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5817 results in Programming Languages and Applied Logic

Page 290 of 291

  • 2 - Machines and semigroups

  • M. Holcombe
  • Book:
    Algebraic Automata Theory
    Published online:
    22 September 2009
    Print publication:
    19 August 1982, pp 25-75

  • Summary

    One of the achievements of modern science has been the realization that very few things in the world are completely static. The behaviour of many systems, both organic and synthetic, is influenced greatly by environmental changes. This interaction between a system and its environment can be vastly complicated and yet it is an area that we must try to understand if we are going to be in a position to predict the behaviour of the system and its effect on its environment.

    The particular type of analysis that we present here is based on techniques that are generally referred to as algebraic. In some cases we will draw on established algebraic results but in general it is a new type of algebra that has arisen from a desire to understand the behaviour of a system in an environment. This is perhaps the most refreshing aspect of the theory. Here, for a change, is a subject whose motivation can be linked to very real problems in the modern world, a subject that has a short but dramatic history and one which has played a large role in the development of the fundamentals of computer science. However its achievements have not been restricted to this case alone and we hope to illustrate this when we examine the examples at the end of this chapter.

    In many of the systems environmental changes alter the behaviour of the system and these changes in behaviour then affect the environment in some way.

  • Introduction

  • M. Holcombe
  • Book:
    Algebraic Automata Theory
    Published online:
    22 September 2009
    Print publication:
    19 August 1982, pp ix-xii

  • Summary

    In recent years there has been a growing awareness that many complex processes can be regarded as behaving rather like machines. The theory of machines that has developed in the last twenty or so years has had a considerable influence, not only on the development of computer systems and their associated languages and software, but also in biology, psychology, biochemistry, etc. The so-called ‘cybernetic view’ has been of tremendous value in fundamental research in many different areas. Underlying all this work is the mathematical theory of various types of machine. It is this subject that we will be studying here, along with examples of its applications in theoretical biology, etc.

    The area of mathematics that is of most use to us is that which is known as modern (or abstract) algebra. For a hundred years or more, algebra has developed enormously in many different directions. These all had origins in difficult problems in the theory of equations, number theory, geometry, etc. but in many areas the subject has taken on its own momentum, the problems arising from within the subject, and as a result there has been a general feeling that much of abstract algebra is of little practical value. The advent of the theory of machines, however, has provided us with new motivation for the development of algebra since it raises very real practical problems that can be examined using many of the abstract tools that have been developed in algebra.

  • 4 - The holonomy decomposition

  • M. Holcombe
  • Book:
    Algebraic Automata Theory
    Published online:
    22 September 2009
    Print publication:
    19 August 1982, pp 114-144

  • Summary

    The aim of this chapter is the description of a method for decomposing an arbitrary transformation semigroup into a wreath product of ‘simpler’ transformation semigroups, namely aperiodic ones and transformation groups. The origin of this theory is the theorem due to Krohn and Rhodes which gave an algorithmic procedure for such a decomposition. There are now various proofs of this result extant, some are set in the theory of transformation semigroups and others are concerned with the theory of state machines. In the light of the close connections between the two theories forged in chapter 2 we can expect a similar correspondence between the two respective decomposition theorems. The proof of the decomposition theorem for state machines has the advantage that it can be motivated the more easily, but at the expense of some elegance. Recently Eilenberg has produced a new, and much more efficient, decomposition and it is this theory that we will now study. It is set in the world of transformation semigroups.

    Before we embark on the details let us pause for a moment and consider how we could approach the problem of finding a suitable decomposition. Let M = (Q, Σ,F) be a state machine and let |Q| = n. Consider the collection π of all subsets of Q of order n−1. Then π is an admissible subset system, and we may construct a well-defined quotient machine M/π.

  • 1 - Semigroups and their relatives

  • M. Holcombe
  • Book:
    Algebraic Automata Theory
    Published online:
    22 September 2009
    Print publication:
    19 August 1982, pp 1-24

  • Summary

    We may as well begin at the beginning and this will involve us in a brief excursion through some of the fundamental concepts essential for any algebraic subject. It will also enable us to become acquainted with the notation used, although the experienced reader could easily skip through this chapter. We will assume that the reader has a knowledge of elementary set theory.

    Relations

    One of the fundamental concepts in mathematics is that of a relation. It can be introduced in a variety of ways but the most useful one for us is the following abstract approach.

    Let A be a non-empty set. A relation, ℛ, on A is a subset ℛ ⊆ A × A. If (a, a′) ∈ A × A and (a, a′) ∈ ℛ we say that a is ℛ-related to a′. Sometimes a natural notation is used in mathematics to express this relationship between two elements of a set, for example if A = ℤ, the set of all integers, then there is a relation ≤ that can be defined on ℤ. We write a ≤ a′ if the number a′ − a is not negative and the set ℛ ⊆ ℤ × ℤ defining this relation consists of all ordered pairs (a, a′)∈ ℤ × ℤ such that a < a≤.

    A relation ℛ on the set A is an equivalence relation if:

    1. (i)(a, a) ∈ ℛ for all a ∈ A

    2. (ii) (a, a′)∈ ℛ ⇒(a′, a) ∈ ℛ for a, a′ ∈ A…

    3. […]

  • 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’.

Page 290 of 291