Question

The following BNF grammar defines a set of binary strings. \(\langle\) string \(\rangle::=\langle\) zero \(\rangle\langle\mathrm{A}\rangle\langle\) zero \(\rangle\) \(\langle A\rangle::=\langle\) zero \(\rangle\langle B\rangle\langle\) zero \(\rangle\) \(\langle\mathrm{B}\rangle::=\langle\) one \(\rangle\langle\mathrm{B}\rangle \mid<\) one \(>\) \(<\) zero \(>::=0\) : := 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 language.

Short answer

The language is binary strings starting and ending with 0, with at least one 1 sandwiched between. Design a Turing machine to check this pattern.

Step-by-step solution

Understanding the BNF Grammar

The grammar provides rules for generating binary strings. The term ⟨string⟩ consists of ⟨zero⟩⟨A⟩⟨zero⟩, where ⟨zero⟩ is 0. ⟨A⟩ further unfolds into ⟨zero⟩⟨B⟩⟨zero⟩. ⟨B⟩ produces a sequence of ⟨one⟩ symbols ending with one ⟨one⟩, with ⟨one⟩ being defined as 1. This means any string defined by this grammar is of the structure 0, followed by 0, followed by at least one 1, followed by 0.

Simplifying the Grammar

For simplicity, the grammar generates strings of the form 0 followed by 0s, then by one or more 1s, and finally followed by a 0. It essentially describes a binary string of the form 0...01...10, where the number of 1s is at least one and is sandwiched between 0s.

Describing the Language

The language generated by this grammar consists of binary strings starting and ending with 0, and having at least one 1. The strings can be expressed in the regular expression form as 00+10.

Designing the Turing Machine

To design the Turing machine, it should take a binary string as input and decide if it fits the grammar: 1. Read the first symbol. If not 0, reject. 2. Read subsequent 0s until finding the first 1. 3. Move right to ensure all subsequent are 1s until finding the next 0. 4. Confirm the final symbol is 0. 5. If the tape satisfies these conditions, erase it to blank. If any condition fails, keep it marked.

Constructing the Turing Machine's State Diagram

Create states for each condition: Start in an initial state, look for the first 0, then transition to checking 1s, and ensure it transitions to checking the final 0. If it matches, enter a final state where the tape is erased. If it doesn't follow this structure, transition to a rejecting state. Use states to manage transitions and track acceptance.

Key concepts

Binary Strings

Binary strings are sequences composed solely of the digits 0 and 1. They are fundamental in computer science because they reflect the underlying binary numeral system used by digital computers. In our context, binary strings can have patterns that align with specific grammatical structures or rules. The BNF grammar described in the problem defines strings starting and ending with the digit 0, while containing at least one 1 in the middle. Thus, the legal binary strings formed from this grammar have a specific structure. Here's what to remember about binary strings in this setup:

• Binary strings use only "0's" and "1's".
• The grammar dictates the order of these numbers.
• In our exercise, the pattern is 00...+1...+0, ensuring at least one 1 between two 0s.

Understanding these properties of binary strings helps in designing algorithms and systems that can process or validate such strings efficiently.

Turing Machine

A Turing Machine is a theoretical model that describes a mathematical machine capable of simulating any algorithm. The machine works with a strip of tape marked into cells, with each cell containing a symbol from a finite alphabet. Turing Machines are critical for exploring the capabilities and limits of what can be computed.

In this exercise, we are tasked with creating a Turing Machine to recognize binary strings that match the BNF grammar. Here's a simplified process of how this Turing Machine operates:

• Start by reading the first symbol on the tape. If it's not 0, the machine will reject the string.
• Progressively move across the tape, confirming there's at least one 1 following the initial 0s.
• After reading 1s, ensure that the sequence ends with 0.
• If all these conditions are met, the machine erases the tape, signifying acceptance.
• If any check fails, the tape retains a mark, indicating rejection.

By focusing on these operations, a Turing Machine can efficiently determine whether a binary string fits the specified grammatical structure.

Regular Expression

Regular expressions (regex) are sequences of characters that form search patterns, used mostly to match or find strings within texts. They play a significant role in computer science for defining search patterns in strings.

In our exercise, the grammar describing our binary strings can be condensed into a regular expression. The structured pattern can be depicted as:

• \[ 0+1+0 \]

This regular expression signifies a string starting with one or more 0s, followed by one or more 1s, and ending with 0. Understanding this pattern helps in swiftly implementing and verifying the properties of the string across different computing applications, making regex a versatile tool in both theoretical and applied computer science.

Formal Language Theory

Formal Language Theory is a comprehensive study of syntax and grammar structures. It forms the theoretical foundation for understanding languages used in computer science and linguistics. Language theory is crucial for constructing compilers, interpreters, and understanding computational linguistics.

In this exercise, the BNF grammar represents a specific formal language, dictating how binary strings should be formed. Formal languages can be characterized by:

• Syntax rules, like those provided in the BNF grammar.
• Expressiveness that defines a language's complexity.
• Correspondence with automata, where languages can be processed through machines like Turing Machines.

This connection between syntax and automata underpins many computing systems' logic, and mastering these concepts is key to understanding how machines interpret and process programming languages and other structured inputs.