Lecture Notes in Computer Science 3472

19 Model Checking 559
The first state π0 of the run is the initial state q0 of M and each next
state πi+1 is reached by reading one letter ai. A run is called accepting, if
πn+1 ∈ F .The letters ai read by a run form a w = a1 ...an. If the run is acceting,
then the word read by the run is also said to be accepting. The language a DFA
M recognizes is denoted by L (M). It is defined as the set of accepting words.
We call a language L regular if there is a DFA accepting L.
A different kind of automaton which operates on infinite words was introduced
by Büchi [Büc62] for obtaining a decision procedure for the monadic
second-order theory of structures with one successor. Later these automata were
called Büchi automata. The main idea of Büchi automata is to operate on infinite
input words w = a0a1 ... ∈ Σ ω ,whereasΣ ω denotes the set of all infinite
words over the alphabet Σ.
Definition 19.3. A Büchi automaton is a 5-tuple A =(Σ,Q,δ,q0, F ), where
• Σ is a finite set of actions or letters,
• Q is a finite set of states,
• δ : Q × Σ → 2 Q is a transition function,
• q0 ∈ Q is a initial state and,
• F ⊆ Q is a set of accepting states.
Starting from its initial state the automaton chooses nondeterministically a
possible successor state in δ (q, a) of the current state q.
Definition 19.4. An infinite run π of a Büchi automaton A on a word w =
a0a1 ...∈ Σ ω is a sequence π = π0π1 ...∈ Q ω , such that
• π0 = q0
• πi+1 ∈ δ (πi, ai).
The first state π0 of the run is the initial state q0 of A and each next state
πi+1 is one of the states reachable by reading one letter ai. The states that
occur infinitely many times in a run are inf(π) ={q | q ∈ Q and q = πi for
infinitely many i ≥ 0}. An infinite run of a Büchi automaton is accepted if it
visits accepting states infinitely often. Formally an infinite run π = π0π1π2 ...
is a accepted iff inf(π) ∩ F �= ∅. An infinite word w = a0a1 ...∈ Σ ω is accepted
by the automaton, if and only if there is an infinite run of the automaton which
accepts the word. The language L (A) accepted by a Büchi automaton A is the set
of all accepted words. The complement of a language L (A) accepted by a Büchi
automaton is the set of all not accepted words, it is defined as L (A) =Σ ω \L(A).
The length of a finite run π = π0π1 ...πn is the number of its elements
denoted by |π| = n + 1. The length of an infinite run is denoted by |π| = ∞.
For 0 ≤ i < |π| the suffix of a run π = π0π1 ...πn starting with element πi is
denoted by π i = πiπi+1 ....