On biautomata

Abstract

We initiate the theory and applications of biautomata. A biautomaton can read a word alternately from the left and from the right. We assign to each regular language L its canonical biautomaton. This structure plays, among all biautomata recognizing the language L, the same role as the minimal deterministic automaton has among all deterministic automata recognizing the language L. We expect that from the graph structure of this automaton one could decide the membership of a given language to certain significant classes of languages. We present the first result of this kind: a language L is piecewise testable if and only if the canonical biautomaton of L is acyclic. From this result the famous Simon's characterization of piecewise testable languages easily follows.

Minimal and Hyper-Minimal Biautomata

“…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]. For such biautomata the following is known from the literature [7,19]:…”

“…Derivatives are used in [19] for the definition of the canonical biautomaton of a regular language, which is similar to the canonical DFA as described in [3]. The set of states of the canonical biautomaton for a regular language L consists of all derivatives of L-this is a finite set because L is regular-and the right language of a state u −1 Lv −1 is the language u −1 Lv −1 .…”

“…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]. For such biautomata the following is known from the literature [7,19]:…”

“…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]. Moreover, in [7] it was shown that classical DFA minimization algorithms can be adapted to biautomata as well.…”

“…Recently, an alternative automaton model to deterministic finite automata, the so called biautomaton (DBiA) [19] was introduced. Roughly speaking, a biautomaton consists of a deterministic finite control, a read-only input tape, and two reading heads, one reading the input from left to right (forward transitions), and the other head reading the input from the opposite direction, i.e., from right to left (backward transitions).…”