Set constraint

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

BoolG	→ false
BoolG	→ true
BListG	→ nil
BListG	→ cons(BoolG,BListG)
BList1G	→ cons(true,BListG)
BList1G	→ cons(false,BList1G)

is transformed to the set inclusion system (constants and variables indicated by lower and upper case initials, respectively):

BoolS	⊇ false
BoolS	⊇ true
BListS	⊇ nil
BListS	⊇ cons(BoolS,BListS)
BList1S	⊇ cons(true,BListS)
BList1S	⊇ cons(false,BList1S)

This system has a minimal solution, viz. ("L(N)" denoting the tree language corresponding to the nonterminal N in the above tree grammar):

BoolS	= L(BoolG)	= { false, true }
BListS	= L(BListG)	= { nil, cons(false,nil), cons(true,nil), cons(false,cons(false,nil)), ... }
BList1S	= L(BList1G)	= { nil, cons(true,nil), cons(true,cons(false,nil)),... }

The maximal solution of the system is trivial; it assigns the set of all terms to every variable.

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