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Concept# Language equation

Summary

Language equations are mathematical statements that resemble numerical equations, but the variables assume values of formal languages rather than numbers. Instead of arithmetic operations in numerical equations, the variables are joined by language operations. Among the most common operations on two languages A and B are the set union A ∪ B, the set intersection A ∩ B, and the concatenation A⋅B. Finally, as an operation taking a single operand, the set A* denotes the Kleene star of the language A. Therefore language equations can be used to represent formal grammars, since the languages generated by the grammar must be the solution of a system of language equations.
Language equations and context-free grammars
Ginsburg and Rice
gave an alternative definition of context-free grammars by language equations. To every context-free grammar G = (V, \Sigma, R, S), is associated a system of equations in variables V. Each variable X \in V is an

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