Determinization of Finite State Weighted Tree Automata

We investigate the determinization of nondeterministic bottom-up/top-down finite state weighted tree automata over some semiring A and compare the resulting four classes of formal tree series with each other. In particular, for every bottom-up finite state weighted tree automaton over some semifield (i.e., a semiring which contains a multiplicative inverse for every nonzero element) we construct a deterministic bottom-up weighted tree automaton. This construction is called partial determinization, because it may lead to an automaton with an infinite set of states. We prove sufficient conditions under which the partial determinization produces a bottom-up finite state weighted tree automaton which is equivalent to the original one. Using this partial determinization we generalize well known theorems on classes of tree languages (cf. [14] Chapter II, Theorems 2.6 and 2.10, Example 2.11), viz if A is a commutative and locally finite semifield, then (i) nondeterministic bottom-up, (ii) deterministic bottom-up, and (iii) nondeterministic top-down finite state weighted tree automata are equally powerful. Moreover, if the input alphabet is not trivial, then deterministic top-down finite state weighted tree automata are strictly less powerful than the aforementioned classes.