The number of similarity relations and the number of minimal deterministic finite cover automata

Authors:
Cezar Câmpeanu;Andrei Paun
Affiliations:
Department of Mathematics and Computer Science, University of Prince Edward Island, Canada;Department of Computer Science, University of Western Ontario, London, Ontario, Canada
Venue:
CIAA'02 Proceedings of the 7th international conference on Implementation and application of automata
Year:
2002

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Abstract

Finite Deterministic Cover Automata (DFCA) can be obtained from Deterministic Finite Automata (DFA) using the similarity relation. Since the similarity relation is not an equivalence relation, the minimal DFCA for a finite language is usually not unique. We count the number of minimal DFCA that can be obtained from a given minimal DFA with n states by merging the similar states in the given DFA. We compute an upper bound for this number and prove that in the worst case (for a non-unary alphabet) it is ⌈4n-9+√8n+1/8⌉!/(2⌈4n-9+√8n+1/8⌉ - n + 1)! We prove that this upper bound is reached, i.e. for any given positive integer n we find a minimal DFA with n states, which has the number of minimal DFCA obtained by merging similar states equal to this maximum.