Lecture Notes in Computer Science 3472

74 Henrik Björklund
output 0. An example of the construction for two small automata, accepting the
common string a is shown in Figure 3.4.
Σ ∪{r, f }/0
s
f /1
f /0
r, b/0 a, b/0
r,a/0
f /1
f /0
a/0
r/0
b/0
r/0
a, b/0
Fig. 3.4. An example of the reduction. The automata A1 and A2 over the alphabet
Σ = {a, b} are combined into a Mealy machine with input alphabet I = {a, b, r, f }
and output alphabet O = {0, 1}. The automata both accept the string a from their
initial states (the leftmost state of each automaton), and the input-output sequence
raf is a UIO sequence for state s of the machine.
The state s in the constructed Mealy machine has a UIO sequence if and only
if the automata accept a common string x . We first show the if direction. Suppose
all the automata accept x = a1 ...ak .Thenra1 ...ak f is a UIO sequence for s.
From state s, input rxf will obviously give a sequence of only zeroes as output.
If the machine is initially in some state t �= s, then the first symbol, r, will take
it to the initial state of the automaton Ai that t belongs to. The sequence x will
then lead to an accepting state of Ai. Finally, the symbol f will give a transition
to swithoutput1. Thus no other state can have an output of only zeroes on
input rxf .
For the other direction, assume that there is a UIO sequence x for s. Then
x must contain an f , because input f is the only symbol that can produce an
output different from 0. Assume that no r appears before the first occurrence
of f in x . Consider the prefix x ′ = a1a2 ...ak f of x that cuts x after the first
f . We claim that λ(q, x ′ )=0 k 1 for all q �= s. Thisisbecauseδ(q, f )=s for
all states, so if the f produces output 0, then all subsequent outputs will be
A1
A2