Lecture Notes in Computer Science 3472

∧ ⊥⊤
⊥ ⊥⊥
⊤ ⊥⊤
∨ ⊥⊤
⊥ ⊥⊤
⊤ ⊤⊤
∧ {⊥} {⊥, ⊤} {⊤}
{⊥} {⊥} {⊥} {⊥}
{⊥, ⊤} {⊥} {⊥, ⊤} {⊥, ⊤}
{⊤} {⊥} {⊥, ⊤} {⊤}
∨ {⊥} {⊥, ⊤} {⊤}
{⊥} {⊥} {⊥, ⊤} {⊤}
{⊥, ⊤} {⊥, ⊤} {⊥, ⊤} {⊤}
{⊤} {⊤} {⊤} {⊤}
5 Preorder Relations 125
∀
{⊥} {⊥}
{⊥, ⊤} {⊥}
{⊤} {⊤}
∃
{⊥} {⊥}
{⊥, ⊤} {⊤}
{⊤} {⊤}
Fig. 5.2. Semantics of logical operators on test outcomes.
Definition 5.4. The set O of test expressions inducing the observable testing
equivalence contains exactly all of the following constructs, with o, o1, ando2
ranging over O:
o def
= Succ (5.1)
| Fail (5.2)
| ao for a ∈ Act (5.3)
| �ao for a ∈ Act (5.4)
| εo (5.5)
| o1 ∧ o2 (5.6)
| o1 ∨ o2 (5.7)
| ∀o (5.8)
| ∃o (5.9)
Intuitively, Expressions (5.1) and (5.2) state that a test can succeed or fail by
reaching two designated states Succ and Fail, respectively. A test may check
whetheranactioncanbetakenwheninto a given state, or whether an action
is not possible at all; these are expressed by (5.3) and (5.4). We can combine
tests by means of boolean operators using expressions of form (5.6) and (5.7).
By introducing tests of form (5.5) we allow a process to “stabilize” itself through
internal actions. Finally, we have universal and existential quantifiers for tests
given by (5.8) and (5.9). Nondeterminism is introduced in the tests themselves
by the Expressions (5.7) and (5.9), the latter being a generalization of the former.
Definition 5.5. With the semantics of logical operators as defined in Figure 5.2,
the function obs inducing the observable testing equivalence, obs : O×Q →
Pconv, isdefinedasfollows:
⊓⊔