Turing machine |x-y| - algorithm

Question

The problem is not about programming, but about algorithm. We need to solve |x-y| in unary code on the Turing machine. For example if we have 111 * 11 on the tape. * as a number separator, then we need to leave 1. I understand that when we have a space left or right near the *, the program should remove the * and terminate, but I always get it to work if only the numbers on the right are larger or vice versa. To make the same algorithm work and if the ribbon is 11 * 1 and 1 * 11 does not work.
If it is possible to write a system command:
q1 1 -> q2 _ R
Or with a function table as in the screenshot

Answer 1

I would apply this algorithm:
Move to the right and ensure that there is a "1" at both sides of the "*"
Wipe out the rightmost "1" and then walk back to wipe out the leftmost "1"
Repeat
When the condition in the first scan is not true, then wipe out the "*" and stop.
This leads to the following transition table:
State
In
Out
Move
New State
start
1
1
→
toright
start
*
_
→
stop
toright
1
1
→
toright
toright
*
*
→
toright
toright
_
_
←
delright
delright
1
_
←
toleft
delright
*
_
←
stop
toleft
1
1
←
toleft
toleft
*
*
←
toleft
toleft
_
_
→
delleft
delleft
1
_
→
start
Here is a little implementation in JavaScript:
class Turing {
constructor(str, idx=0) {
this.tape = Array.from(str);
this.idx = idx;
}
read() {
return this.tape[this.idx] || " ";
}
writeAndMove(chr, move) {
this.tape[this.idx] = chr;
this.idx += (move > 0) - (move < 0);
}
toString() {
return this.tape.join("").trim();
}
static run(input, transitions, state, endState) {
const turing = new Turing(input);
let move, chr;
while (state !== endState) {
chr = turing.read();
[,,chr, move, state] = transitions.find(t => t[0] === state && t[1] === chr);
turing.writeAndMove(chr, move);
}
return turing.toString();
}
}
const transitions = [
// STATE IN OUT MOVE NEWSTATE
// --------- ---- --- -- ----------
["start" , "1", "1", +1, "toright" ],
["start" , "*", " ", +1, "stop" ],
["toright" , "1", "1", +1, "toright" ],
["toright" , "*", "*", +1, "toright" ],
["toright" , " ", " ", -1, "delright"],
["delright", "1", " ", -1, "toleft" ],
["delright", "*", " ", -1, "stop" ],
["toleft" , "1", "1", -1, "toleft" ],
["toleft" , "*", "*", -1, "toleft" ],
["toleft" , " ", " ", +1, "delleft" ],
["delleft" , "1", " ", +1, "start" ],
];
// Demo run
const output = Turing.run("1*111", transitions, "start", "stop");
console.log(output);