Lecture Notes in Computer Science 3472

570 Therese Berg and Harald Raffelt
node designated as the root, denoted by ε ∈ r. Therootε has no parent and
every other node n �= ε has a unique parent. The children of a node c(n) are
the nodes n ′ which have n as parent. c(n) :={n ′ | n = p(n ′ )}. Thelevel |n| of
anoden is the distance from the root ε to the node: the root’s level is |ε| =0
and |n| =1+|p(n)|. Abranch β = n0n1 ... of a tree is a infinite sequence of
nodes such that n0 is the root ε and for all other nodes ni (i > 0) of the branch
the predecessor of a node in the branch is its parent:ni−1 = p(ni).
Definition 19.7. A run of an alternating Büchi automaton on a word w =
a1a2 ...Σω is a state-labeled tree (R, L), where R is a tree and L is a mapping
from the nodes of the tree r to the states, such that r(ε) =q0 and the following
holds:
• Each node n with level |n| = i |π| of the tree r has k children n1,...,nk such
that {L(n1),...,L(nk )} satisfies¡ δ (L(n), ai )
Note that the maximal level of a node in R is at most |π|. Not all branches
need to reach such depth, since if δ (L(n), a) =true, thenn does not need to
have any children. On the other hand we can not have δ (L(n), a) =false ,since
false is not satisfiable.
For an alternating Büchi automaton a run (r, L) is accepting, iff every infinite
branch visits accepting states infinitely often. Note that true and false are special
states. For any action both states have only a single transition to itself. The
state true is accepting and the state false is not accepting. Therefore a run with a
branch visiting a false-state can not be accepting and a run with a branch visiting
a true-state is accepting if all other branches visit accepting states infinitely often.
The language L (A) of an alternating Büchi automaton A is determined by all
words for which an accepting run exists. Note that for a word w there may be
more than one accepting run.
Example. Figure 19.6 outlines a run of the Büchi automation of Figure 19.5 on
the infinite word w =({coin}{}) ω .
¬U (true, coin ∧ X (coin))
{coin} {coin}
¬U (true, coin ∧ X (coin))
{} {}
... ...
¬U (true, coin ∧ X (coin))
{coin} {coin}
¬coin
{}
true true
Fig. 19.6. Example run of an alternating Büchi automaton