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node labelled b is accepted by a PTA with two states Q =
, with
initial state q b , priorities pri ( q b )=1, pri ( q ) = 2, and the transitions
q b ,q
( q b ,a,q b ,q ) , ( q b ,a,q,q b ) , ( q b ,b,q,q ) , ( q, a, q, q ) , ( q, b, q, q ) .
The state q b is used to non-deterministically select a path that leads to
a node with label b . This is done by the first two transitions. A simple
argument shows that this language cannot be accepted by a deterministic
parity tree automaton.
We now analyse algorithmic and closure properties of PTAs. Some closure
properties are rather easy to obtain.
Proposition 2.13 The class of regular languages of infinite trees is closed
under union, intersection, and projection (relabelling).
Proof For the closure under union and intersection one can use a classical
product construction where the two given automata are executed in parallel.
The acceptance condition becomes a Muller condition that expresses that
both automata accept for the intersection, or that at least one of the automata
accepts for the union. As mentioned above, one can use the LAR construction
to turn a Muller automaton into an equivalent parity automaton.
For the projection let h Γ be a relabelling. It is applied to trees by
applying it to each label of the tree. Given a PTA A for a tree language T
we want to construct a PTA for the tree language h ( T )= {h ( t ) | t ∈ T} .
For this purpose we simply replace every transition ( q, a, q 0 ,q 1 )in A by the
transition ( q, h ( a ) ,q 0 ,q 1 ).
The connection to games is used for the closure under complementation
and the emptiness test, as explained in the following.
2.4.1 Complementation
Let us first analyse why the complementation problem for PTAs is a di cult
problem. By definition, a tree is accepted if
path . (path satisfies acceptance condition) .
By negation of this statement we obtain that a tree is not accepted if
path . (path does not satisfy acceptance condition) .
This exchange of quantifiers makes the problem di cult. But there are two
observations to make that lead towards the solution. First of all, statements
of the form
are very close to the nature of games and strategies: there
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