ISSN 0032-9460, Problems of Information Transmission, 2013, Vol. 49, No. 2, pp. 149–162.
Pleiades Publishing, Inc., 2013.
Original Russian Text
An.A. Muchnik, 2013, published in Problemy Peredachi Informatsii, 2013, Vol. 49, No. 2, pp. 58–72.
Determinization of Ordinal Automata
An. A. Muchnik
Received September 5, 2011; in ﬁnal form, February 5, 2013
Abstract—It is proved that for each nondeterministic ordinal automaton there exists a deter-
ministic ordinal automaton which is equivalent to the original one for all countable ordinals.
An upper bound for the number of states of the deterministic automaton is double exponential
in the number of states of the nondeterministic automaton.
A word in a given alphabet is a ﬁnite sequence of symbols of this alphabet; an ω-word is an
inﬁnite sequence of such symbols. There exist two types of ﬁnite automata whose inputs are words
in a given alphabet. Those are deterministic and nondeterministic automata (the latter are a gen-
eralization of the former). As is proved in , these two types of automata are equivalent; namely,
each nondeterministic ﬁnite automaton is equivalent to a deterministic ﬁnite automaton, i.e., the
deterministic automaton accepts the same words as the original one. Moreover, a deterministic
automaton equivalent to a nondeterministic automaton with N states requires 2
states or less.
This bound cannot be improved: for each N there exists a nondeterministic automaton with N
states which is not equivalent to any deterministic automaton with less than 2
Not only a word but also an ω-word can be an input of a deterministic ﬁnite automaton. When
reading an ω-word, an automaton produces an inﬁnite sequence of states, which is called a run of
the automaton on the given ω-word. We say that an automaton accepts a given ω-word if at least
one of its accepting states occurs in the run inﬁnitely many times. Thus we obtain the deﬁnition
of B¨uchi automata on ω-words [2, 3].
Similarly, one can deﬁne nondeterministic B¨uchi automata on ω-words. A nondeterministic
B¨uchi automaton can have more that one run on a given ω-word, or no runs at all. By deﬁnition,
it accepts a given ω-word if there exists a run of this automaton on this ω-word in which one of the
accepting states occurs inﬁnitely many times. It is known that the class of recognizable languages
for deterministic B¨uchi automata is strictly included in the class of recognizable languages for
nondeterministic automata (see, e.g., ).
In this paper we consider a more general form of automata on ω-words, namely, Muller au-
tomata . By a macro-state of such an automaton we mean any subset of the set of its states.
A Muller automaton is deﬁned by a ﬁnite automaton on ﬁnite words and a family of its macro-
states. Elements of this family are called accepting macro-states. A deterministic Muller automaton
accepts an ω-word if the set of states which occur in the run of the automaton on this ω-word is an
accepting macro-state. Similarly, for nondeterministic Muller automata we say that an automaton
accepts an ω-word if there exists a run of the automaton on this ω-word for which the set of states
occurring inﬁnitely many times in this run is an accepting macro-state.
Supported in part by the Russian Foundation for Basic Research, project nos. 09-01-00709 and 12-01-