Transfinite Semantics in the Form of Greatest Fixpoint

Transfinite semantics is a semantics according to which program executions can continue working after an infinite number of steps. Such a view of programs can be useful in the theory of program transformations.So far, transfinite semantics have been succesfully defined for iterative loops. This paper provides an exhaustive definition for semantics that enable also infinitely deep recursion.The definition is actually a parametric schema that defines a family of different transfinite semantics. As standard semantics also match the same schema, our framework describes both standard and transfinite semantics in a uniform way.All semantics are expressed as greatest fixpoints of monotone operators on some complete lattices. It turns out that, for transfinite semantics, the corresponding lattice operators are cocontinuous. According to Kleene’s theorem, this shows that transfinite semantics can be expressed as a limit of iteration which is not transfinite.