Chapter II ω-Automata
1. ω-Automata and ω-Languages
ω-automata accept (or reject) words of infinite length
ω-languages consisting of infinite words appear:
— in verification, as encodings of non-terminating executions of a program
— in arithmetic, as encodings of sets of real numbers
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ω-Languages
• An ω-word is an infinite sequence of letters.
• The set of all ω-words is denoted by Σ .
• An ω-language is a set of ω-words, i.e., a subset
of Σ .
• A language can be concatenated with an ωlanguage to yield the ω-language
, but
two ω-languages cannot be concatenated.
• The ω-iteration of a language ⊆ Σ ⋆ , denoted
by , is an ω-language.
• Observe: ∅ = ∅.
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ω-Regular Expressions
• ω-regular expressions have syntax
∷=
|
|
+
where is an (ordinary) regular expression.
• The ω-language ( ) of an ω-regular expression is
inductively defined by
=
+
=
=
∪
( )
( )
• A language is ω-regular if it is the language of some
ω-regular expression .
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Büchi Automata
• Invented by J.R. Büchi, swiss logician.
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Büchi Automata
• Same syntax as DFAs and NFAs, but different
acceptance condition.
• A run of a Büchi automaton on an ω-word is an
infinite sequence of states and transitions.
• A run is accepting if it visits the set of final states
infinitely often.
– Final states renamed to accepting states.
• A DBA or NBA accepts an ω-word if it has an
accepting run on it; the ω-language
of is
the set of ω-words it accepts.
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Some examples
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From ω-Regular Expressions to NBAs
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From ω-Regular Expressions to NBAs
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From ω-Regular Expressions to NBAs
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From NBAs to ω-Regular Expressions
• Lemma: Let
be a NFA, and let ,
be states
of . The language
of words with runs
leading from to and visiting exactly
once is regular.
• Let
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denote a regular expression for
1 ω-Automata and ω-Languages
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From NBAs to ω-Regular Expressions
• Example:
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From NBAs to ω-Regular Expressions
• Given a NBA
, we look at it as a NFA, and
compute regular expressions
.
• We show:
= (
)
∈
– An ω-word belongs to
iff it is accepted by a
run that starts at and visits some accepting
state infinitely often.
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From NBAs to ω-Regular Expressions
• Example:
=
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