, we point out that we shall use the word function to mean a partial function in general. We discuss this more fully below.

    Sets

    Generally we shall use capital letters A, B, C,… to denote sets. We write x ∈ A to mean that x is a member of A, and we write x ∉ A to mean that x is not a member of A. The notation {x: … x…} where … x … is some statement involving x means the set of all objects x for which … x … is true. Thus {x : x is an even natural number} is the set {0,2,4,6,…}.

    If A, B are sets, we write A ⊆ B to mean that A is contained in B (or A is a subset of B); we use the notation A ⊂ B to mean that A ⊆ B but A ≠ B (i.e. A is a proper subset of B). The union of the sets A, B is the set {x : x ∈ A or x ∈ B (or both)}, and is denoted by A ∪ B; the intersection of A, B is the set {x: x ∈ A and x ∈ B} and is denoted by A ∩ B. The difference (or relative complement) of the sets A, B is the set {x : x ∈ A and x ∉ B} and is denoted by A \B.

  • 12 - Complexity of computation

  • Nigel Cutland
  • Book:
    Computability
    Published online:
    28 May 2018
    Print publication:
    19 June 1980, pp 212-235

  • Summary

    In the real world of computing, the critical question about a function f is not Is f computable?, but rather Is f computable in practical terms? In other words, Is there a program for f that will compute f in the time (or space) we have available? The answer depends partly on our skill in writing programs and the sophistication of our computers; but intuitively we feel that there is an additional factor which can be described as the ‘intrinsic complexity’ of the function f itself. The theory of computational complexity, which we introduce in this chapter, has been developed in order to be able to discuss such questions and to aid the study of the more practical aspects of computability.

    Using the URM approach, we can measure the time taken to compute each value of a function f by a particular program, on the assumption that each step of a URM computation is performed in unit time. The time of computation thus defined is an example of a computational complexity measure that reflects the complexity or efficiency of the program being used. (Later we shall mention other complexity measures.)

    With a notion of complexity of computation made precise, it is possible to pursue questions such as How intrinsically complex is a computable function f? and Is it possible to find a ‘best’ program for computing f?

    The theory of computational complexity is a relatively new field of research; we shall present a small sample of results that have a bearing on the questions raised above. At the end of the chapter we shall provide suggestions for the reader wishing to pursue this topic further.

    We begin in § 1 by defining some notation; after some discussion we proceed to show that there are arbitrarily complex computable functions. Section 2 is devoted to the surprising and curious Speed-up theorem of M. Blum, which shows in particular that there are computable functions having no ‘best’ program.

  • Preface

  • Nigel Cutland
  • Book:
    Computability
    Published online:
    28 May 2018
    Print publication:
    19 June 1980, pp viii-x

  • Summary

    The emergence of the concept of a computable function over fifty years ago marked the birth of a new branch of mathematics: its importance may be judged from the fact that it has had applications and implications in fields as diverse as computer science, philosophy and the foundations of mathematics, as well as in many other areas of mathematics itself. This book is designed to be an introduction to the basic ideas and results of computability theory (or recursion theory, as it is traditionally known among mathematicians).

    The initial purpose of computability theory is to make precise the intuitive idea of a computable function; that is, a function whose values can be calculated in some kind of automatic or effective way. Thereby we can gain a clearer understanding of this intuitive idea; and only thereby can we begin to explore in a mathematical way the concept of computability as well as the many related ideas such as decidability and effective enumerability. A rich theory then arises, having both positive and negative aspects (here we are thinking of non-computability and undecidability results), which it is the aim of this book to introduce.

    We could describe computability theory, from the viewpoint of computer science, as beginning with the question What can computers do in principle (without restrictions of space, time or money)? – and, by implication – What are their inherent theoretical limitations? Thus this book is not about real computers and their hardware, nor is it about programming languages and techniques. Nevertheless, our subject matter is part of the theoretical background to the real world of computers and their use, and should be of interest to the computing community.

    For the basic definition of computability we have used the ‘idealised computer’ or register machine approach; we have found that this is readily grasped by students, most of whom are aware of the idea of a computer.

  • 13 - Further study

  • Nigel Cutland
  • Book:
    Computability
    Published online:
    28 May 2018
    Print publication:
    19 June 1980, pp 236-238

  • Summary

    Our basic study of computability has been designed so that it could serve as a stepping stone to more advanced or more detailed study in any of several directions. In this brief postlude, we shall mention some of the areas in which further study could be pursued, and we offer some suggestions for further reading. The divisions below are not hard and fast, and there are many interrelations between the various areas we mention.

    Computability Further study of the theoretical notion of computability (the starting point of this book) could be pursued in two directions: (a) more detailed examination of other equivalent approaches to computability (which we surveyed in chapter 3); (b) examination of more restricted notions of effective computability, involving, for instance, finite automata and similar devices.

    Some references (several historical) for (a) were given in chapter 3. For both (a) and (b) we suggest the books of Minsky [1967] (a very comprehensive treatment), Arbib [1969], or Engeler [1973].

    Recursion theory We use this traditional title under which to mention more advanced ideas arising out of the notion of computability on ℕ, such as we began to pursue in chapters 7, and 9 to 11. Specific areas include:

    Hierarchies: there are various ways to extend the sequence beginning ‘recursive, r.e., …’ to obtain a hierarchy of kinds of set, each kind of set having more difficult decision problem than the preceding one. Among the important hierarchies that have been studied are the arithmetic hierarchy, the hyperarithmetic hierarchy, and the analytical hierarchy.

    Reducibilities and degrees: between ≤m and ≤T there is a spectrum of reducibilities that could be investigated. For the student wishing to delve further into Turing reducibility, the next step would be to master a proof of the Friedberg-Muchnik solution to Post's problem, before proceeding to further results and proofs in this area, some of which we mentioned in chapter 9.

Page 318 of 322