Regular Grammars and Regular Languages: Defining Simple Formal Languages

Explore regular grammars and regular languages in formal language theory. This guide defines regular grammars, explains their characteristics, and shows how they generate regular languages, including examples and diagrams of finite automata.

Regular Grammars in Discrete Mathematics

What is a Regular Grammar?

A regular grammar is a formal grammar used to generate regular languages. A regular language is a relatively simple type of formal language that can be recognized by a finite automaton (a simple type of computer).

Rules for Regular Grammars

Regular grammars have these characteristics:

  • The left-hand side (LHS) of each production rule contains only one non-terminal symbol.
  • The right-hand side (RHS) can have one of the following forms:
    • A terminal symbol followed by a non-terminal symbol (xY)
    • A single terminal symbol (x)
    • A non-terminal symbol followed by a terminal symbol (Yx)
  • Where X and Y are non-terminal symbols, and x is a string of terminal symbols.

Types of Regular Grammars

  • Left Linear Grammar (LLG): Production rules are of the form X → xY or X → x.
  • Right Linear Grammar (RLG): Production rules are of the form X → Yx or X → x.

Example: Finite Automaton and Right Linear Grammar

Consider a finite automaton (FA) that accepts strings starting with the symbol 'y' (Σ = {x, y}).

(A diagram of this finite automaton would be included here.)

The corresponding right linear grammar is:

  • X → yY
  • Y → ε | xY | yY

(ε represents the empty string.) This grammar generates the language L = {y, yx, yy, yxx, yxy, ...}.

Reversing the productions of the RLG yields a left-linear grammar that generates the reverse of L, L' = {y, yy, xy, ...}.

Converting Between Right Linear and Left Linear Grammars

You can convert a right-linear grammar (RLG) to a left-linear grammar (LLG) by reversing the finite automaton that recognizes the language, constructing the RLG for the reversed automaton, and then reversing the productions of that RLG.

(An example demonstrating this conversion process would be included here, including diagrams of the finite automata.)

Converting Right Linear Grammars to Finite Automata

To convert an RLG to a finite automaton, follow these steps:

  1. Start with the non-terminal symbol on the LHS of the first production as the start state.
  2. Create transitions based on the productions: for a production of the form X → aY, make a transition from state X to state Y on input symbol 'a'.
  3. Final states are the states that correspond to productions with an empty string on the RHS.

(Illustrative examples showing this conversion would be added here.)

Converting Left Linear Grammars to Finite Automata

To convert a left-linear grammar to a finite automaton, first convert the LLG into an equivalent RLG (by reversing the language), then create the FA for the RLG (as described above), and finally, reverse the FA. This reversed FA recognizes the original LLG language.

(An example demonstrating this conversion process would be included here, including diagrams of the finite automata and grammars.)

Conclusion

Regular grammars are fundamental in the theory of computation, providing a formal way to define regular languages which are closely tied to finite automata, enabling the design of efficient algorithms for language processing.