Descendants of a recognizable tree language for prefix constrained linear monadic term rewriting with position cutting strategy

For a tree language L, a finite set Z of regular Σ-path languages, and a set S of Z-prefix constrained linear monadic term rewriting rules over Σ, the position cutting descendant of L for S is the set S↑⁎(L) of trees reachable from a tree in L by rewriting in S by position cutting strategy. If L is recognizable, then S↑⁎(L) is recognizable as well. Moreover, if S is finite, then we can construct a tree automaton recognizing S↑⁎(L). For a recognizable tree language L and a finite set Z of regular Σ-path languages, we study the set DZ,↑(L) of position cutting descendants of L for all sets of Z-prefix constrained linear monadic term rewriting rules over Σ. We show that DZ,↑(L) is finite, and that if L is given by a tree automaton A and each element of Z is given by an automaton, then we can construct a set {R1,…,Rk} of Z-prefix constrained linear monadic term rewriting systems over Σ such that DZ,↑(L)={R1⁎↑(L),…,Rk⁎↑(L)}.