Partial inductive definitions

Abstract

An attempt to consider partial definitions of semantically oriented data types will be described. We will in a certain sense think of such data types as inductively defined. A class of inductive definitions will be interpreted as partial definitions: partial inductive definitions. The presentation of such a definition is in itself elementary and the true complexity of the definition will show itself in questions concerning the isolation of totally defined objects. It is the same situation as in the case of partial recursive functions. The basic aim is to investigate the possibility to give direct inductive definitions of semantical notions exploring, so to speak, the structure of the given notion rather than to think of such notions as indirectly presented by a formal system or given by a definition, together with a proof of its correctness, in terms of recursion on some well-founded structure.