The relational data file and the decision problem for classes of proper formulas

• Authors: Robert A. DiPaola
• Venue: SIGIR '71 Proceedings of the 1971 international ACM SIGIR conference on Information storage and retrieval
• Year: 1971

The Relational Data File (RDF) of The Rand Corporation is among the most developed of question-answering systems. The "information language" of this system is an applied predicate calculus. The atomic units of information are binary relational sentences. The system has an inference-making capacity.As part of the actual construction and implementation of the RDF, a theory was developed by J. L. Kuhns to identify those formulas of the predicate calculus which represent the "reasonable" inquiries to put to this system. Accordingly, the classes of definite and proper formulus were defined, and their properties studied. The definite formulas share a semantic property Kuhns judged as necessarily possessed by a reasonable question to be processed by the RDF. The author has previously shown that the decision problem for the class of definite formulas is recursively unsolvable. The proper formulas are definite, and satisfy additional syntactic conditions intended to make them especially suitable for machine processing. The class of proper formulas depends on which logical primitives are employed. Different primitives give rise to different classes of formulas. A formula which can be effectively transformed into a proper equivalent is admissible. Kuhns conjectures that with respect to one particular class of proper formulas, all definite formulas are admissible. In the paper it is shown that the decision problem for several classes of proper formulas is solvable. The following results are established. Theorem 1: The class of proper formulas in prenex form on any complete set of connectives is recursive. Theorem 2: The class of proper formulas on ¬, ∨, ∃ is recursive. Theorem 3: The class of proper formulas on ¬, ⊃, ∃ is recursive. Theorem 4: The class of proper formulas on ¬, ⊃, ∨, ∃, is recursive. Thus, there is a mechanical decision procedure which determines whether an arbitrary formula is a member of the class. It follows that the analogues of Kuhns' conjecture for these classes are false.