   
 4 Algebraic semantics for PITLCan 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
  

 