Lecture Notes in Computer Science 3472

s0
c = ∞
8 Test Derivation from Timed Automata 215
on?, c = ∞,
c := 0
off !, c =5,
c := ∞
s1
c � 5
Fig. 8.6. TIOA specification for a Light Switch [SVD01]
• ∀ 0≤d ′ ≤d : ν⊕d ′ |=Inv(s)
(s,ν) d
↣ (s,ν⊕d)
where the actions in R >0 are referred to as time delays and
(ν ⊕ d)(c) def
�
ν(c)+d if (ν(c)+d) ∈ intv(c)
=
∞ otherwise
on?, c < 5,
c := 0
The following lemma, which is a direct corollary of the definitions, gives four
basic properties of the operational semantics of a timed I/O automaton.
Lemma 8.19. Let A be a TIOA, then
• OS(A ) is deterministic
• OS(A ) possesses Wang’s time additivity property:
q d+d′
↣ q ′ iff ∃ q ′′ : q d
↣ q ′′ d′
′′ ∧ q ↣ q ′
• Each state of OS(A ) has either
– a single outgoing transition labeled with
✑
an output action, or
✑✑✸ o!
✉
– both outgoing delay transitions and outgoing
✑
input transitions (one for each input action),
but no outgoing output transitions
✑✑✸ d
✉
i1? �❅
in?
�✠ ··· ❅❘
States of the second type are called stable
• For each state q ∈ QOS(A ), there exists a unique finite sequence of output
actions σ and a unique stable state q ′ such that q σ
↣ q ′ .
8.4.2 Discretization
The construction of a finite subautomaton used, for the discretization of the
state space, is based on the fundamental concept of a region due to Alur and