Lecture Notes in Computer Science 3472

Ψ(S ′ )={PE ′ | E ′ ∈ 2E(S ′ �
) ∧ PE ′ =
ϕ∈Γ (E ′ )
8 Test Derivation from Timed Automata 209
ϕ ∧
�
ϕ∈Γ (E(S ′ )−E ′ )
¬ϕ}
Then, the set of guards ϕi whose disjunction equals the disjunctive normal form
is denoted as GDNF, i.e,
GDNF(PE ′)={ϕi ∈ Φ(C ) | �
i ϕi = DNF (PE ′)}
and finally Ψdnf (S ′ )is:
�
Ψdnf (S ′ )=
PE ′ ∈Ψ (S ′ )
GDNF (PE ′).
To make this definition more understandable we show the next example.
Using our example of the automatic Light Switch, we present the procedure for
find the equivalences classes for S ′ = {s1}.
Example. Let S ′ = {s1}, then the transitions from S ′ are:
E(S ′ )={(s1, on?, con < 5, s1), (s1, off !, con =5, s0)}
the guards of E(S ′ )are:
Γ (E(S ′ )) = {con < 5, con =5}
only for simplicity we will present 2 Γ (E(S ′ )) instead of 2 E(S ′ ) :
2 Γ (E(S ′ )) = {∅, {con < 5, con =5}, {con < 5}, {con =5}}
and:
Ψ(S ′ )={(con ≥ 5) ∧ (con �= 5), (con < 5) ∧ (con �= 5),
(con ≥ 5) ∧ (con =5), (con < 5) ∧ (con =5)}
the disjunctive normal form of Ψ(S ′ )is:
Ψdnf (S ′ )={con > 5, con < 5, con =5, ∅}
Then we have: [s1, con > 5], [s1, con < 5] and [s1, con = 5] as states for our
equivalence class graph.
The state space of the ERA specification is a graph of equivalence classes.
A node in this graph corresponds to an equivalence class. A transition between
two nodes is labeled with an action, and represents the possibility of execute
an action in a state in the source node, wait some amount of time, and thereby
enter in a state in the target node. The graph is constructed by start from
an existing node [S ′ , p] (initially the equivalence class of the initial location),
and then for each enabled action a, compute the set of locations S ′′ that can
be entered by execute the a action from the current equivalence class. Then
the partitions p ′ of location S ′′ can be computed according to Definition 8.14.
Every [S ′′ , p ′ ]isthenana successor of [S ′ , p]. Only equivalence classes whose
constraints have solutions need to be represented. The equivalence class graph
is defined inductively in the Algorithm 11.
Each equivalence class [S ′ , p] is decorated with the action sets M , C , R from
the testing preorder, as it is shows in definition 8.15.