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[1] gave an alternative definition of context-free grammars by language equations.

, is associated a system of equations in variables

Ginsburg and Rice used a fixed-point iteration argument to show that a solution always exists, and proved that the assignment

Language equations with added intersection analogously correspond to conjunctive grammars.

[citation needed] Brzozowski and Leiss[2] studied left language equations where every concatenation is with a singleton constant language on the left, e.g.

with one variable on the right-hand side.

Every nondeterministic finite automaton has such corresponding equation using left-concatenation and union, see Fig.

If intersection operation is allowed, equations correspond to alternating finite automata.

Baader and Narendran[3] studied equations

using left-concatenation and union and proved that their satisfiability problem is EXPTIME-complete.

Conway[4] proposed the following problem: given a constant finite language

This problem was studied by Karhumäki and Petre[5][6] who gave an affirmative answer in a special case.

A strongly negative answer to Conway's problem was given by Kunc[7] who constructed a finite language

such that the greatest solution of this equation is not recursively enumerable.

Kunc[8] also proved that the greatest solution of inequality

Language equations with concatenation and Boolean operations were first studied by Parikh, Chandra, Halpern and Meyer [9] who proved that the satisfiability problem for a given equation is undecidable, and that if a system of language equations has a unique solution, then that solution is recursive.

Later, Okhotin[10] proved that the unsatisfiability problem is RE-complete and that every recursive language is a unique solution of some equation.

For a one-letter alphabet, Leiss[11] discovered the first language equation with a nonregular solution, using complementation and concatenation operations.

Later, Jeż[12] showed that non-regular unary languages can be defined by language equations with union, intersection and concatenation, equivalent to conjunctive grammars.

By this method Jeż and Okhotin[13] proved that every recursive unary language is a unique solution of some equation.