The Equivalence, Unambiguity and Sequentiality Problems of Finitely Ambiguous Max-Plus Tree Automata are Decidable

A max-plus automaton is a finite automaton with transition weights in the real numbers. To each word, it assigns the maximum weight of all accepting paths on the word, where the weight of a path is the sum of the path’s transition weights. Max-plus automata and their min-plus counterparts are weighted automata [19, 18, 13, 2, 4] over the max-plus or min-plus semiring. Under varying names, max-plus and min-plus automata have been studied and employed many times in the literature. They can be used to determine the star height of a language [7], to decide the finite power property [20, 21] and to model certain timed discrete event systems [5, 6]. Additionally, they appear in the context of natural language processing [14]. For practical applications, the decidable properties of an automaton model are usually of great interest. Typical problems considered include the emptiness, universality, inclusion, equivalence, sequentiality and unambiguity problems. We consider the last three of these problems for finitely ambiguous automata, which are automata in which the number of accepting paths for every word is bounded by a global constant. If there is at most one accepting path for every word, the automaton is called unambiguous. It is called deterministic or sequential if for each pair of a state and an input symbol, there is at most one valid transition into a next state. It is known [11] that finitely ambiguous max-plus automata are strictly more expressive than unambiguous max-plus automata, which in turn are strictly more expressive than deterministic max-plus automata. Let us quickly recall the considered problems and the related results. The equivalence problem asks whether two automata are equivalent, which is the case if the weights assigned by them coincide on all words. In general, the equivalence problem is undecidable [12] for max-plus automata, but for finitely ambiguous max-plus automata it becomes decidable [22, 9]. The sequentiality problem asks whether for a given automaton, there exists an equivalent deterministic automaton. This was shown to be decidable by Mohri [14] for unambiguous max-plus automata. Finally, the unambiguity problem asks whether for a given automaton, there exists an equivalent unambiguous automaton. This problem is known to be decidable for finitely ambiguous and even polynomially ambiguous max-plus automata [11, 10]. In conjunction with Mohri’s results, it follows that the sequentiality problem is decidable for these classes of automata as well. In this paper, we show that these three problems are decidable for finitely ambiguous max-plus tree automata, which are max-plus automata that operate on trees instead of words. In the form of probabilistic context-free grammars, max-plus tree automata are commonly employed in natural language processing [17]. Our approach to the decidability of the equivalence problem uses ideas from [9]. We use a similar induction argument and also reduce the equivalence problem to the same decidable problem, namely the decidability of the existence of an integer solution for a system of linear inequalities [15]. On words, the proof relies on the decomposition of words into subwords of bounded length, of which one is removed in the induction step. This argument cannot be applied to trees as easily. A tree can be decomposed into contexts of bounded height, but this requires contexts with multiple variables. Removing such a context does usually not yield a tree. Consequently, our induction is much more involved. We also point out and correct an important oversight in the main theorem of [9]. The decidability of the unambiguity problem employs ideas from [11]. Here, we show how the dominance property can be generalized to max-plus tree automata. To show the decidability of the sequentiality problem, we first combine results from [3] and [14] to show the decidability of this problem for unambiguous max-plus tree automata, and then combine this result with the decidability of the unambiguity problem. Our solution of the equivalence problem can be applied to weighted logics. In [16], a fragment of a weighted logic is shown to have the same expressive power as finitely ambiguous weighted tree automata.