Interpretations of Recursion under Unbounded Nondeterminacy

A language is constructed that supports arbitrary atomic statements, composition, alternatives, and mutual recursion in the presence of unbounded nondeterminacy. The concept of interpretation is defined axiomatically. By operational means a standard interpretation is constructed, which is proved to be the smallest interpretation with respect to the Egli-Milner ordering. The corresponding weakest precondition and weakest liberal precondition are characterized as smallest and largest preparator functions. We characterize the interpretations which upon divergence cannot deliver unjustified meaningful values. These interpretations form a spectrum that ranges from standard semantics, over various forms of fair semantics, to the so-called friendly semantics where error is signalled only if divergence can become inevitable after finitely many steps. We determine the corresponding preparator functions