Question

Write a BNF grammar that describes an arbitrarily long string of the characters \(a, b\), and \(c\). The string can contain any number of occurrences of these three letters (including none) in any order. The strings "empty", \(a\), accaa, abcabccba, and bbbbb are all valid members of this language.

Short answer

The BNF grammar is: \[ ::= '' | 'a' | 'b' | 'c' \]

Step-by-step solution

Understand BNF Grammar

BNF (Backus-Naur Form) is a notation for expressing the grammar of a language in a formal way. It consists of production rules that describe how strings in the language can be constructed. A BNF grammar is made up of terminal symbols (actual characters or strings from the language), non-terminal symbols (which can be broken down into terminal symbols), and production rules.

Identify the Terminals and Non-terminals

In our case, the terminals are the characters 'a', 'b', and 'c'. These are the actual symbols that can appear in the strings of the language. The non-terminal symbol we need can be named , which will represent the entire set of possible strings including the empty string.

Define the BNF Production Rules

The main rule in the BNF grammar will be for the non-terminal . It can be recursively defined to either:1. End with no characters (represented by an empty production) to allow for an empty string.2. Be any of the terminal characters 'a', 'b', or 'c', followed by another for constructing longer strings. The rule can be written as: \[ ::= '' | 'a' | 'b' | 'c' \]

Key concepts

Production Rules

In BNF Grammar, production rules are the foundation for generating strings in a language. Each rule describes how symbols in the language can be combined or replaced with other symbols.
In our exercise, we have a single production rule that defines how strings constructed from the characters 'a', 'b', and 'c' can form.
The production rule is denoted by the non-terminal (for this task it's ) and details the possible derivations or expansions it can take.

• The production rule starts with the non-terminal and is followed by ::=. This is read as: " consists of".
• Followed by a series of alternatives separated by the vertical bar '|', which represents a choice. Each alternative is a possible form that can take.

The specific production rule in this case is:

   ::= '' | 'a'  | 'b'  | 'c' 

This rule allows to either be an empty string or to include an 'a', 'b', or 'c' character followed by another . This is what allows the definition to generate arbitrarily long strings of these characters.

Terminals and Non-terminals

In BNF grammar, understanding the distinction between terminals and non-terminals is crucial. Terminals are the basic symbols from which strings are formed, and non-terminals act as placeholders or variables that can expand into terminals and other non-terminals.
For the strings defined in this exercise:

• Terminals: These are the specific symbols that appear directly in the output string. In our example, the terminals are the characters 'a', 'b', and 'c'. These are the elements that are used to compose the final strings.
• Non-terminals: This symbol does not appear in the output string but guides the production process. In our exercise, the non-terminal is . It directs the construction of the string by referencing other rules which may introduce terminals or other non-terminals.

With these rules, the non-terminal provides a way to keep the grammar flexible, allowing a wide variety of strings composed of 'a', 'b', and 'c' in any sequence or length, including the empty string.

Recursive Definition

The concept of recursion is central to defining BNF grammars that allow infinite or arbitrarily long strings. A recursive definition includes a rule that references itself, allowing for repeated application of the rule.
In the context of our BNF grammar, the non-terminal uses a recursive production rule. It refers to itself, resulting in:

• An option to produce a string consisting of terminals 'a', 'b', or 'c' followed by another instance of .
• An option to terminate this recursive process by choosing the empty string option, effectively stopping further expansion.

This recursion means you can generate sequences of any length using just a few simple rules. For instance:

 ::= '' | 'a'  | 'b'  | 'c' 

The recursive rule can grow by repeating itself, with each character ('a', 'b', 'c') adding to the length of the final string.
By continuously repeating and expanding this pattern, the recursive definition elegantly handles both short and extremely long strings efficiently.