Nondeterministic Biautomata and Their Descriptional Complexity

Abstract

We investigate the descriptional complexity of nondeterministic biautomata, which are a generalization of biautomata [O. Klíma, L. Polák: On biautomata. RAIRO -Theor. Inf. Appl., 46(4), 2012]. Simply speaking, biautomata are finite automata reading the input from both sides; although the head movement is nondeterministic, additional requirements enforce biautomata to work deterministically. First we study the size blow-up when determinizing nondeterministic biautomata. Further, we give tight bounds on the number of states for nondeterministic biautomata accepting regular languages relative to the size of ordinary finite automata, regular expressions, and syntactic monoids. It turns out that as in the case of ordinary finite automata nondeterministic biautomata are superior to biautomata with respect to their relative succinctness in representing regular languages.

Minimal and Hyper-Minimal Biautomata

“…The biautomata as introduced in [19] always had to satisfy both these properties, while in [7,8] also biautomata that lack one or both of these properties, as well as nondeterministic biautomata were studied. Throughout the current paper, when writing of biautomata, or DBiAs, we always mean deterministic biautomata that satisfy both the ⋄-property, and the F -property, i.e., the model as introduced in [19].…”

“…Later we call the former property the ⋄-property and the latter one the F -property. In [19] and a series of forthcoming papers [7,8,15,18] it was shown that biautomata share a lot of properties with ordinary finite automata. For instance, as minimal DFAs, also minimal DBiAs are unique up to isomorphism [19].…”

Minimization and Characterizations for Biautomata

“…This notion of biautomata was recently introduced in [5] and moreover resembles that of nondeterministic linear automata as defined in [8], which characterize the family of linear context-free languages. A nondeterministic biautomaton is a sixtuple A = (Q, Σ, ·, •, I, F ), where Q is a finite set of states, Σ is an alphabet, · : Q × Σ → 2 Q is the forward transition function, • : Q × Σ → 2 Q is the backward transition function, I ⊆ Q is the set of initial states, and F ⊆ Q is the set of final or accepting states.…”

“…Moreover, simple structural characterizations based on biautomata for language families such as the piecewise testable or prefix-suffix testable languages were given in [7]. Recently in [6] also descriptional complexity issues for biautomata were addressed and in [5] biautomata were generalized to deal with nondeterminism. The role of the ⋄and the F -property for descriptional complexity issues is briefly discussed in [5], but in particular, a complete understanding of the influence of the latter property is missing.…”

“…Whether this elegant minimization method can also be applied to a biautomaton A is not completely clear, since the above mentioned two properties to enforce determinism may be lost by computing A R or P(A R ). To this end we utilize the recently introduced nondeterministic biautomata [5]. It is known that these machines already accept non-regular languages and characterize the family of linear context-free languages [8], but as a side result we prove that nondeterministic biautomata with the ⋄-property accept regular languages only.…”