Curried least general generalization: A framework for higher order concept learning

Continued progress with research in inductive logic programming relies on further extensions of their underlying logics. The standard tactics for extending expressivity include a generalization to higher order logics, which immediately forces attention to the computational complexity of higher order reasoning. A major thread of inductive logic programming research has focussed on the identification of preferred hypothesis sets, initiated by Plotkin's work on least general generalizations (LGGs). Within higher order frameworks, a relevant extension of LGG is Furukawa's hyper least general generalization (HLGG) [FIG97]. We present a relevant higher order extension of Furukawa's HLGG based on currying, which we call Curried Least General Generalization (CLGG). The idea is that the formal difficulties with the reasoning complexity of a higher order language can be controlled by forming new hypothetical terms restricted to those obtainable by Currying. This technique subsumes the inductive generalization power of HLGG, provides a basis for a significant extension of first order ILP, and is theoretically justified within a well understood formal foundation.