Language equation

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.

Ginsburg and Rice gave an alternative definition of context-free grammars by language equations. To every context-free grammar ${\displaystyle G=(V,\Sigma ,R,S)}$, is associated a system of equations in variables ${\displaystyle V}$. Each variable ${\displaystyle X\in V}$ is an unknown language over ${\displaystyle \Sigma }$ and is defined by equation ${\displaystyle X=\alpha _{1}\cup \ldots \cup \alpha _{m}}$ where ${\displaystyle X\to \alpha _{1}}$, ..., ${\displaystyle X\to \alpha _{m}}$ are all productions for ${\displaystyle X}$. Ginsburg and Rice used a fixed-point iteration argument to show that a solution always exists, and proved that the assignment ${\displaystyle X=L_{G}(X)}$ is the least solution to this system, i.e. any other solution must be a componentwise subset of this one.

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