- if symbol = 1, write 3, move to the right, change to state 2.
- otherwise: move to the right, keep state.
- if symbol = 1, write 3, move right, change to state 2.
- otherwise: move right, keep state.
state 2:
- it looks for a 2 to replace by a 4.
- if symbol = 2, write 4, change to state 3.
- else: move to the right, keep state
- else: move right, keep state
state 3:
- if it doesn't find a 0, go to the left, keep state
- if it finds a 0, write 0, move to the right, go to state 1.
- if it finds a 0, write 0, move right, go to state 1.
// As I write this, I realize that if I replace state 0 by state 3, nothing happens.
// Excursus: After competing this project, I searched for similar ones, and found one by a William Bernoudy. My finding the nth prime TM had 14 states and 12 symbols, while his had 14 states and only 10 symbols. But if I replace state 0 by state 3, I have one state less! Anyways, from now on no state is state 3.
// Excursus: After competing this project, I searched for similar ones, and found one by a William Bernoudy. My finding the nth prime TM had 14 states and 12 symbols, while his had 14 states and only 10 symbols. But if I replace state 0 by state 3, I have one state less! Anyways, from now on no state is state 3. We modify state 2, which refers to it.
state 0:
- If symbol = 0, go to state 1.
- Else: move to the right, keep state.
- If symbol = 0, move right, change to state 1.
- else: move left, keep state.
state 2:
- it looks for a 2 to replace by a 4.
- if symbol = 2, write 4, change to state 0.
- else: move right, keep state
Now, once all the 1s are turned into 3s, state 1 would go on searching, so we want to modify it to notice that it has run out of threes.
state 1:
- if symbol = 1, write 3, move to the right, change to state 2.
- if symbol = 8, write 8, move to the right, change to state 4.
- otherwise: move to the right.
- if symbol = 1, write 3, move right, change to state 2.
- if symbol = 8, write 8, move right, change to state 4.
- otherwise: move right.
If there were no 2s left to turn to 4s, then n|m (n divides m). But if there are, n can still divide m, so we keep on going.
@ -68,7 +73,7 @@ state 4:
We also notice that if state 2 finds no 2s to turn into 4s, then ¬ (n|m), so we add that option.
state 2:
- if symbol = 2, write 4, change to state 3.
- if symbol = 2, write 4, change to state 0.
- if symbol = 9, ACCEPT.
- else: move to the right.
@ -88,8 +93,7 @@ End.
(divisor 1.1) Accepts if n|m or if n=m.
Now the input is: 05111...11822..229
The 5 will be to a 6 once we change the 2 that corresponds to the 8. If there is no such 2, n = m.
So we modify state 4 and 5 and create a state 6.
The 5 will be cahnged to a 6 once we change the 2 that corresponds to the 8. If there is no such 2, n = m. So we modify state 4 and 5 and create a state 6.
state 5:
- if it reads a 3, write 1, move left, keep state.
@ -103,6 +107,94 @@ state 4:
state 6:
- if it reads a 5, accept.
- if it erads a 6, reject.
- if it reads a 6, reject.
- else: move to the left.
End.
(is prime 1.1) Accepts if n is prime.
We start with: 0518777...77722..229
This reads: The initial n=2, after which there are enough 7s to increase n to find divisors of m.
Q: But, how many 7s?
A: Well, a priori at least sqrt(n), but up to n, if you one.
Q: But Nuño, if you put ceil(sqrt(n)) '7's, aren't you offloading some of the calculations to yourself instead of making the machine calculate it?
A: Yes, I am.
Anyways, right now the only state which can accept is state 2, which does so if ¬(n|m) for a given n. Instead of accepting, we want to increase n by 1. We modify state 2, and create 2 new states: state 7 and state 8.
state 2:
- if it reads a 2: write 4, move left, change to state 0.
- if it reads a 9: write 9, move left, change to state 7.
- else: move to the right
state 7:
- if it reads a 6: write 5, move left, keep state.
- if it reads a 4: write 2, move left, keep state.
- if it reads a 3: write 1, move left, keep state.
- if it reads a 0: write 0, move right, change to state 8.
state 8:
-if it reads an 8: write 1, move right, keep state.
-if it reads a 7: write 8, move right, change to state 3.
- if it reads a 2: ACCEPT. There is no space left, at least one of each pair of divisors has been tried
End.
(Find the nth prime 1.1)
Initial input: 0AA...AA51829␣␣␣...␣␣
It will replace an A by a B each time it finds a prime, so if n-1 is the number of 'A's, it will find the nth prime.
State 9 will change an A to a B. States 10, 11 and 12 initialize n to 2. 12, 13 and 14 move m one step to the right and increase it to m+1. By moving it one step to the right, n is bounded only by m+1.
The states that can accept are state 6 and state 8, and only state 6 can reject.
state 6
- if it reads a 5, write 5, change to state 9.
- if it reads a 6: write 6, change to state 10.
- else: move to the left.
state 8:
-if it reads an 8: write 1, move right, keep state.
-if it reads a 7: write 8, move right, change to state 3.
-if it reads a 2: write 2, move left, change to state 9.
state 9:
- if it reads an A: write B, move right, change to state 10.
- if it reads a 0: ACCEPT. There are no more As to change.
- else: move left.
state 10:
- if is reads a 1: write 1, move right, change to state 11.
- if is reads a 3: write 1, move right, change to state 11.
- else: move right.
state 11:
- if it reads a 1: write 8, move right, change to state 12
- if it reads a 3: write 8, move right, change to state 12
- if it reads a X: write 8, move right, change to state 12
- else: REJECT // Shouldn't be seeing anything else.
state 12:
- if it reads a 1: write 7, move right, keep state
- if it reads a 3: write 7, move right, keep state
- if it reads a 8: write 7, move right, keep state
- if it reads a 2: write 7, move right, change to state 13.
- if it reads a 4: write 7, move right, change to state 13.
- else: move right.
state 13:
- if it reads a 9: write 2, move right, keep state
- if it reads a 4: write 2, move right, keep state
- if it reads a ␣: write 2, move right, change to state 14
- else: move right.
state 14:
- if it reads a ␣: write W, move left, change to state 0.
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