SLIDE 1
2
2 3 4 HALT HALT 5 For all n >2 there are no - - PowerPoint PPT Presentation
2 3 4 HALT HALT 5 For all n >2 there are no - - PowerPoint PPT Presentation
2 3 4 HALT HALT 5 For all n >2 there are no natural a,b,c such that a n +b n = c n . I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain. 6 FERMAT()
SLIDE 2
SLIDE 3
- Σ
Σ Σ
3
SLIDE 4
4
SLIDE 5
5
HALT HALT
SLIDE 6
6
For all n>2 there are no natural a,b,c such that an+bn= cn. I have a truly marvelous demonstration
- f this
proposition which this margin is too narrow to contain.
SLIDE 7
7
FERMAT() 𝑢 ← 3 while true for all 𝑜 ∈ 3, … , 𝑢 and 𝑦, 𝑧, 𝑨 ∈ 1, … , 𝑢 if 𝑦𝑜 + 𝑧𝑜 = 𝑨𝑜 then return 𝑦, 𝑧, 𝑨, 𝑜 end for 𝑢 ← 𝑢 + 1 end while
SLIDE 8
8
SLIDE 9
- HALT 𝑞𝑠𝑝𝑠𝑏𝑛, 𝑗𝑜𝑞𝑣𝑢
- Halt(Turing,Turing)
- Halt(Turing,Turing)
Turing(Turing)
- Halt(Turing,Turing)
Turing(Turing)
9
Turing 𝑞𝑠𝑝𝑠𝑏𝑛 if HALT(𝑞𝑠𝑝𝑠𝑏𝑛, 𝑞𝑠𝑝𝑠𝑏𝑛) then loop forever else return true
SLIDE 10
〈𝑁〉 𝑁 𝑁𝐼𝐵𝑀𝑈 〈𝑁, 𝑁〉
- HALT = { 𝑁, 𝑦 ∶ 𝑁 is a TM that halts on 𝑦}
- 𝑁𝐼𝐵𝑀𝑈
HALT
- 𝑁𝑈𝑉𝑆𝐽𝑂𝐻
10
SLIDE 11
- HALT = { 𝑁, 𝑦 ∶ 𝑁 is a TM that halts on 𝑦}
- 𝑁𝐼𝐵𝑀𝑈
HALT
- 𝑁𝑈𝑉𝑆𝐽𝑂𝐻
11
〈𝑁〉 𝑁𝐼𝐵𝑀𝑈 〈𝑁, 𝑁〉
✔ ✕
✔ 𝑁𝑈𝑉𝑆𝐽𝑂𝐻 ∞
SLIDE 12
12
〈𝑁〉 𝑁𝐼𝐵𝑀𝑈 〈𝑁, 𝑁〉
✔
∞
✕
✔
〈𝑁𝑈𝑉𝑆𝐽𝑂𝐻〉 𝑁𝑈𝑉𝑆𝐽𝑂𝐻?
𝑁𝑈𝑉𝑆𝐽𝑂𝐻
SLIDE 13
⋯
13
⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
𝑁1 𝑁2 𝑁3 𝑁4 𝑁5 〈𝑁1〉
⋯
𝑁𝑈𝑉𝑆𝐽𝑂𝐻
〈𝑁2〉 〈𝑁3〉 〈𝑁4〉 〈𝑁5〉
SLIDE 14
14
SLIDE 15
- 𝐵 ≤ 𝐶
𝐶 𝐵
- 𝐶
⟹ 𝐵
- 𝐵
⟹ 𝐶 𝐶
15
𝐵
SLIDE 16
- 𝐵
𝐶 𝐵 𝐶 𝐵 ≤ 𝐶 𝐵 𝐶
16
𝑁
𝐵
𝑦 𝑁𝐶 𝑧
SLIDE 17
17
𝐶 HALT ≤ 𝐶
SLIDE 18
- ACCEPTS = { 𝑁, 𝑦 ∶ 𝑁 is a TM that
accepts 𝑦}
- 𝑁, 𝑦 ∈ ACCEPTS ⟹ 𝑦
𝑁
- 𝑁, 𝑦 ∉ ACCEPTS ⟹ 𝑦
𝑁
- ACCEPTS
18
SLIDE 19
19
𝑁𝐼𝐵𝑀𝑈 〈𝑁, 𝑦〉
✔ ✕
✔ 𝑁
𝐵𝐷𝐷𝐹𝑄𝑈𝑇
𝑁
𝐵𝐷𝐷𝐹𝑄𝑈𝑇
〈𝑁, 𝑦〉 〈𝑁〉 〈𝑁′〉 〈𝑁′, 𝑦〉
✔ ✕
✔ ✕
SLIDE 20
- HALT ≤ ACCEPTS
- 𝑁
𝐵𝐷𝐷𝐹𝑄𝑈𝑇
ACCEPTS
- HALT
- 〈𝑁, 𝑦〉
𝑁𝐵𝐷𝐷𝐹𝑄𝑈𝑇 〈𝑁, 𝑦〉
- 𝑁
𝑁′
- 𝑁𝐵𝐷𝐷𝐹𝑄𝑈𝑇 〈𝑁′, 𝑦〉
- 𝑁, 𝑦 ∈ HALT
- 𝑁, 𝑦 ∉ HALT
∎
20
SLIDE 21
- EMPTY = { 𝑁 ∶ 𝑁 is a TM that accepts
nothing}
- EMPTY
21
E F
SLIDE 22
- ACCEPTS ≤ EMPTY
- 〈𝑁, 𝑦〉
𝑁𝑦 𝑧 𝑁(𝑦)
- 𝑁𝐵𝐷𝐷𝐹𝑄𝑈𝑇
𝑁𝑦 𝑁𝐹𝑁𝑄𝑈𝑍(〈𝑁𝑦〉)
- 𝑁
𝑦 ⟹ 𝑀 𝑁𝑦 = Σ∗ ⟹ 𝑁𝐹𝑁𝑄𝑈𝑍 〈𝑁𝑦〉
- 𝑁
𝑦 𝑦 ⟹ 𝑀 𝑁𝑦 = ∅ ⟹ 𝑁𝐹𝑁𝑄𝑈𝑍 𝑁𝑦 ∎
22
SLIDE 23
23
SLIDE 24
24
𝑏 𝑏𝑐 𝑏 𝑑𝑏𝑐𝑑 𝑐𝑑𝑑 𝑑 𝑏 𝑏𝑐 𝑐𝑑𝑑 𝑑 𝑏 𝑑𝑏𝑐𝑑 𝑐𝑑𝑑 𝑑
SLIDE 25
25
SLIDE 26
- https://youtu.be/RG2uPLG5K48
- ¬∃𝑦, 𝑧, 𝑨, 𝑜 ∈ ℕ: 𝑜 ≥ 3 ∧ 𝑦𝑜 + 𝑧𝑜 = 𝑨𝑜
- 3𝑦2 − 2𝑦𝑧 − 𝑧2𝑨 − 7 = 0 (𝑦 = 1, 𝑧 = 2, 𝑨 = −2)
- 26
SLIDE 27
- 1.
EQ = 𝑁, 𝑁′ ∶ 𝑁, 𝑁′ TMs, 𝑀 𝑁 = 𝑀(𝑁′)
2.
GRAVITON = ∅ {1}
27
SLIDE 28
28
SLIDE 29
- HALT, ACCEPTS, EMPTY
- HALT, ACCEPTS, EMPTY
- 29