Question

Describe the language defined by the following grammar: \(\langle\) goal \(\rangle::=\langle\) letter \(\rangle \mid\langle\) letter \(\rangle\langle\) next \(\rangle\) \(\langle\) next \(\rangle::=,<\) letter \(>\) ::= A

Short answer

The language consists of the strings 'A' and 'A,A'.

Step-by-step solution

Understand the Grammar Structure

The grammar provided defines a language with a production rule starting from . The rules are: 1. ::= or 2. ::= , 3. ::= A

Examine the Symbol

The symbol resolves to 'A' based on the production rule ::= A. This implies that wherever appears, it should be replaced by the character 'A'.

Analyze the Symbol

The symbol resolves to ',', which further simplifies to ',A'. Therefore, always produces the string ',A'.

Analyze the Symbol

The can produce two types of strings according to the rules: 1. ::= can directly produce 'A'. 2. ::= produces 'A,next', which simplifies to 'A,A' due to the resolution.

Define the Language

The language defined by this grammar consists of two possible strings: 'A' and 'A,A'. These are the only combinations produced by the provided grammar rules.

Key concepts

Production Rule

Production rules form the backbone of any formal grammar. They outline how symbols in a language evolve and transform into one another. These rules showcase how to replace symbols in derivation.

In the given exercise, the production rules include different symbols like \(\), \(\), and \(\). For instance:

• \( ::= \mid \)
• \( ::= ,\)
• \( ::= A\)

These rules state that the \(\) can transform into a \(\) or into a \(\) followed by \(\). The \(\) symbol then transforms into ",A", and \(\) transforms into "A". This restricts the language to specific sequences, like "A" or "A,A". Understanding these rules allows us to see how starting from a single symbol, a string is fully derived and produced as per the grammar's structure.

Grammar Structure

Grammar structure in any language provides the framework that defines how strings are formed within that language. It dictates the permissible composition of symbols and characters, indicating valid word or sentence formations.

In the grammar provided, the structure is laid out using production rules which determine the paths from one symbol to another. Specifically:

• \(\) initiates as the root symbol of the grammar.
• The transformation follows through \(\) and \(\), lining up the elements to create sequences like "A" and "A,A".

This strict structure ensures that any derivation starts from the \(\), adhering to the given rules to form strings, maintaining order and coherency within the language.

Language Definition

Language definition in formal grammar refers to the specific set of strings or words that the grammar generates. It's the culmination of production rules applied systematically starting from the initial symbol.

For the exercise, the language defined is tightly constrained by its rulesets.

• The production \( ::= \) results in a simple string "A".
• Following \( ::= \) expands to "A,A".

Thus, the language comprises exactly two strings: "A" and "A,A". These finite possibilities directly stem from the combination and sequence of defined symbols, making the language simple yet tightly controlled. Each string manifests as a direct outcome of the grammar's production procedure, clearly illustrating the boundaries and scope of the defined language.