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
32 - Turing machines and recursiveness
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
We are not going to write any more programs to show, case by case, that this or that particular function is Turing-computable, not just because it gets painfully tedious, but because we can now fairly easily establish that every µ-recursive function is Turing-computable and, conversely, every Turing-computable function is µ-recursive. This equivalence between our two different characterizations of computable functions is of key importance, and we'll be seeing its significance in the remaining chapters.
µ-Recursiveness entails Turing computability
Every µ-recursive function can be evaluated ‘by hand’, using pen and paper, prescinding from issues about the size of the computation. But we have tried to build into the idea of a Turing computation the essentials of any hand-computation. So we should certainly hope and expect to be able to prove:
Theorem 32.1Every µ-recursive function is Turing-computable.
Proof sketch We'll say that a Turing program is dextral (i.e. ‘right-handed’) if
i. in executing the program – starting by scanning the leftmost of some block(s) of digits – we never have to write in any cell to the left of the initial scanned cell (or scan any cell more than one to the left of that initial cell); and
ii. if and when the program halts standardly, the final scanned cell is the same cell as the initial scanned cell (in other words, the input block(s) of digits at the beginning of a computation and the final output block start in the same cell).
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- An Introduction to Gödel's Theorems , pp. 298 - 304Publisher: Cambridge University PressPrint publication year: 2007