Inductive inference of monotonic formal systems from positive data

Abstract

A formal system is a finite set of expressions, such as a grammar or a Prolog program. A semantic mapping from formal systems to concepts is said to be monotonic if it maps larger formal systems to larger concepts. A formal system Γ is said to be reduced with respect to a finite setX if the concept defined by Γ containsX but the concepts defined by any proper subset Γ′ of Γ cannot contain some part ofX. Assume a semantic mapping is monotonic and formal systems consisting of at mostn expressions that are reduced with respect toX can define only finitely many concepts for any finite setX and anyn. Then, the class of concepts defined by formal systems consisting of at mostn expressions is shown to be inferable from positive data. As corollaries, the class of languages defined by length-bounded elementary formal systems consisting of at most,n axioms, the class of languages generated by context-sensitive grammars consisting of at mostn productions, and the class of minimal models of linear Prolog programs consisting of at mostn definite clauses are all shown to be inferable from positive data.