Turing Machine Variant – Multiple Tapes

A multi-tape Turing Machine (TM) is a TM with access to a fixed finite number of tapes. A k-tape TM has access to k tapes.

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Initial Configuration

The input string is placed on the first tape. All other tapes are blank. All tape heads point to the start of their respective tapes.

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Transition Function

The transition function \delta is mapped as follows

    \begin{equation*} Q \times \Gamma^k \rightarrow Q \times (\Gamma \times \{L, R, S\})^k \end{equation*}

Based on the current state and the current-tape cell read by each of the k tapes, the transition specifies the next state, what to write on each of the k tapes and the direction to move the tape-head on each of the k tapes.

The direction S implies staying on the current cell.

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    \begin{equation*} \delta(q, (x_1, y_2, z_3)) = (q, (X_1, R), (Y_2, L), (Z_3, S))  \end{equation*}

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Equivalence to Single Tape

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Any multi-tape Turing Machine computation can be carried out by an equivalent single-tape Turing Machine.

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The single-tape TM has the tape contents of the individual tapes separated by a separator symbol (\#). The tape-head of each individual tape is marked by the dotted equivalent of the symbol it points to.

To carry out one transition of a k-tape TM on the single-tape

  1. Scan right to the first dotted symbol. Remember this symbol in the current state.
  2. Repeat step 1 to scan and remember k symbols in the current state.
  3. The current state and the k symbols will have a finite number of possibilities.
  4. Use the transition function to find the next configuration.
  5. Bring tape-head back to the start.
  6. Scan right to the first dotted symbol. Apply the configuration for the first tape. This involves writing a symbol, and moving the tape-head (reapplying the dot). If the dotted symbol is one cell to the left of a separator, a new blank cell must be added to accommodate for a R transition.
  7. Repeat step 6 to scan and apply the configuration for each of the k tapes.

Remembering symbols

A Turing Machine can remember by using specific states. Say the tape alphabet is \{0, 1\} and there are 3 tapes. There can be a state qp_{010} to remember that the current state is q, the tape-heads point to 0 on the first-tape, 1 on the second-tape and 0 on the third-tape. From this state it can make an appropriate transition by following the transition function.

Finite number of possibilities

To remember multiple symbols, a single-tape TM equivalent to a k-tape TM (with Q states) will need |\Gamma|^k new states for each of the Q states for a total of |Q||\Gamma|^k states.

Adding a new blank cell

Adding a new blank cell involves shifting all the contents of the tape one cell to the right. This has to be done in a right-most first fashion.


A language L is recognized by a single-tape TM if and only if L is recognized by a k-tape TM.

A language L is decided by a single-tape TM if and only if L is decided by a k-tape TM.

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