Lecture Notes in Computer Science 3472

19 Model Checking 569
has a transition function including δ (q0, a) ={q1, q2, q3}. This mapping can be
written as δ (q0, a) =q1 ˙∨q2 ˙∨q3 using positive Boolean formulas. In an alternating
Büchi automaton one can have mappings like δ (q0, a) =(q1 ˙∧q2) ˙∨(q3 ˙∧q4),
meaning that the automaton starts from its initial state q0 with an a-transition
and can continue in both states q1, q2 or in both states q3, q4. Notethatan
alternating Büchi automaton can continue in more than one state at the same
time.
Definition 19.6. An alternating Büchi automaton (Σ,Q,δ,q0, F )isaBüchi
automaton where the transition function is defined as follows:
δ : Q × Σ → B + (Q)
˙∧
{coin}
¬trueU (coin ∧ X (coin))
{coffee, coin}
{}
{coffee}
˙∨ ˙∨ ˙∨ ˙∨
{coin}
¬coin
false true
{coffee, coin}
{}
{coffee}
Fig. 19.5. Example Alternating Büchi Automaton
In Figure 19.5 a graphical representation of an alternating Büchi automaton
is presented, visualizing the transition function of an automaton in an abstract
way. The dotted lines are used to identify edges which belong to positive Boolean
combination of states.
Because of the universal choice in alternating transitions, a run of an alternating
automaton is a tree rather than a sequence. A tree R =(r, p) is an infinite
directed acyclic graph where r is a set of nodes and p is a parent function. One
˙∧
2 AP