Lecture Notes in Computer Science 3472

s0
on, true, {c}
off , c =5, ∅
8 Test Derivation from Timed Automata 203
s1
c ≤ 5
on, c < 5,
{c}
Fig. 8.1. A timed automaton specification of an automatic Light Switch
Remark 8.3. Timed automata were introduced by Alur and Dill [AD94] as a generalization
of finite-state machines over infinite words [Tho90]. We only consider
timed automata without acceptance conditions which are usually referred to as
timed safety automata [HNSY92]. An introduction to acceptance is given in
Section 19.2, whereas a discussion of acceptance conditions in the context of
timed automata can be found elsewhere [HKWT95].
The behavior of a timed automaton A depends on both its current location
and the actual values of all its clocks.
Definition 8.4. A clock valuation over a set of clocks C is a map ν that
assigns to each clock c ∈ C avalueinR ≥0 .WithV (C )wedenotethesetof
clock valuations over C .Ford ∈ R ≥0 , ν + d denotes the clock interpretation
which maps every clock c to the value ν(c) +d. Forλ ⊆ C , ν[λ := 0] denotes
the clock interpretation for C which assigns 0 to each c ∈ λ, and agrees with ν
over the rest of the clocks.
A labeled transition system M with uncountably many states can be used
to define the possible behavior of a timed automata A .AstateofM has to
be a pair 〈s,ν〉 such that s is a location of A and ν is a clock valuation for C
satisfying invariant InvA (s). Transitions of M represent either an elapse of time
or a transition of A .
Definition 8.5. The semantics of a timed automaton A is given by the LTS
M = 〈Q, Q0, L, →〉, where
• Q = {〈s,ν〉∈SA × V (CA ) | ν |= InvA (s)}
• Q0 ⊆ Q with 〈s,ν〉∈Q0 iff s ∈ S0A and ν(c) = 0 for all clocks c ∈ CA
• L = ΣA ∪ R≥0 •→⊆Q × L × Q, which could be either
– (〈s,ν〉, d, 〈s,ν+ d〉) iffd ∈ R≥0 and for all 0 ≤ d ′ ≤ d, ν + d ′ |= InvA (s)
– (〈s,ν〉, a, 〈s ′ ,ν[λ := 0]〉) iff(s, a,ϕ,λ,s ′ ) ∈ EA and ν |= ϕ
Due to dense-time clocks, the transition system M for a timed automaton A
has infinitely many states and operates on infinitely many symbols. Analysis of