Set constraint

[note 2] They arise in program analysis, abstract interpretation, and type inference.

Each regular tree grammar can be systematically transformed into a system of set inclusions such that its minimal solution corresponds to the tree language of the grammar.

For example, the grammar (terminal and nonterminal symbols indicated by lower and upper case initials, respectively) with the rules is transformed to the set inclusion system (constants and variables indicated by lower and upper case initials, respectively): This system has a minimal solution, viz.

("L(N)" denoting the tree language corresponding to the nonterminal N in the above tree grammar): The maximal solution of the system is trivial; it assigns the set of all terms to every variable.

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