Between Deterministic and Nondeterministic Quantitative Automata

There is a challenging trade-off between deterministic and nondeterministic automata, where the former suit various applications better, however at the cost of being exponentially larger or even less expressive. This gave birth to many notions in between determinism and nondeterminism, aiming at enjoying, sometimes, the best of both worlds. Some of the notions are yes/no ones, for example initial nondeterminism (restricting nondeterminism to allowing several initial states), and some provide a measure of nondeterminism, for example the ambiguity level. We analyze the possible generalization of such notions from Boolean to quantitative automata, and suggest that it depends on the following key characteristics of the considered notion N– whether it is syntactic or semantic, and if semantic, whether it is word-based or language-based. A syntactic notion, such as initial nondeterminism, applies as is to a quantitative automaton A, namely N(A). A word-based semantic notion, such as unambiguity, applies as is to a Boolean automaton t-A that is derived from A by accompanying it with some threshold value t ∈ R, namely N(t-A). A language-based notion, such as history determinism, also applies as is to t-A, while in addition, it naturally generalizes into two different notions with respect to A itself, by either: i) taking the supremum of N(t-A) over all thresholds t, denoted by Threshold-N(A); or ii) generalizing the basis of the notion from a language to a function, denoted simply by N(A). While in general N(A) ⇒ Threshold-N(A) ⇒ N(t-A), we have for some notions N(A) ≡ Threshold-N(A), and for some not. (For measure notions, ⇒ stands for ≥ with respect to the nondeterminism level.) We classify numerous notions known in the Boolean setting according to their characterization above, generalize them to the quantitative setting and look into relations between them. The generalized notions open new research directions with respect to quantitative automata, and provide insights on the original notions with respect to Boolean automata. 2012 ACM Subject Classification Theory of computation → Logic and verification