Question

Describe the behavior of the Turing machine \((1,1,1,1, R)\) \((1,0,0,2, L)\) \((2,1,0,2, L)\) \((2, b, 1,3, L)\) \((3, b, b, 1, R)\) when run on the tape \(\ldots b 101 b \ldots\)

Short answer

The Turing machine toggles '1's to '0's and inserts '1' on blank reaching a new state.

Step-by-step solution

Initial Setup and Understanding the Turing Machine Format

The Turing machine instructions are given in the format \((q_i, x, y, q_j, D)\) where \(q_i\) is the current state, \(x\) is the current tape symbol, \(y\) is the symbol to write, \(q_j\) is the next state, and \(D\) is the direction (L for left, R for right) to move the tape head. The initial tape content is \(\ldots b 101 b \ldots\), with the head starting on the first '1'.

Process First State Transition

Starting at state 1 on symbol '1', the rule \((1,1,1,1,R)\) applies. The head remains on '1', state stays at 1, and moves right to the next tape position.

Apply State Transition Again

The head is now on symbol '0', still in state 1. The rule \((1,0,0,2,L)\) applies. The machine writes '0', transitions to state 2, and moves left.

Transition in State 2 on Symbol '1'

The head is back on the first '1', now in state 2. The rule \((2,1,0,2,L)\) applies. The machine writes '0', remains in state 2, and moves left again.

Transition in State 2 on Blank

The head moves left to a blank 'b', still in state 2. The rule \((2,b,1,3,L)\) applies. The machine writes '1', changes to state 3, and moves left again.

Transition to Final State 1 from State 3

The head is now on another blank 'b', in state 3. The rule \((3,b,b,1,R)\) applies. The machine writes a blank, changes to state 1, and moves right. The sequence repeats until the machine halts or a new pattern emerges.

Key concepts

Computational Theory

Computational theory forms the foundation for understanding how Turing machines operate. These abstract machines are a central concept in computer science, helping us analyze what can be computed.
At its core, computational theory seeks to understand the limits of what computers can achieve. It's concerned with algorithms, which are the step-by-step procedures or formulas for solving problems. When we look at Turing machines, they become an essential part of this theory because they model the computational process.
Turing machines help define the boundaries of computation. By simulating any calculation that any computer could perform, they provide insight into both what is solvable and what is not solvable by mechanical processes. This, in turn, is indispensable for understanding machine capabilities and restrictions.

State Transition

State transition is an essential concept when discussing Turing machines. It describes how the machine moves from one state to another, depending on the current state and the symbol it encounters.
In a Turing machine, states are like "modes" of operation. Each state determines the set of rules or instructions the machine follows. State transitions occur as the machine reads and processes symbols on its tape.
For example, when the Turing machine in the described problem encounters the sequence (1,1,1,1,R), it knows that while it's in state 1 and sees a '1', it should remain in state 1, keep the '1' intact, and move right.
State transitions are similar to decision-making steps in programming, where the machine decides its next move based on rules predefined by its state and the read input. Ultimately, successful computation depends on a series of precise state transitions that direct the machine toward a desired outcome.

Machine Instructions

Machine instructions serve as the blueprint guiding a Turing machine's operation. These instructions are formatted as tuples in the form (q_i, x, y, q_j, D) where each element has a specific function:

• q_i: The current state of the machine
• x: The symbol the tape head is reading
• y: The symbol to write back on the tape
• q_j: The state into which the machine should transition
• D: The direction (Left or Right) in which to move the tape head

Each instruction plays a crucial role in the computational process. For instance, if the machine is in state 1 and reads a '0', it uses an instruction like (1,0,0,2,L) to modify the tape, change its state to 2, and adjust the tape head's position.
These instructions collectively form a program that dictates the machine's operations.

Tape Manipulation

Tape manipulation is critical to how a Turing machine performs calculations. The tape acts as the memory of the machine and can be infinitely long, holding symbols that the machine processes.
Manipulation involves reading current symbols, modifying them, and moving the tape head across the tape. Each step of manipulation is determined by the machine instructions given.
In the exercise example, initial tape contents are \( \ldots b 101 b \ldots \). The instructions guide actions like leaving the '1' untouched; writing '0' instead of a '1'; or placing a '1' where a blank 'b' exists.
Moving the tape head in left (L) or right (R) directions after each symbol adjustment allows the Turing machine to work across the tape systematically.
This manner of tape manipulation is essential since it mimics the data handling and memory allocation processes in modern computers, showing how a series of simple changes leads to complex results.