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Distinguishability Operations and Closures

Câmpeanu C., Moreira N., Reis R.

Fundamenta Informaticae

2016

Vol. 148, nr 3/4

243--266

Given a language L, we study the language of words D(L), that distinguish between pairs of different left quotients of L. We characterize this distinguishability operation, show that its iteration has always a fixed point, and we generalize this result to operations derived from closure operators and Boolean operators. For the case of regular languages, we give an upper bound for the state complexity of the distinguishability operation, and prove its tightness. We show that the set of minimal words that can be used to distinguish between different left quotients of a regular language L has at most n - 1 elements, where n is the state complexity of L, and we also study the properties of its iteration. We generalize the results for the languages of words that distinguish between pairs of different right quotients and two-sided quotients of a language L.

Results on Transforming NFA into DFCA

Câmpeanu C., Kari L., Păun A.

Fundamenta Informaticae

2005

Vol. 64, nr 1-4

53-63

In this paper we consider the transformation from (minimal) non-deterministic finite automata (NFAs) to deterministic finite cover automata (DFCAs). We want to compare the two equivalent accepting devices with respect to their number of states; this becomes in fact a comparison between the expression power of the nondeterministic device and the expression power of the deterministic with loops device. We prove a lower bound for the maximum state complexity of deterministic finite cover automata obtained from non-deterministic finite automata of a given state complexity n, considering the case of a binary alphabet. We show, for such binary alphabets, that the difference between maximum blow-up state complexity of DFA and DFCA can be as small as [..]compared to the number of states of the minimal DFA. Moreover, we show the structure of automata for worst case exponential blow-up complexity from NFA to DFCA. We conjecture that the lower bound given in the paper is also the upper bound. Several results clarifying some of the structure of the automata in the worst case are given (we strongly believe they will be pivotal in the upper bound proof).