On the Herbrand Kleene universe for nondeterministic computations

Abstract

For nondeterministic recursive equations over an arbitrary signature of function symbols including the nondeterministic choice operator "or" the interpretation is factorized. It is shown that one can either associate an infinite tree with the equations, then interprete the function symbol "or" as a nondeterministic choice operator and so mapping the tree onto a set of infinite trees and then interprete these trees. Or one can interprete the recursive equation directly yielding a set-valued function. Both possibilities lead to the same result, i.e. we obtain a commuting diagram. This explains and solves a problem posed in [Nivat 80]. Basically the construction gives a generalisation of the powerdomain approach applicable to arbitrary nonflat (nondiscrete) algebraic domains.