Abstract
We consider weighted tree automata (wta) over strong bimonoids and their initial algebra semantics and their run semantics. There are wta for which these semantics are different; however, for bottom-up deterministic wta and for wta over semirings, the difference vanishes. A wta is crisp-deterministic if it is bottom-up deterministic and each transition is weighted by one of the unit elements of the strong bimonoid. We prove that the class of weighted tree languages recognized by crisp-deterministic wta is the same as the class of recognizable step mappings. Moreover, we investigate the following two crisp-determinization problems: for a given wta ${\cal A}$, (a) does there exist a crisp-deterministic wta which computes the initial algebra semantics of ${\cal A}$ and (b) does there exist a crisp-deterministic wta which computes the run semantics of ${\cal A}$? We show that the finiteness of the Nerode algebra ${\cal N}({\cal A})$ of ${\cal A}$ implies a positive answer for (a), and that the finite order property of ${\cal A}$ implies a positive answer for (b). We show a sufficient condition which guarantees the finiteness of ${\cal N}({\cal A})$ and a sufficient condition which guarantees the finite order property of ${\cal A}$. Also, we provide an algorithm for the construction of the crisp-deterministic wta according to (a) if ${\cal N}({\cal A})$ is finite, and similarly for (b) if ${\cal A}$ has finite order property. We prove that it is undecidable whether an arbitrary wta ${\cal A}$ is crisp-determinizable. We also prove that both, the finiteness of ${\cal N}({\cal A})$ and the finite order property of ${\cal A}$ are undecidable.
Highlights
The determinization problem shows up if one wants to specify a problem in a nondeterministic way and to calculate its solution in a deterministic way
It is well known that the determinization problem is solved positively if T is the class of all finite-state automata, i.e., for each nondeterministic finite-state automaton A
The same holds true for the class T of all finite-state tree automata [TW68, Thm. 1]
Summary
The determinization problem shows up if one wants to specify a problem (e.g., a formal language) in a nondeterministic way and to calculate its solution (e.g., membership) in a deterministic way. There exists a wsa such that there is no equivalent deterministic wsa (see, e.g., [BV03, Lm. 6.3] for a weighted tree automaton over a monadic alphabet with this property)(i) On the other side, there are subclasses of T for which the determinization problem can be solved positively: the subclass of all wsa over locally finite semirings [KM05, p. We will follow the lines of [CDIV10] and identify subclasses of T for which the crisp-determinization problem is solvable, i.e., for every wta A of that subclass, there exists a crisp-deterministic wta B such that A and B are i-equivalent, i.e., A and B have the same initial algebra semantics, and B can be constructed effectively. In this paper we show that for arbitrary wta A it is undecidable whether im(hV(A)) is finite By restricting this result to the case of monadic input trees (i.e., strings), we have solved partially the open problem stated in [CDIV10, Sect. An arbitrary bottom-up deterministic wta is crisp-determinizable (Theorem 8.5), for an arbitrary bottom-up deterministic wta, its Nerode algebra is finite (Theorem 8.7), and an arbitrary bottom-up deterministic wta has the finite order property (Theorem 8.9)
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