Question

The following BNF grammar defines a set of binary strings. \(\langle\) string \(\rangle::=\langle\) zero \(\rangle\langle\) one \(\rangle \mid\langle\) zero \(\rangle\langle\) string \(\rangle\langle\) one \(\rangle\) ::= 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 consists of balanced binary strings with equal 0s and 1s, each 0 having a corresponding 1.

Step-by-step solution

Understanding the BNF Grammar

The given BNF grammar specifies a language for binary strings. The grammar defines as either or . A corresponds to 0 and a corresponds to 1.

Deriving the Language Pattern

From the grammar, every valid string begins with a 0 and ends with a 1, and there can be more 0s and 1s nested within as . This results in strings that are balanced in the number of 0s and 1s, and each 0 has a corresponding 1 at the end.

Describing the Language

The language consists of binary strings where the number of 0s equals the number of 1s, and each 0 precedes its corresponding 1, possibly with other balanced substrings in between.

Designing a Turing Machine - Step 1: Tape Structure and Movement

To design a Turing machine for this language, set the tape to read each character. The tape starts at the leftmost symbol and moves right. As it reads a 0, it marks it, then searches for a corresponding 1 to mark, indicating pairing. It continues this process to ensure all 0s have corresponding 1s.

Designing a Turing Machine - Step 2: State Transitions

Create states to mark each 0 as read (let's say by replacing 0 with X), and transition to a state that searches for a 1 to mark with another symbol (Y). Continue this process until either all are marked and end on a blank tape or unmarked 0s or 1s remain.

Designing a Turing Machine - Step 3: Halting Conditions

If the Turing machine reads the entire string and marks all 0s and 1s without finding unpaired symbols, it halts with a blank tape, indicating the string is valid. If unmarked symbols remain or mismatches are detected, it results in a non-blank tape.

Key concepts

BNF Grammar

Backus-Naur Form (BNF) grammar is a notation technique used to describe the syntax of languages, particularly programming and formal languages. In the context of binary strings, BNF provides a framework to define what constitutes a valid string through a series of recursive definitions.

- **Rule Definitions**: Each line in BNF describes how a specific non-terminal symbol can be derived. For example, the non-terminal can be produced from either or . - **Application**: By applying these rules repeatedly, we can generate valid strings of the language.

BNF is valuable because it offers a clear, structured way to define complex language patterns simply. It uses rules to replace non-terminal symbols with other non-terminals or terminal symbols. This method is especially useful for languages where recursion plays a significant role in the structure, like binary strings that need to be balanced and recursive in pattern definition.

Turing Machine

A Turing Machine is a theoretical computational model used to simulate the logic of algorithm execution. It's a powerful concept that's crucial for understanding computation and decision problems in formal languages. In deciding the binary strings created by our BNF grammar, the Turing machine operates under these principles:

- **Tape and Head**: Imagine an infinitely long tape divided into cells, each cell can hold a symbol. The tape head reads and writes symbols and moves left or right. - **State Transitions**: The machine operates states that dictate what to do next based on the current situation — what the head reads under it. For example, if it reads a 0, it marks it (turns it to X), transitions to another state, and seeks a corresponding 1 to mark as Y. - **Halting Conditions**: The machine stops when all possible actions are exhausted — either when the string is validated with a clean tape or errors are discovered, leaving some symbols unprocessed.

By designing a Turing machine, one can determine if a binary string follows the grammar rules laid out by the BNF: i.e., whether it is balanced and correctly matched.

Binary Strings

Binary strings are sequences that consist solely of 0s and 1s. These strings are foundational in computer science since they essentially represent data in binary code, which underpins digital computing.

- **Simple Structure**: Despite consisting of just two symbols, binary strings hold complex information, as each position represents a power of two. - **Use in Grammars**: In formal languages, binary strings can be generated using grammar rules, such as BNF, to define specific sequences as valid. In our context, strings that start with a 0, end with a 1, and balance numbers of 0s and 1s are of interest.

Binary strings can be manipulated and tested for properties such as balance and structure, making them an excellent subject for study in both theoretical and practical computing approaches.

Balanced Strings

Balanced binary strings are those where the number of 0s equals the number of 1s, and every 0 has a corresponding 1 that follows it. This balance ensures that each substring maintains the proportional relationship of these characters.

- **Characteristics**: In a properly balanced string, apply recursive rules like in BNF to ensure each 0 is matched by a 1 at some later position in the string. - **Checking Balance**: You might check the balance by comparing the number of 0s against 1s as a simple method, or use a Turing machine for automation and decision-making of acceptance.

Balanced strings highlight how strict ordering and quantity can influence the validity of a language construct, serving as a basis for computational verification through automated machines like the Turing machine.