DFA with a Bounded Activity Level

Lookahead DFAs are used during parsing for sake of resolving conflicts (as described in more detail in the introduction). The parsing of an input string w may require many DFA-explorations starting from different letter positions. This raises the question how many of these explorations can be active at the same time. If there is a bound on this number depending on the given DFA M only (i.e., the bound is valid for all input strings w), we say that M has a bounded activity level. The main results in this paper are as follows. We define an easy-to-check property of DFAs named prefix-cyclicity and show that precisely the non prefix-cyclic DFAs have a bounded activity level. Moreover, the largest possible number `M of mutually overlapping explorations of a given non prefix-cyclic DFA M with t+1 states, the so-called maximum activity level of M , is bounded from above by 2 − 1, and this bound is tight. We show furthermore that the maximum activity levels of equivalent DFAs coincide so as to form an invariant of the underlying regular language, which leads us to a characterization of prefix-cyclicity in terms of the Nerode relation. We finally establish some complexity results. For instance, the problem of computing `M for a given non prefix-cyclic DFA M is shown to be PSPACE-complete.