The Solvability of the Decision Problem for Classes of Proper Formulas and Related Results

Abstract

In connection with the development of an actual question-answering system, the Relational Data File of The Rand Corporation, a class of formulas of the predicate calculus, the definite formulas of Kuhns, was proposed as the class of symbolic representations of the “reasonable” questions to put to this retrieval system. Roughly speaking, the definite formulas are those formulas F such that the set of true substitution instances of F in a finite interpretation I are preserved on passage to a certain special extension I ′ of I . The proper formulas are those definite formulas which are supposedly especially suitable for machine processing. It has previously been shown by the author that the decision problem for the class of definite formulas is recursively unsolvable. In this paper, however, it is shown that the decision problem for various classes of proper formulas is solvable. Thus, for each of these classes there is a mechanical procedure to decide of an arbitrary formula whether it is a member of the class. It follows that for each of these classes there is no effective transformation @@@@ which takes an arbitrary definite formula F into a proper equivalent @@@@( F ). In addition, it is shown that the decision problems for a number of classes of formulas which bear a structural similarity to any class of proper formulas are recursively unsolvable.