Turing machine to solve a^(0+1+2+3+....+n)

Question

I am trying to solve the following Turing machine challenge:

Design a Turing machine that accepts the language L, where L is Xaⁿ, with n = 0 + 1 + 2 + 3 +....+ M

Examples:

• Xa: accepted because M is 1, and 0+1 is 1.
• Xaaa: accepted because M is 2, and 0+1+2 is 3.
• Xaaaaa: rejected, because there is no M for which the sum is 5.

I tried to take this approach:

• turn one a to Y (I picked Y, nothing special about it).

• turn two a into Y's, but this is where I get stuck: How can I find out how many a's will be converted into Y's?

Answer 1

You could use an algorithm where before the "X" you store a number of "a" which have to be paired with the same number of "a" after the "X". During successful pairing, the left-side "a" are removed and at the right of the "X" and the matched "a", a new "X" is placed, while the old "X" becomes an "a" as well, so that now these matched "a" are at the left of the "X" and the process can repeat.

We can achieve that by using "X" to temporarily mark those "a" that have been paired. Here is a more detailed algorithm description:

• If after the (last) "X" there is a blank: accept

• While before the (first) "X" there is an "a":

  • Find the leftmost "a" and remove it (replace it with a blank)
  • Find the first character after the last "X":
    • If it is a blank: reject
    • Otherwise it is an "a", and replace it with "X".

• Otherwise (there is a blank before the first "X"):

  • Replace all "X" with "a"
  • If the character after the last replacement is a blank: reject
  • It is an "a": replace it with "X"

• Repeat from the top

Here is a transition table for the above algorithm ("_" represents a blank):

state	read	write	move head	next state
start	X		right	check
check	_		left	accept
check	a		left	toLeft
toLeft	a or X		left	toLeft
toLeft	_		right	wipeLeft
wipeLeft	a	_	right	wipeRight
wipeLeft	X	a	right	replaceX
replaceX	X	a	right	replaceX
replaceX	a	X	right	check
replaceX	_		left	reject
wipeRight	a		right	wipeRight
wipeRight	X		right	wipeRight2
wipeRight2	X		right	wipeRight2
wipeRight2	a	X	left	toLeft
wipeRight2	_		left	reject

And finally a snippet that runs this transition table on your input:

createTuring({
    transitions: [
        { state: "start",      read: "X",             move: "R", nextState: "check" },
        { state: "check",      read: "_",             move: "L", nextState: "accept" },
        { state: "check",      read: "a",             move: "L", nextState: "toLeft" },
        { state: "toLeft",     read: "aX",            move: "L", nextState: "toLeft" },
        { state: "toLeft",     read: "_",             move: "R", nextState: "wipeLeft" },
        { state: "wipeLeft",   read: "a", write: "_", move: "R", nextState: "wipeRight" },
        { state: "wipeLeft",   read: "X", write: "a", move: "R", nextState: "replaceX" },
        { state: "replaceX",   read: "X", write: "a", move: "R", nextState: "replaceX" },
        { state: "replaceX",   read: "a", write: "X", move: "R", nextState: "check" },
        { state: "replaceX",   read: "_",             move: "L", nextState: "reject" },
        { state: "wipeRight",  read: "a",             move: "R", nextState: "wipeRight" },
        { state: "wipeRight",  read: "X",             move: "R", nextState: "wipeRight2" },
        { state: "wipeRight2", read: "X",             move: "R", nextState: "wipeRight2" },
        { state: "wipeRight2", read: "a", write: "X", move: "L", nextState: "toLeft" },
        { state: "wipeRight2", read: "_",             move: "L", nextState: "reject" },
    ],
    initState: "start",
    tape: "Xa",
});

<script src="https://trincot.github.io/turing.js"></script>