语言图灵机 L={a^mb^na^mb^n ∣ m,n≥0}

Question

I am having trouble making a Turing machine for language L={a^mb^na^mb^n ∣ m,n≥0}

What I have thought so far is:

If we start with a blank, the string is empty and it should accept, if not, start reading as and I thought to mark the a's with X and the b's with Y would be OK

Answer 1

A high-level strategy for designing a TM for this is the following:

1. Check to see whether you're looking at a string of the form a^2k or b^2k (including the empty string). In any of these cases, halt-accept. Otherwise, continue to step 2.
2. Cross off pairs of a , one each from the first and third sections, respectively, until one of the sections runs out of a . If one runs out while the other still has a , halt-reject. Otherwise, continue to step 3.
3. Cross off pairs of b , one each from the second and fourth sections, respectively, until one of the sections runs out of b . If one runs out while the other still has b , halt-reject. Otherwise, halt-accept.

Answer 2

There are 4 cases in this question:

1. Blank String can be straight away accepted.
2. List contains only a's so if their total number is even we accept the string.
3. Like point 2 if the input contains only b's.
4. The input is a combination of both a and b's.

I will mark the first set of a's as X and second set of a's as Z and the first set of b's as U and second set of b's as V.

The Turing machine designed is:

Here the states {q0,q10} deals with the first case, {q0, q1, q11, q12, q13, q14} deals with the second case, {q0, q4, q15, q16, q17, q18} deals with the third case and {q0, q1, q2, q3, q4, q5, q6, q7, q8, q9} deals with the final case.

I have also designed the corresponding code in python for this Turing machine.

#function to perform action of states
def action(inp, rep, move):
    global tapehead
    if tape[tapehead] == inp:
        tape[tapehead] = rep
        if move == 'L':
            tapehead -= 1
        else:
            tapehead += 1
        return True
    return False

tape = ['B']*50 
string = input("Enter String: ")
i = 5
tapehead = 5
for s in string: #loop to place string in tape
    tape[i] = s
    i += 1

state = 0
a, b, X, Z, U, V, R, L, B = 'a', 'b', 'X', 'Z', 'U', 'V', 'R', 'L', 'B'
oldtapehead = -1
accept = False
while(oldtapehead != tapehead): #if tapehead not moving that means terminate Turing machine
    oldtapehead = tapehead

    if state == 0:
        if action(a, X, R):
            state = 1
        elif action(B, B, R):
            state = 10
        elif action(Z, Z, R):
            state = 7
        elif action(b, U, R):
            state = 4

    elif state == 1:
        if action(a, a, R):
            state = 1
        elif action(b, b, R):
            state = 2
        elif action(B, B, L):
            state = 11

    elif state == 2:
        if action(b, b, R) or action(Z, Z, R):
            state = 2
        elif action(a, Z, L):
            state = 3

    elif state == 3:
        if action(b, b, L) or action(Z, Z, L) or action(a, a, L):
            state = 3
        elif action(X, X, R):
            state = 0

    elif state == 4:
        if action(b, b, R):
            state = 4
        elif action(Z, Z, R):
            state = 5
        elif action(B, B, L):
            state = 15

    elif state == 5:
        if action(Z, Z, R) or action(V, V, R):
            state = 5
        elif action(b, V, L):
            state = 6

    elif state == 6:
        if action(Z, Z, L) or action(V, V, L) or action(b, b, L):
            state = 6
        elif action(U, U, R):
            state = 0

    elif state == 7:
        if action(Z, Z, R):
            state = 7
        elif action(V, V, R):
            state = 8

    elif state == 8:
        if action(V, V, R):
            state = 8
        elif action(B, B, R):
            state = 9

    elif state == 11:
        if action(a, a, L):
            state = 11
        elif action(X, X, R):
            state = 12

    elif state == 12:
        if action(a, Z, R):
            state = 13

    elif state == 13:
        if action(a, X, R):
            state = 12
        elif action(B, B, R):
            state = 14

    elif state == 15:
        if action(b, b, L):
            state = 15
        elif action(U, U, R):
            state = 16

    elif state == 16:
        if action(b, V, R):
            state = 17

    elif state == 17:
        if action(b, U, R):
            state = 16
        elif action(B, B, R):
            state = 18

    else:
        accept = True


if accept:
    print("String accepted on state = ", state)
else:
    print("String not accepted on state = ", state)

You can check any states which are not clear from the figure or test it against any inputs. Output for some inputs:

Enter String: aaaaabbaaaaabb
String accepted on state =  9

Enter String: aaaaaa
String accepted on state =  14

Enter String: 
String accepted on state =  10

Enter String: aaabaaa
String not accepted on state =  5

Enter String: bbb
String not accepted on state =  16