4 Algebraic semantics
for PITL
Algebraic semantics for PITL (12)
Can we give the semantic domain an algebraic structure? [5mm] Let denote the set
of intervals for which , i.e.,
The of two PITL formula is then
Algebraic semantics for PITL (13)
So we need algebraic operators that correspond to and
Algebraic semantics for PITL (14)
What about chop (‘ ’)? [2mm] Let denote the fusion of two intervals
, i.e.,
Let ( and are not the same), and
Let then
Algebraic semantics for PITL (15)
Algebraic semantics for PITL (16)
What about ?
Algebraic semantics for PITL (17)
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
Algebraic semantics for PITL (18)
What about a state formula, i.e., a formula without temporal operators? [5mm] A state
formula only constrains the first state of an interval. Let be a state formula. Then the
following holds
where
Algebraic semantics for PITL (19)
What about chopstar ‘ ’? [2mm] 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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