Lecture Notes in Computer Science 3472

3.4.2 Naik’s Algorithm
3 State Verification 79
Naik’s algorithm for computing UIO sequences was presented in 1997 [Nai97],
when the problem was already known to be PSPACE-complete [LY94]. The
algorithm is complete, in the sense that it computes a UIO sequence for every
vertex that has one, and always terminates. The basic idea is still to examine
all possible input sequences, but it is augmented by a number of other ideas,
including:
(1) Techniques for ensuring termination.
(2) Inference rules to extend a sequence for one state to sequences for other
states.
(3) Heuristics that under some circumstances provide UIO sequences for some
states in polynomial time.
(4) A modified depth-first search to find UIO sequences for some states.
(5) Techniques for shortening found sequences.
We will examine the first four items in this list.
Naik uses UIO trees, which are trees with edges labeled by input-output
pairs, and nodes labeled by vectors that represent paths in the machine that
exhibit the input-output behavior indicated by the edges from the root to the
node.
Definition 3.8 (Path Vectors). Given a Mealy machine M, apath vector is a
> satisfying the following. There
vector of state pairs PV =< s1/s ′ 1,...,sk /s ′ k
is an input sequence x and an output sequence y such that λ(si, x )=y and
δ(si, x )=s ′ i for every pair si/s ′ i in PV. The vector IV (PV)=< s1,...,sk >
consisting of the first components of the pairs in PV is called the initial vector of
PV. The vector CV (PV)=< s ′ 1 ,...,s′ k > consisting of the second components
is called the current vector.
Given a Mealy machine M =(I , O, S,δ,λ), its IO tree is an infinite tree in
which every node has |I |·|O| children, one for each possible input-output pair in
M. The edges to the children are labeled by distinct input-output pairs. Thus
the nodes of the tree correspond to every possible input-output sequence.
Definition 3.9 (Full UIO Tree). Let M =(I , O, S,δ,λ) be a Mealy machine.
The full UIO tree of M is a labeling of the IO tree by path vectors. The root of
the tree is labeled by < s1/s1,...,sn/sn >, wheren is the number of states of
M. If a node in the tree is reachable from the root by a path labeled by input
sequence x and output sequence y, and has label PV =< s1/s ′ 1 ,...,sk /s ′ k >,
then for every si/s ′ i in PV,itmustholdthatλ(si, x )=y and δ(si, x )=s ′ i .
Furthermore, PV contains all pairs si/s ′ i such that λ(si, x )=y and δ(si, x )=s ′ i .
Notice that some nodes may be labeled by empty path vectors.
It immediately follows that if a node, labeled by a singleton < s/s ′ >,is
reachable from the root by a path labeled with input sequence x and output
sequence y, thenx is a UIO sequence for s. Thisisbecauses must be the only