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Design Turing machine for multiplication
A Turing machine is a seven tuples
(Q, Σ, Γ, δ, q0, qacc, qrej)
Where,
Q is a finite set of states;
Σ is the input alphabet does not contain the blank symbol t;
Γ is the tape alphabet, where t ∈ Γ and Σ ⊆ Γ;
δ: (Q × Γ) → (Q × Γ × {L, R}) is the transition function;
q0 ∈ Q is the start state;
qacc ∈ Q is the accept state;
qrej ∈ Q is the reject state, where qrej ≠ qacc.
Problem
Construct Turing machine (TM) for multiplication of two unary integers.
Solution
Multiplication of two unary integers 3*2=6 In Turing Machine 3 represents − 111 2 represents − 11
Let ‘M’ be a symbol used to separate two integers
| B | 1 | 1 | 1 | M | 1 | 1 | B |
Here B= blank
M= Symbol used two separate two integers
⇑= Head
After performing multiplication, the output is as follows −
| B | 1 | 1 | 1 | 1 | 1 | 1 | B |
The TM for multiplication is given below −
Explanation
q0 − Looking at the first unused symbol in input.
q0 − Looking at the first unused symbol in input.
q1 − Found a 1 in the first factor; skipping to the x symbol
q2 − Found x
q4 − Found a 1 in the second factor; changed it to Y; skipping to the =
q3 − Used all the 1s in the second factor; convert the Ys back to 1s
q5 − Skipping to end of tape
q6 − Skipping back to =
q7 − Skipping back over second factor
q8 − Found Y marking previously used 1 from second factor
q9 − Skip back over 1ʼs in first factor
q12 − Consumed the entire first factor; convert xs back to 1s
q11 − Found ⊔
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