Book contents
- Frontmatter
- Contents
- Preface
- 1 What Gödel's Theorems say
- 2 Decidability and enumerability
- 3 Axiomatized formal theories
- 4 Capturing numerical properties
- 5 The truths of arithmetic
- 6 Sufficiently strong arithmetics
- 7 Interlude: Taking stock
- 8 Two formalized arithmetics
- 9 What Q can prove
- 10 First-order Peano Arithmetic
- 11 Primitive recursive functions
- 12 Capturing p.r. functions
- 13 Q is p.r. adequate
- 14 Interlude: A very little about Principia
- 15 The arithmetization of syntax
- 16 PA is incomplete
- 17 Gödel's First Theorem
- 18 Interlude: About the First Theorem
- 19 Strengthening the First Theorem
- 20 The Diagonalization Lemma
- 21 Using the Diagonalization Lemma
- 22 Second-order arithmetics
- 23 Interlude: Incompleteness and Isaacson's conjecture
- 24 Gödel's Second Theorem for PA
- 25 The derivability conditions
- 26 Deriving the derivability conditions
- 27 Reflections
- 28 Interlude: About the Second Theorem
- 29 µ-Recursive functions
- 30 Undecidability and incompleteness
- 31 Turing machines
- 32 Turing machines and recursiveness
- 33 Halting problems
- 34 The Church–Turing Thesis
- 35 Proving the Thesis?
- 36 Looking back
- Further reading
- Bibliography
- Index
31 - Turing machines
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- 1 What Gödel's Theorems say
- 2 Decidability and enumerability
- 3 Axiomatized formal theories
- 4 Capturing numerical properties
- 5 The truths of arithmetic
- 6 Sufficiently strong arithmetics
- 7 Interlude: Taking stock
- 8 Two formalized arithmetics
- 9 What Q can prove
- 10 First-order Peano Arithmetic
- 11 Primitive recursive functions
- 12 Capturing p.r. functions
- 13 Q is p.r. adequate
- 14 Interlude: A very little about Principia
- 15 The arithmetization of syntax
- 16 PA is incomplete
- 17 Gödel's First Theorem
- 18 Interlude: About the First Theorem
- 19 Strengthening the First Theorem
- 20 The Diagonalization Lemma
- 21 Using the Diagonalization Lemma
- 22 Second-order arithmetics
- 23 Interlude: Incompleteness and Isaacson's conjecture
- 24 Gödel's Second Theorem for PA
- 25 The derivability conditions
- 26 Deriving the derivability conditions
- 27 Reflections
- 28 Interlude: About the Second Theorem
- 29 µ-Recursive functions
- 30 Undecidability and incompleteness
- 31 Turing machines
- 32 Turing machines and recursiveness
- 33 Halting problems
- 34 The Church–Turing Thesis
- 35 Proving the Thesis?
- 36 Looking back
- Further reading
- Bibliography
- Index
Summary
In this chapter, we introduce Turing's classic analysis of algorithmic computability. And then – in the next chapter – we will establish the crucial result that the Turing-computable total functions are exactly the µ-recursive functions. This result is fascinating in its own right; it is hugely important historically; and it enables us later to establish some further results about recursiveness and incompleteness in a particularly neat way. So let's dive in without more ado.
The basic conception
Think of executing an algorithmic computation ‘by hand’, using pen and paper. We follow strict rules for writing down symbols in various patterns. To keep things tidy, let's write the symbols neatly one-by-one in the squares of some suitable square-ruled paper. Eventually – assuming that we don't find ourselves carrying on generating output for ever – the computation process stops and the result of the computation is left written down in some block of squares on the paper.
Now, Turing suggests, using a two-dimensional grid for writing down the computation is not of the essence. Imagine cutting up the paper into horizontal strips a square deep, and pasting these together into one long tape. We could use that as an equivalent workspace.
Using a rich repertoire of symbols is not of the essence either. Suppose some computational system uses 27 symbols. Number these off using a five-binarydigit code (so the 14th symbol, for example, gets the code ‘01110’).
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- An Introduction to Gödel's Theorems , pp. 287 - 297Publisher: Cambridge University PressPrint publication year: 2007