Class 22: On Computable Numbers

cs2102: Discrete Mathematics
University of Virginia, Fall 2016

See course site for notes:
https://uvacs2102.github.io

Transcript

1. Class  22:   On  Uncomputable Numbers cs2102:  Discrete  Mathematics  |  F16 uvacs2102.github.io   0 David  Evans   University  of  Virginia
2. Plan Today:   Cantor’s  Continuum  “Hypothesis” Computable  Numbers (Turing) Tuesday:  undecidable  problems Slack  poll  on  additional  topics 1 PS9  due  next  Wednesday
3. Recap:  Cardinalities 2 |  ℕ  | |  pow(ℕ)  | Integers  (ℤ) Rationals (ℚ) ℕ×ℕ Finite  bit  strings cists Countable Reals  in  [0,  1)   Infinite  bit  strings
4. [0, 1) vs   0,1 ×[0, 1) 3
5. 4 bijection  to  
6. 5 Je  le  vois,  mais je  ne  le  crois pas! Georg  Cantor,  letter  to  Richard  Dedekind  (about  this  proof) bijection  to  
7. Anything  between  these? 6 |  ℕ  | |  pow(ℕ)  | Integers  (ℤ) Rationals (ℚ) ℕ×ℕ Finite  bit  strings cists Countable Reals  in  [0,  1)   Infinite  bit  strings [0,  1]  × [0,  1]
8. Aleph-­‐Naught 7 ℵ = |ℕ| “smallest  infinite  cardinal  number” =  smallest  infinite  ordinal  number (Recall  from  Class  19)
9. Second  Smallest  Infinite  Cardinal? 8 ℵ =    ? ℵ = |ℕ| ℕ = ℝ =  ℵ
10. Second  Smallest  Infinite  Cardinal? 9 ℵ =? ℵ ℵ = |ℕ| Cantor’s  Continuum   Hypothesis Proven  that  it  cannot  be  settled  with  ZFC  axioms!
11. 10 Proof  of  non-­‐answerability  with  ZFC  axioms:   ℵ =? ℵ
12. 11 Proof  of  non-­‐answerability  with  ZFC  axioms:   ℵ =? ℵ Gödel  (1940):  Proof  that  there  is  no  proof that  ℵ > ℵ Cohen  (1963):  Proof  that  there  is  no  proof that  ℵ ≤ ℵ
13. On  Computable  Numbers 12
15. Alan  Turing 1912-­‐1954
17. Modelling  Computation 16 The  execution  of  a  state  machine, = (, ⊆  ×, B ∈ ) is  a  (possibly  infinite)  sequence  of  states,  (B , E , … , G ) that: 1.  B = B (it  begins  with  the  start  state) 2.  ∀ ∈ 0, 1, … , − 1  .   M  → MOE ∈ (if   and   are   consecutive  states  in  the  sequence,  there  is  an  edge   → ∈ . Class  13: Is  this  a  “good”  model  of  computation?
18. Is   computable? 17 =  
19. Is   computable? 18 =   = 2
20. Is   computable? 19 =   = 2
21. Is   computable? 20 =   = 2 = 4   −   4 3 + 4 5 −   4 7 + 4 9 +  … Gottfried  Wilhelm  Leibniz
22. 1679 21 Gottfried  Wilhelm  Leibniz Digital  Mechanical  Calculator:  +,  -­‐,  *,  /
23. 22 ...a  general  method  in   which  all  truths  of   reason  would  be   reduced  to  a  kind  of   calculation.    At  the  same   time,  this  would  be  a   sort  of  universal   language  or  script,  … Gottfried  Wilhelm  Leibniz
24. State  Machine  Model? 23 The  execution  of  a  state  machine, = (, ⊆  ×, B ∈ ) is  a  (possibly  infinite)  sequence  of  states,  (B , E , … , G ) that: 1.  B = B (it  begins  with  the  start  state) 2.  ∀ ∈ 0, 1, … , − 1  .   M  → MOE ∈ (if   and   are   consecutive  states  in  the  sequence,  there  is  an  edge   → ∈ .
25. What  Computers  was   Turing  modelling?
26. Cray-­‐1  (1976) Pebble  (2014) Apple  II  (1977) Palm  Pre  (2009) MacBook  Air  (2008) Surface  (2016)
27. Colossus  (1944) Apollo   Guidance   Computer   (1969) Honeywell  Kitchen  Computer  (1969) ($10,600  “complete  with  two-­‐week   programming  course”)  
28. Pre-­‐WWII  “Computers”
29. Modeling   “Scratch  Paper”   (and  Input and  Output)
30. Two-­‐Dimensional  Paper  =  One-­‐Dimensional  Tape
31. Modeling  Pencil  and  Paper   # C S S A 7 2 3 How  long  should  the  tape  be? ... ...
32. Modeling  Processing
33. Modeling  Processing
34. Modeling  Processing Look  at  the  current   state of  the   computation Follow  simple  rules   about  what  to  do  next Scratch  paper  to  keep   track
35. Modeling  Processing  (Brains) Follow  simple  rules Remember  what  you  are  doing “For  the  present  I  shall  only   say  that  the  justification  lies   in  the  fact  that  the  human   memory  is  necessarily   limited.” Alan  Turing 34
36. Modelling  Processing 35 = (, ⊆  ×, B ∈ ) is  a  finite  set.
37. Modelling  Processing 36 = (, ⊆  ×, B ∈ ) is  a  finite  set. = (, ⊆  ×  Γ   → , B ∈ ) is  a  finite  set,   Γ is  finite  set  of  symbols  that  can   be  written  in  memory # C S S A 7 2 3 ... ...
38. Turing’s  Model 37 = (,   a  finite  set  of  (head)  states ⊆  ×  Γ →  ×  Γ  ×  ,       transition  function      B ∈ , start  state jkklmn ⊆ accepting  states ) is  a  finite  set,   Γ is  finite  set  of  symbols  that  can  be  written  in  memory = {Left,  Right,  Halt}
39. Cardinality  of  Turing  Machines = (, ⊆  ×  Γ →  ×  Γ  ×  ,  B ∈ , jkklmn ⊆ ) is  a  finite  set,   Γ is  finite  set  of  symbols  that  can  be  written  in  memory = {Left,  Right,  Halt}
40. Cardinality  of  Computable  Numbers
41. Are  there  Uncomputable Numbers? 40
42. Lingering  Questions • Is  Turing’s  model  the  right  one? • Is  there  an  interesting  uncomputable number? • What  does  this  tell  us  about  what  can  be   computed  in  practice? We’ll  answer  some  of  these  Tuesday…