Question

The following BNF grammar defines a set of binary strings. \(\langle\) string \(\rangle::=\langle\) one \(\rangle \mid<\) one \(>\langle\) string \(\rangle\) : : = 1 a. Describe the language defined by this grammar. b. Write a Turing machine to decide whether any binary string is a string in this language by halting with a blank tape if the string is in the language and halting with a nonblank tape if the string is not in the lanquage.

Short answer

The language is all non-empty strings of '1's, and the Turing machine halts with a blank tape if the string is valid.

Step-by-step solution

Understanding BNF Grammar

The BNF grammar provided defines a binary language. The symbol \( \langle \text{string} \rangle \) can either be replaced by \( \langle \text{one} \rangle \), which is '1', or by a sequence \( \langle \text{one} \rangle \langle \text{string} \rangle \). This means any valid string is just a sequence of '1's, such as '1', '11', '111', etc.

Defining the Language

The language defined by the BNF grammar is the set of all strings consisting solely of the digit '1'. Therefore, any string that belongs to this language is of the form '1^n', where \( n \geq 1 \), indicating a positive number of repetitions of '1'.

Designing a Turing Machine

To decide if a given binary string belongs to this language, design a Turing machine with the following behavior: 1. Read the first character of the input string. 2. If the first character is a '0', immediately write a non-blank symbol (e.g., 'X') and halt, indicating the string is not in the language. 3. If the character is '1', move to the next character. 4. Repeat steps 1-3 until a blank (end of input) is reached. 5. If the tape is blank at the end of reading, erase the tape to blank and halt, indicating the string is in the language.

Explaining the Turing Machine Process

Let's detail the machine states: - **Start State**: Read the first symbol. - **State 1**: If the symbol is '1', move right and stay in State 1; if '0', move to Reject State. - **Reject State**: Replace the current symbol with 'X' and halt. - **Accept State**: If the tape is blank at the current spot (end of tape), erase the tape and halt.

Key concepts

BNF grammar

BNF (Backus-Naur Form) is a notation technique used to express the grammar of a language in a formal way. It is commonly used in computer science to define the syntax of languages and protocols. The basic structure of BNF is made up of production rules that describe how symbols can be combined to form valid strings in a language.

The given BNF grammar defines a simple language composed solely of binary strings made of the character '1'. The grammar can be broken down as follows:

• ****: This is the start symbol that represents the entire language.
• **** ::= '1': This rule defines as the terminal symbol '1'.
• ** ::= | **: This rule means a can be a single '1' or a '1' followed by another .

Ultimately, this set of production rules ensures that any resulting string in the language consists only of one or more '1's, such as '1', '11', '111', and so forth.

Turing machine

A Turing machine is a theoretical computational model that manipulates symbols on a strip of tape according to a set of rules. It is composed of a tape, a head to read and write on the tape, and a finite set of states, including a start state, possible accept and reject states, and a set of transition functions. It is a fundamental concept in computer science used to understand the limits of what can be computed.

For our exercise, a Turing machine is designed to decide if a given binary string is part of the language defined by the BNF grammar. The language consists only of strings of '1's. Here's how it works:

• First, it reads the input string from the left.
• If it encounters a '0', it writes a non-blank symbol (like 'X'), signaling the string is invalid, and halts.
• If it finds a '1', it moves to the next character, repeating the process until a blank section of the tape is reached (indicating the end of the string).
• If it reads through the entire string without encountering '0', it will erase the contents of the tape and halt, signifying the string is valid.

This process effectively distinguishes strings of '1's from all other combinations.

Binary strings

Binary strings are sequences composed entirely of the digits '0' and '1'. These strings are fundamental in computing and digital electronics, representing numbers in the binary numeral system.

The binary numeral system is base-2, contrasting with the more familiar decimal system, which is base-10. In many computing systems, binary strings serve crucial roles, including:

• Representation of data at the most basic level in computers.
• Encoding of various types of information, such as instructions or text.
• Forming the basis for more complex data structures and interpretations, such as integers or ASCII code.

The exercise focuses on a special type of binary string, consisting entirely of '1's. These strings are distinct because they do not include '0'. In this context, binary strings of the form '1', '11', '111', etc., are essential for illustrating the syntactical properties expressed in the provided BNF grammar.