Finite Sequentiality of Unambiguous Max-Plus Tree Automata

Abstract

We show the decidability of the finite sequentiality problem for unambiguous max-plus tree automata. A max-plus tree automaton is called unambiguous if there is at most one accepting run on every tree. The finite sequentiality problem asks whether for a given max-plus tree automaton, there exist finitely many deterministic max-plus tree automata whose pointwise maximum is equivalent to the given automaton.

Finite Sequentiality of Finitely Ambiguous Max-Plus Tree Automata

“…For the converse, if the separation property is not satisfied, we show how to construct finitely many unambiguous max-plus-WTA whose pointwise maximum is equivalent to A and which all do not satisfy the tree fork property. As the behavior of an unambiguous max-plus-WTA which does not satisfy the tree fork property is finitely sequential [34], we obtain that A is also finitely sequential.…”

“…Theorem 1 [34] The behavior of a trim unambiguous max-plus-WTA A is finitely sequential if and only if A does not satisfy the tree fork property. If A is finitely sequential, a finitely sequential representation of A can be effectively constructed.…”

“…On the one hand, we endow our new property with a criterion accounting for the non-linear structure of trees. This new criterion is inspired by the criterion we added in [34] to the fork property [25], the property characterizing finite sequentiality of unambiguous max-plus word automata, in order to obtain the tree fork property, the property characterizing finite sequentiality of unambiguous max-plus tree automata. On the other hand, we strengthen the A-Fork property as with only the first modification, our new property would wrongly characterize some finitely sequential automata as not being finitely sequential.…”

“…Finally, we show that if the separation property is not satisfied for a given max-plus tree automaton, then we can construct finitely many unambiguous max-plus tree automata which all do not satisfy the tree fork property and whose pointwise maximum is equivalent to the automaton. By [34], these unambiguous automata then possess finitely sequential representations. Combining these, we obtain a finitely sequential representation of the original automaton.…”

“…An extended abstract of this paper appeared as [39]. Here, we provide full proofs together with detailed illustrations for these.…”