An Algorithm for Comparing Deterministic Regular Tree Grammars

An algorithm to decide inclusion for languages de ned by deterministic regular tree grammars is presented. The algorithm is shown to run in polynomial time based on the number of constructors, constructor arity, and number of non-terminals. Correctness proofs are included. 1 Regular tree grammars A regular tree grammar, G, is a grammar for constructing trees. Formally, G = (C;N;P; S) where: 1. C is a nite, non-empty set of constructors. Each constructor c is of arity n. Constants are 0-arity constructors. 2. N is a nite, non-empty set of non-terminals where N \ C = ;. 3. P is a nite set of productions of the form A ! c(X1; : : : ;Xn) where A is a non-terminal, c is a constructor, and each Xi is a non-terminal. If the constructor is a constant, then we write c as a shorthand for c(). 4. S is a special element of N called the start symbol. A regular tree is constructed from a regular tree grammar in the following manner: 1. If A is a non-terminal, then the tree whose only node is A is a regular tree. 2. If t is a regular tree, B is an occurrence of a non-terminal in t, and there is a production B ! c(X1; : : : ;Xn), then the result of replacing in t the occurrence of B with a tree whose root is c and whose children in left to right order are X1; : : : ;Xn is a regular tree. If the regular tree t is constructed starting from a non-terminal A then we say that A derives t. A tree is a proper tree if it contains only constructors and is an improper tree if it contains a non-terminal. For a grammar G = (C;N; P; S), the language generated by G, L(G), is the set of all proper trees derivable from S. A regular tree grammar is deterministic if for each non-terminal A and constructor c, there is at most one production of the form A ! c(X1; : : : ;Xn). In this paper we will consider only deterministic regular tree grammars and their languages. Deterministic regular tree grammars provide a convenient way of describing recursive data. [Liu98] presents a method of analyzing recursive data that uses these grammars as results of a xed point computation. A xed point is found when consecutive grammars G1 and G2 are iterated such that L(G2) contains L(G1).