4 Algebraic semantics
for PITL
Can we give the semantic domain an algebraic structure?
Let denote the set of intervals for which , i.e.,
The of two PITL formula is then
So we need algebraic operators that correspond to and
What about chop (‘ ’)?
Let denote the fusion of two intervals , i.e.,
Let ( and are not the same), and
Let then
What about ?
What about ?
can be defined as
is the set of intervals containing states
is the set of intervals containing states
is the set of intervals containing states
is the set of intervals containing exactly 2 states
What about a state formula, i.e., a formula without temporal operators?
A state formula only constrains the first state of an interval. Let be a state formula. Then
the following holds
where
What about chopstar ‘ ’?
In the semantics of ‘ ’ both finite and infinite iteration are considered simultaneously. Let’s
define separate algebraic operators for them.
Let and denote respectively finite and infinite iteration of a set
and can be defined as follows
Then we have
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