Lecture Notes in Computer Science 3472

218 Laura Brandán Briones and Mathias Röhl
• Q ′ = Q n
• L ′ = Σ ∪{2−n }
• q ′ 0 = q0
• for all q, q ′ ∈ Q ′ and a ∈ L ′ , q a
↣ ′ q ′ iff q a
↣ q ′ .
The grid automaton is the restriction of OS(A )tothetimestepsin2 −n ,
therefore G (A , n) is finite.
Example. In Figure 8.8 the grid automaton of our example of the Light Switch
for n = 2 is presented. Here we denote the initial state as >, for distinguish
it from the double circle denoting the initial state in a TIOA.
〈s 1 ,
c =5〉
1
2
〈s 1 ,
c = 9 2 〉
1
2
〈s 1 ,
c =4〉
on?
1
2
off !
〈s 1 ,
c = 7 2 〉
on?
1
2
〈s 1 ,
c =3〉
on?
1
2
〈〈s0, c = ∞〉〉
〈s 1 ,
c = 5 2 〉
1
2
1
2
on?
〈s 1 ,
c =2〉
1
2
on?
〈s 1 ,
c = 3 2 〉
1
2
on?
〈s 1 ,
c =1〉
on?
1
2
〈s 1 ,
c = 1 2 〉
Fig. 8.8. The grid automaton G (A , n) withA as the Light Switch automaton and
n =2
Corollary 8.26. Let A and B be TIOA with the same input actions, and let
n be at least the number of regions of SA × SB, then
A � B iff G (A , n) � G (B, n).
on?
on?
1
2
on?
〈s 1 ,
c =0〉
Using the grid automaton with the appropriate degree of granularity the
problem of decide bisimulation equivalence of TIOA is reduced to the problem
of decide bisimulation equivalence of their finite subautomata.
8.4.3 Testing
A test sequence for a TIOA A is a finite sequence of delays and input actions
of A (we denoted the set of this sequences as Exp). A test sequence σ can be