Update README.md

README.md

@@ -14,3 +14,95 @@ Anyways, to grok how Turing machines, as described in Automata and Computability
 Early versions, deprecated, start with 0.
 - A Turing Machine that accepts if a number n doesn't divide another number m and rejects otherwise.
 - A Turing Machine that detects whether a number >=2 is prime.
+
+Here is a walkthrough of how the TMs work. The states are gradually defined as the Turing machine moves, and then modified when it makes sense to do so. I think that this will be easier to understand than a table.
+
+(divisor 1.0) Accepts if n doesn't divide m, rejects otherwise.
+
+Gets as an input 0111...11822..229␣␣␣␣␣...., where:
+
+- 0 is the left endmarker
+- There are (n-1) '1's, and 1 '8'. The 8 signals the end of the '1's.
+- Similarly, there are (m-2) '2's and 1 '9'. The 9 signals the end of the '2's.
+- The ␣ are empty characters, and the TM has an infinite number of them to the left.
+
+state 0:
+- The TM starts on this state.
+- Inmediatly, it moves to the right, to state 1.
+
+state 1:
+- Looks for a 1 to replace by a three
+- if symbol = 1, write 3, move to the right, change to state 2.
+- otherwise: move to the 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
+
+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.
+
+// 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.
+
+state 0:
+- If symbol = 0, go to state 1.
+- Else: move to the 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 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.
+
+state 4:
+- if symbol = 2, write 4, move to the right, change to state 5.
+- if symbol = 9, REJECT, n | m
+
+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 = 9, ACCEPT.
+- else: move to the right.
+
+If state 4 does find a 2, then the situation looks, for example, like this; take n=3, m=6, then the tape cpuld have:
+
+0338444229
+
+3 '2's were turned into '4's. and we are in state 5, which turns the '3's into '1's and keeps going
+
+state 5:
+- if symbol = 3, write 1, move to the left, keep state.
+- if symbol = 0, write 0, move to the right, change to state 1.
+- else: move to the left, keep state.
+
+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.
+
+state 5:
+- if it reads a 3, write 1, move left, keep state.
+- if it reads a 5, write 6, keep state. // It has found such a 2.
+- if it reads a 0, write 0, change to state 1.
+
+state 4:
+- if it reads a 2, write 4, move to the left, change to state 5
+- if it reads a 9, write 9, move to the left, change to state 6.
+- else: move to the right, keep state.
+
+state 6:
+- if it reads a 5, accept.
+- if it erads a 6, reject.
+- else: move to the left.
+