On Determinism and Unambiguity of Weighted Two-way Automata

Abstract

In this paper, we first study the conversion of weighted two-way automata to one-way automata. We show that this conversion preserves the unambiguity but does not preserve the determinism. Yet, we prove that the conversion of an unambiguous weighted one-way automaton into a two-way automaton leads to a deterministic two-way automaton. As a consequence, we prove that unambiguous weighted two-way automata are equivalent to deterministic weighted two-way automata in commutative semirings.Comment: In Proceedings AFL 2014, arXiv:1405.527

Tropical Two-Way Automata

“…Actually, if a run of a two-way automaton contains an unmoving circuit, the signature of this run contains a vector where two entries with an index with the same parity are equals. In [2], we prove that the restriction of the slice automaton of A to states labelled by vectors that do not contain this kind of entry results in a finite one-way automaton where every computation corresponds to a reduced computation of A with the same weight and that every reduced computation of A has a representative in this finite one-way automaton. Finally, by Lemma 2, Proposition 2.…”

“…We give here the formal definition of the slices. A more complete description of the slice automaton is given in [2]. Definition 3.…”

Two-Way Automata over Locally Finite Semirings

“…This process produces the list of transitions in position i in the run. It is formally described in [2].…”

Two-way representations and weighted automata

“…We consider, like for instance in [3], that, during the computation of two-way automata, the move on the input only depends on the state and not on the transition. These two-way automata are as powerful as the usual ones; they correspond to the family of δ-local automata defined in [5]. The set of transitions is the support of E, the set of initial states is the support of I and the set of final states is the support of T .…”

“…In [5], it is proven that a deterministic two-way weighted automaton can emulate any unambiguous one-way weighted automaton i.e. a one-way weighted automaton with only one computation for each accepted word.…”