**Exercise 2**a) Specify a 2-tape Turing machine that sorts the bits of a binary input (listing all the 1s first, followed by the 0s).b) Specify a 1-tape Turing machine that makes a copy of the input. (The input of word $w$ results in the word $ww$ on the tape.)

Turing machine:A Turing machine is a theoretical model of computation that consists of a few key components: a tape, a read/write head, a set of states, and a transition function. The tape is one of the fundamental components of a Turing machine.Turing Machine Components:Tape: The tape is a long, one-dimensional strip divided into cells. Each cell can hold a symbol from a finite set of symbols, which includes the input alphabet, blank symbols, and any additional symbols used by the machine. The tape extends infinitely in both directions.Read/Write Head: This head is positioned at a particular cell on the tape at any given time. It can read the symbol currently under it and write a new symbol onto the tape.States: The machine operates in a finite set of states. The current state, along with the symbol being read from the tape, determines the machine's behavior.Transition Function: The transition function defines how the machine changes its state and moves the tape head based on the current state and the symbol read from the tape. It specifies the next state, the symbol to write, and the direction to move the tape head (left or right).Operation of a Turing Machine:Initialization: The input is placed on the tape, and the machine is in an initial state with the tape head positioned at a particular cell on the input.Execution: The machine reads the symbol under the tape head, consults its transition function based on the current state and the symbol read, and follows the defined transition to move to the next state, write a symbol onto the tape, and move the tape head left or right.Transition: The machine repeatedly performs these steps—reading, writing, changing state, and moving the tape head—according to the transition rules specified by the transition function.Halting: The machine halts when it reaches a predefined halting state or a state with no defined transition for the current symbol read. At this point, the computation ends.Importance of the Tape:The tape serves as the working memory of the Turing machine. It provides the space needed for the machine to perform its computations. The ability to read, write, and move along the tape while altering symbols allows the machine to perform various computational tasks, making the tape a crucial aspect of the Turing machine model.Here's a description of a 2-tape Turing machine that sorts the bits of a binary input by listing all the 1s first, followed by the 0s:Turing Machine Description:The Turing machine uses two tapes: Input tape and Output tape.Algorithm Steps:Initialization:Input tape contains the binary input.Output tape is initially blank.Step-by-Step Procedure:Read the symbol from the input tape.If the symbol is '1', write '1' on the output tape and move right on both tapes.If the symbol is '0', move right on the input tape only.Repeat this process until the input tape is fully traversed.Completion:Once the input tape is completely traversed:Move back to the beginning of the input tape.Traverse the input tape again.If '0' is encountered, write '0' on the output tape and move right on the output tape.Continue until the end of the input tape is reached.Halt:The machine halts when both tapes have been fully traversed.Example Execution:For input '101001':Initial Input Tape: 101001 (Head at the start)Initial Output Tape: __ (Blank)Step 1: Read '1' from the input tape, write '1' on the output tape.Input Tape: 101001 (Move head to the right)Output Tape: 1_Step 2: Read '0', move right on the input tape.Input Tape: 101001 (Move head to the right)Output Tape: 1_Step 3: Read '1', write '1' on the output tape.Input Tape: 101001 (Move head to the right)Output Tape: 11_Step 4: Read '0', move right on the input tape.Input Tape: 101001 (Move head to the right)Output Tape: 11_Step 5: Read '0', move right on the input tape.Input Tape: 101001 (Move head to the right)Output Tape: 11_Step 6: Read '1', write '1' on the output tape.Input Tape: 101001 (Move head to the right)Output Tape: 111_Step 7: Read '0', move right on the input tape.Input Tape: 101001 (Move head to the right)Output Tape: 111_Step 8: Input tape fully traversed. Move back to the start of the input tape.Input Tape: 101001 (Move head to the left)Step 9: Traverse input tape again.Input Tape: 101001 (Move head to the right)Output Tape: 111_Step 10: Read '0', write '0' on the output tape.Input Tape: 101001 (Move head to the right)Output Tape: 1110_Step 11: Read '0', write '0' on the output tape.Input Tape: 101001 (Move head to the right)Output Tape: 11100_Step 12: Input tape fully traversed. Halt. Here's a high-level description of a 1-tape Turing machine that duplicates the input on the tape:Turing Machine Description:This Turing machine will read the input word on the tape and duplicate it by writing the original word followed by itself.Algorithm Steps:Initialization:The input word is placed on the tape.The tape head is positioned at the beginning of the input word.Step-by-Step Procedure:Read the input symbols one by one while moving to the end of the input word.Once the end of the input word is reached, move the head back to the beginning of the input word.Duplication:Starting from the beginning of the input word, write the symbols again immediately after the original word.Continue this process until the entire input word has been duplicated.Halt:The machine halts after successfully duplicating the input word.Example Execution:Suppose the input word is '0101':Initial Tape: 0101 (Tape head at the start)Step 1: Read '0', move right.Step 2: Read '1', move right.Step 3: Read '0', move right.Step 4: Read '1', move right.End of Input Reached.Move the head back to the start of the input word: 0101 (Tape head at the start)Start duplicating:Write '0' after '0101': 010101Move right.Write '1' after '101': 01010101Move right.Write '0' after '010': 010101010Move right.Write '1' after '1010': 0101010101Input word '0101' has been successfully duplicated.This process continues until the entire input word has been duplicated on the tape. The tape will contain the input word followed by an exact copy of itself.The 2-tape Turing machine sorts a binary input by reading the symbols and segregating '1's onto the first tape while '0's are placed onto the second tape. It iterates through the input once, categorizing each symbol accordingly onto the designated tapes, resulting in a segregated output with '1's on the first tape followed by '0's on the second tape.The 1-tape Turing machine duplicates an input word by reading the symbols while moving to the end of the input word. Once it reaches the end, the machine returns to the start of the input and writes the symbols again immediately after the original word. This process continues until the entire input word has been duplicated, resulting in the input word followed by an exact copy of itself on the tape.

Engineering

Computer Science

a) Specify a 2-tape Turing machine that sorts the bits of a binary input ()listing all the 1s first, followed by the Os.) b) Specify a 1-tape Turing machine that makes a copy of the input. (The input of word w results in the word ww on the tape.)

a) Specify a 2-tape Turing machine that sorts the bits of a binary input ()listing all the 1s first, followed by the Os.) b) Specify a 1-tape Turing machine that makes a copy of the input. (The input of word w results in the word ww on the tape.)

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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