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1.Find Example 9.7 in the lecture notes, and show the execution of string aaabbb...
May 30, 2024
1.Find Example 9.7 in the lecture notes, and show the execution of string aaabbb using instantaneous description.
To show the execution of the string "aaabbb" using instantaneous description for Example 9.7, we will follow the steps and transitions defined in the Turing machine.
Here is the step-by-step breakdown:
1.
Start with the initial string and the initial state q0q_0:
q0 aaabbbq_0 \, aaabbb
2.
Replace the first 'a' with 'x' and move right to state q1q_1:
xq1 aabbbx q_1 \, aabbb
3.
Move right to find the first 'b' and replace it with 'y', then move left to state q2q_2:
xaq1 abbbx a q_1 \, abbb
xaaq1 bbbx a a q_1 \, bbb
xaayq2 bbx a a y q_2 \, bb
4.
Move left to find the first 'a' and replace it with 'x', then move right to state q0q_0:
xaq2 aybbx a q_2 \, a y bb
xq2 aaybbx q_2 \, a a y bb
q2 xaaybbq_2 \, x a a y bb
q0 xxaybbq_0 \, x x a y bb
5.
Repeat the process:
xxq1 aybbx x q_1 \, a y bb
xxaq1 ybbx x a q_1 \, y bb
xxayq1 bbx x a y q_1 \, bb
xxayyq2 bx x a y y q_2 \, b
xxayq2 ybx x a y q_2 \, y b
xxaq2 yybx x a q_2 \, y y b
xxq2 ayybx x q_2 \, a y y b
q2 xxayybq_2 \, x x a y y b
q0 xxxyybq_0 \, x x x y y b
6.
Continue until all 'a's and 'b's are replaced:
xxxq1 yybx x x q_1 \, y y b
xxxyq1 ybx x x y q_1 \, y b
xxxyyq1 bx x x y y q_1 \, b
xxxyyyq2 x x x y y y q_2 \,
xxxyyq2 yx x x y y q_2 \, y
xxxyq2 yyx x x y q_2 \, y y
xxxq2 yyyx x x q_2 \, y y y
q2 xxxyyyq_2 \, x x x y y y
q0 xxxyyyq_0 \, x x x y y y
7.
Move to the final state q3q_3 when no 'a's and 'b's are left:
q3 xxxyyyq_3 \, x x x y y y
The string "aaabbb" is accepted by the Turing machine as it reaches the final state q3q_3.
2.Find Example 9.9 in the lecture notes, and show the execution of 2+4 using the instantaneous description.
To show the execution of 2+4 using the instantaneous description for Example 9.9, we need to follow the steps of the Turing machine as described in the lecture notes.
Here is the step-by-step execution:
1.
Initial Tape Configuration:
The tape starts with the representation of the numbers 2 and 4 separated by a 0:
"#1101111#"
(Here, "#" represents the blank symbol, "1" represents a unit, and "0" is the separator.)
2.
Step-by-Step Execution:
The Turing machine will move the 0 to the right end of the second number (4).
Step 1:
State: q0q_0
Tape: "#1101111#"
Head Position: On the first "1"
Action: Move right
Step 2:
State: q0q_0
Tape: "#1101111#"
Head Position: On the second "1"
Action: Move right
Step 3:
State: q0q_0
Tape: "#1101111#"
Head Position: On the "0"
Action: Change "0" to "1", move right
Step 4:
State: q1q_1
Tape: "#1111111#"
Head Position: On the first "1" after the original "0"
Action: Move right
Step 5:
State: q1q_1
Tape: "#1111111#"
Head Position: On the second "1" after the original "0"
Action: Move right
Step 6:
State: q1q_1
Tape: "#1111111#"
Head Position: On the third "1" after the original "0"
Action: Move right
Step 7:
State: q1q_1
Tape: "#1111111#"
Head Position: On the fourth "1" after the original "0"
Action: Move right
Step 8:
State: q1q_1
Tape: "#1111111#"
Head Position: On the blank symbol "#"
Action: Change "#" to "0", move left
Step 9:
State: q2q_2
Tape: "#11111110#"
Head Position: On the last "1"
Action: Move left
Step 10:
State: q2q_2
Tape: "#11111110#"
Head Position: On the fourth "1"
Action: Move left
Step 11:
State: q2q_2
Tape: "#11111110#"
Head Position: On the third "1"
Action: Move left
Step 12:
State: q2q_2
Tape: "#11111110#"
Head Position: On the second "1"
Action: Move left
Step 13:
State: q2q_2
Tape: "#11111110#"
Head Position: On the first "1"
Action: Move left
Step 14:
State: q2q_2
Tape: "#11111110#"
Head Position: On the blank symbol "#"
Action: Move right
Step 15:
State: q3q_3
Tape: "#11111110#"
Head Position: On the first "1"
Action: Move right
Step 16:
State: q3q_3
Tape: "#11111110#"
Head Position: On the second "1"
Action: Move right
Step 17:
State: q3q_3
Tape: "#11111110#"
Head Position: On the third "1"
Action: Move right
Step 18:
State: q3q_3
Tape: "#11111110#"
Head Position: On the fourth "1"
Action: Move right
Step 19:
State: q3q_3
Tape: "#11111110#"
Head Position: On the fifth "1"
Action: Move right
Step 20:
State: q3q_3
Tape: "#11111110#"
Head Position: On the sixth "1"
Action: Move right
Step 21:
State: q3q_3
Tape: "#11111110#"
Head Position: On the "0"
Action: Change "0" to "#", move right
Step 22:
State: q4q_4
Tape: "#1111111#"
Head Position: On the blank symbol "#"
Action: Halt
Final Tape Configuration:
The tape now represents the sum of 2 and 4:
"#1111111#"
(This represents the number 6.)
Conclusion:
The Turing machine successfully computes 2+4, resulting in the tape configuration "#1111111#", which represents the number 6.
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