Lecture Notes in Computer Science 3472

80 Henrik Björklund
state that produces output y on input x . If we search the full UIO tree, looking
for nodes labeled by singletons, we will eventually find a UIO sequences for every
state that has one, but unless all states have UIO sequences, the search will never
terminate.
Ensuring Termination. A vector < s1/s ′ , s2/s ′ ,...,sk /s ′ >, where all the second
components of the pairs are identical, is called homogeneous. In particular, any
singleton path vector is homogeneous. If a node v in the UIO tree, reachable by
a path labeled with input sequence x and output sequence y from the root, is
labeled by a homogeneous vector PV, then there is no need to search the subtree
rooted at v. IfPV is a singleton, then a UIO sequence for the only vertex s such
that λ(s, x )=y has already been found. If, on the other hand, PV is not a
singleton, then no extension of x can be a UIO sequence for any state of M,
since for all s such that λ(s, x )=y, the machine ends up in the same state after
this sequence.
Also, if node v in the UIO tree is labeled by PV and on the path from the root
to v, thereisanodev ′ labeled by PV ′ such that PV ⊆ PV ′ , there is no need to
searchthesubtreerootedatv. Suppose the path from the root to v ′ is labeled
by the input sequence x ′ ,andthepathfromv ′ to v by x .Lets1/s ′ 1 and s2/s ′ 2 be
two pairs that appear in PV, and suppose that λ(s1, x ′ ·x ·x ′′ ) �= λ(s2, x ′ ·x ·x ′′ ).
Then λ(s1, x ′ · x ′′ ) �= λ(s2, x ′ · x ′′ ). In particular, if x ′ · x · x ′′ is a UIO sequence
for s1, thensoisx ′ · x ′′ .Thusifs1 has a UIO sequence with x ′ as a prefix, such a
sequence can be found by searching the subtree rooted at v ′ , without descending
into the subtree rooted at v.
Call the label of node v in the full UIO tree a repeated label if a superset of
the label appears in a node on the path from the root to v.
Any node in the full UIO tree with a homogeneous or repeated label is called
terminal. Since it is now clear that there is no point in searching the subtrees
rooted at terminal nodes, we may cut away all children of terminal nodes. We
also remove all subtrees rooted at nodes with empty path vectors as labels. All
vertices in such subtrees have empty labels. The result is called the pruned UIO
tree of the machine. Note that the pruned UIO tree is finite, since there are
only finitely many possible path vectors, and no path vector appears more than
once on any path from the root. Figure 3.6 shows the pruned UIO tree for the
machine in Figure 3.3.
If we search the pruned UIO tree for nodes labeled by singletons, we are still
guaranteed to find a UIO sequence for every state that has one, and can also be
sure that the search terminates.
To further improve efficiency, we can also define vertices labeled with path
vectors such that no state appears exactly once in the current vector as terminal.
It is clear that the input sequence labeling the path to such a vertex cannot be
a prefix of a UIO sequence for any state.
The pruned UIO trees described here bear some resemblance to the splitting
trees used to compute distinguishing sequences; see [LY96] and Chapter 2. The
details of the two data structures are quite different, however. The UIO tree
is labeled by a set of pairs of states, rather than a set of states. Thus it does