Question

A Turing machine contains only the following instructions: $$ \begin{aligned} &(1,1,1,1, R) \\ &(1, b, 1,2, R) \end{aligned} $$ Can this machine ever reach the following configuration? Explain your answer. $$ \ldots b 01 b \ldots $$

Short answer

No, the Turing machine cannot reach the configuration because there is no instruction to write or encounter a '0'.

Step-by-step solution

Interpret the Instructions

The Turing machine's instruction set consists of two instructions. The first instruction \((1,1,1,1,R)\) means if the machine is in state 1 and sees a '1', it writes a '1', remains in state 1, and moves right. The second instruction \((1,b,1,2,R)\) means if the machine is in state 1 and sees a blank 'b', it writes a '1', transitions to state 2 and moves right.

Analyze the Instructions

Notice the machine starts by moving right on encountering '1's, writing '1' over any '1' it sees without changing the tape. It continues moving right indefinitely while staying in state 1 until it encounters a blank 'b'. Once it encounters a 'b', it performs the second instruction, writing '1', changing state to 2, and moving right.

Evaluate the Possible Configurations

Given the instructions, the Turing machine will never encounter a '0' because the input or initial tape configuration is made of only the symbols as dictated by the instructions encountered. The machine writes '1' over any 'b' as it moves into state 2 and continues further right. Thus, it never sees '0' in the tape or writes a '0' on it.

Reason the Impossibility of the Configuration

The configuration \(\ldots b 01 b \ldots\) implies a '0' encountered or written at some point, but the machine starts in state 1 moving over any '1' it encounters, and upon seeing a blank 'b', writes a '1'. There is no instruction to produce a '0', making this configuration unattainable.

Key concepts

State Transition

In a Turing machine, a state transition occurs when the machine changes its state based on the current input symbol it reads and applicable instructions. Each transition determines what action the machine should take next. State transitions are core to the Turing machine function as they guide the machine's computational process step-by-step.

The given machine has defined transitions within the instructions provided:

• When in state 1 and encountering a '1', it writes '1', stays in state 1, and moves right.
• Upon seeing a blank space 'b', it writes '1', transitions to state 2, and continues moving right.

These transitions define how the machine processes data on its tape and how it operates until it reaches a designated end. The transitions are vital to determining possible outcomes and configurations achievable by the machine.

Tape Symbol

Tape symbols in a Turing machine are the characters or symbols that the machine reads and writes on the tape, which serves as its memory. Each position on the tape holds exactly one symbol, and the set of symbols available to the machine determines the variety of operations it can perform.

In the provided exercise:

• '1' is a symbol the machine encounters and can overwrite with the same symbol.
• 'b' represents a blank space that the machine may also overwrite with '1'.

The absence of a '0' in the instruction set signifies that the machine can't generate the desired tape configuration \(\ldots b 01 b \ldots\). If a '0' needs to appear, it must be pre-existing before the machine starts, as the instructions allow only for writing '1' over any encountered symbol.

Instruction Set

A Turing machine's instruction set is a collection of rules that dictate what actions the machine takes in response to the tape symbol it is currently reading. Each instruction specifies:

• Current state of the machine.
• Symbol it reads from the tape.
• Symbol to write.
• New state to transition into.
• Direction to move (left or right).

For the machine in question, its instruction set is quite simple:

• Instruction 1: When in state 1 and seeing a '1', write '1', remain in state 1, and move right.
• Instruction 2: When in state 1 and seeing a blank 'b', write '1', move to state 2, and proceed right.

This concise instruction set limits the machine to only execute certain actions, impacting the range of configurations the machine can achieve.

Machine Configuration

The configuration of a Turing machine refers to the overall state the machine is in at any given point, including the state of the read-write head, the tape contents, and the current state of the machine.

For the configuration to be "\(\ldots b 01 b \ldots\)", certain conditions must be met, including transitions that allow the reading and writing of a '0'. However, based on the provided instructions, such a configuration is unachievable:

• The machine transitions towards writing '1's when encountering either '1's or blank spaces.
• It starts in state 1, can only move to state 2, and neither state provides instructions to produce '0'.

Thus, any configuration with a '0' cannot be formed, as the machine's current instructions do not permit writing '0' or transforming '1' or 'b' into '0' on the tape.