The Turing Guide

524 | NOTES TO PAGES 417–431

4. B. J. Copeland, ‘Computable numbers: a guide’, The Essential Turing, pp. 5–57.


5. J. P. Jones, H. Wada, D. Sato, and D. Wiens, ‘Diophantine representation of the set of prime numbers’,
   American Mathematical Monthly, 83 (1976), 449–64.

CHAPTER 38 BANBURISmUS REVISITED: DEPTHS AND BAyES (SImPSON)

1. Hinsley & Stripp (1993).


2. R. Whelan, ‘The use of Hollerith equipment in Bletchley Park’, NA, HW25/22.


3. Sybil Cannon (nee Griffin), private communication.


4. Whelan described this visit with gusto (Note 2).


5. C. H. O’D. Alexander, ‘Cryptographic history of work on the German Naval Enigma’, (c.1945),
   NA, HW25/1 and The Turing Archive for the History of Computing (http://www.AlanTuring.
   net/alexander_naval_enigma).


6. What I call ‘alignment’, Alexander called ‘distance’.


7. Alexander (Note 5).


8. Alexander (Note 5).


9. I. J. Good, Probability and the Weighing of Evidence, Griffin (1950).


10. Christine Brose (nee Ogilvie-Forbes), private communication.


11. A. M. Turing, ‘Mathematical theory of ENIGMA machine’ ,(c.1940) (also known as ‘Turing’s treatise
    on the Enigma’), NA, HW25/3 and The Turing Archive for the History of Computing (http://www.
    AlanTuring.net/profs_book).


12. A. M. Turing, ‘Visit to National Cash Register Corporation of Dayton, Ohio’ (1942), The Turing
    Archive for the History of Computing (http://www.AlanTuring.net/turing_ncr).


13. Joan Clarke in Hinsley & Stripp (1993).


14. Turing (Note 11).


15. Brose (Note 10).


16. Eileen Johnson (nee Plowman), private communication.


17. C. H. O’D. Alexander, ‘The factor method’ (c.1945), NA, HW43/26, Study 1.


18. E. H. Simpson, ‘Bayes at Bletchley Park’, Significance (June 2010).

CHAPTER 39 TURING AND RANDOmNESS (DOwNEy)

1. É. Borel, ‘Les probabilités dénombrables et leurs applications arithmétiques’, Rendiconti del Circolo
   Matematico di Palermo, 27 (1909), 247–71.


2. Mathematically, saying that a number is normal means that the collection of absolutely normal num-
   bers has a ‘Lebesgue measure’ of 1; this corresponds to saying that if we throw a dart at the real line,
   then with probability 1 it would hit an absolutely normal number.


3. A. Copeland and P. Erdős, ‘Note on normal numbers’, Bulletin of the American Mathematical Society,
   52 (10) (1946), 857–60. Their proof relies on the ‘density’ of primes in base 10.


4. Turing (c. 1936).


5. C. Schnorr, ‘A unified approach to the definition of a random sequence’, Mathematical Systems Theory,
   5 (1971), 246–58.


6. It turns out that if this feature is not allowed and you can only bet in discrete amounts with a minimum
   bet, a completely different notion of randomness comes about called ‘integer-valued randomness’. It is
   a question of physics as to whether this is the correct notion of randomness for the universe, since it
   depends upon whether space–time is a continuum or discrete.


7. Turing (c. 1936).


8. V. Becher, S. Figueira, and R. Picchi, ‘Turing’s unpublished algorithm for normal numbers’, Theoretical
   Computer Science, 377 (2007), 126–38.


9. V. Becher, ‘Turing’s normal numbers: towards randomness’, in S. B. Cooper, A. Dawar, and B. Löwe
   (eds), CiE 2012, Springer Lecture Notes in Computer Science 7318, Springer (2012), 35–45.