Inductive Definitions Over Finite Structures

We give a simple proof of a theorem of Gurevich and Shelah, that the inductive closure of an inflationary operator is equivalent, over the class of finite structures, to the inductive closure (i.e. minimal fixpoint) of a positive operator. A variant of the same proof establishes a theorem of Immerman, that the class of inductive closures of positive first order operators is closed under complementation. Research partially supported by ONR grant N00014-84-K-0415 and by DARPA grant F33615-87-C-1499, ARPA Order 4976, Amendment 20. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of ONR, DARPA, or the U.S. Government. Inductive definitions over finite structures Daniel Leivant Carnegie-Mellon University