Question

Describe the behavior of the Turing machine $$ \begin{aligned} &(1,1,1,2, R) \\ &(2,0,0,1, L) \end{aligned} $$ when run on the tape \(\ldots b 101 b \ldots\)

Short answer

The Turing machine enters an infinite loop between state 1 and state 2, alternating direction with each step.

Step-by-step solution

Understand the Turing Machine Configuration

The given Turing machine transitions are defined as follows: * (1,1,1,2,R): If in state 1 and reading a 1, write a 1, move to state 2, and make a right move. * (2,0,0,1,L): If in state 2 and reading a 0, write a 0, move to state 1, and make a left move.

Determine Initial State and Input

The initial state of the Turing machine is 1. The input tape is defined as ...b 101 b ..., meaning that the machine starts on a blank (b) cell and then moves to the first digit reading 1.

Execute First Transition

The machine starts in state 1, reading a 1. Using the transition (1,1,1,2,R), it writes a 1, changes to state 2, and moves right. Now, the machine reads 0.

Execute Second Transition

In state 2 and reading 0, the machine uses the transition (2,0,0,1,L). It writes 0, changes back to state 1, and moves left. The machine now reads the first 1 again.

Identify Repeating Pattern

Realizing that the machine returns to state 1 reading 1, it will repeatedly apply the transitions: in state 1, read 1, move to state 2; in state 2, read 0, move to state 1. This pattern continues indefinitely on this tape loop.

Key concepts

State Transition

The concept of a state transition is fundamental to understanding Turing machines. Each state transition represents a set of instructions that dictate how the machine reacts to specific inputs from its tape. For a Turing machine, a state transition is formatted as (current state, read symbol, write symbol, next state, move direction). This comprehensive instruction guides the machine step-by-step, altering not only its internal state but also possibly modifying the content on its tape.

For instance, consider our Turing machine's transition (1, 1, 1, 2, R). This tells the machine that if it's in state 1 and the symbol under its read/write head is 1, it should write 1 (leaving the tape unchanged in this case), transition to state 2, and move right. State transitions are crucial as they form the logic of the machine, defining its computing processes and behaviors.

Tape Configuration

The tape of a Turing machine serves as both its input and its workspace. It is theoretically infinite in length, consisting of cells that can each hold a single symbol. The configuration of the tape at any given moment--often referred to simply as the tape configuration--can profoundly influence the machine's behavior.

In our exercise, the tape begins as \(... b 101 b ...\), positioning the read/write head at the first blank (b). Each step of the Turing machine modifies this configuration based on its current state and the rules defined by its state transitions. For example, when the machine reads a 1 at state 1, it moves accordingly. The arrangement of input symbols and the sequence of transformations dictated by these transitions determine the machine's computations. Understanding the current tape configuration is essential for predicting future steps and possible outcomes of the Turing machine's computation.

Infinite Loop

When discussing Turing machines, the concept of an infinite loop is particularly relevant and intriguing. An infinite loop occurs when a machine enters a sequence of state transitions that repeats indefinitely without reaching a halting state. This is influenced by the combination of state transitions and tape configurations that fail to alter the machine's behavior outside of this cycle.

In our example, the Turing machine follows the transitions (1,1,1,2,R) and (2,0,0,1,L), which lead it back and forth between states 1 and 2, moving across the same symbols on the tape and never arriving at a conclusion. The machine thus enters an infinite loop, performing repetitive operations without progressing beyond this pattern. Recognizing an infinite loop is vital, as it indicates an unsolvable problem or a need for adjustments in the machine's design or input to reach a terminal state.

Computation Model

The Turing machine stands as a foundational computation model, representing the mechanics of decision-making and problem-solving within mathematical logic and computer science. It operates on a theoretical framework that uses states, transitions, and an infinite tape to simulate any algorithm's execution.

This model illustrates the nature of computation: how machines interpret instructions, manipulate data, and execute processes. By systematically applying state transitions, a Turing machine processes inputs methodically, showcasing the principles of computation through deliberate steps and operations. Its capacity to execute repeated loops and manage state transitions illustrates fundamental aspects of logical processing.

Understanding the Turing machine as a computation model helps in grasping the overarching principles behind computing devices and algorithms. It highlights how abstract models can represent and perform complex operations, providing a profound perspective of what it means for something to be computable.