  Many forms of inductive logic programming (ILP) use \emph{metarules},
second-order Horn clauses, to define the structure of learnable programs and
thus the hypothesis space. Deciding which metarules to use for a given learning
task is a major open problem and is a trade-off between efficiency and
expressivity: the hypothesis space grows given more metarules, so we wish to
use fewer metarules, but if we use too few metarules then we lose expressivity.
In this paper, we study whether fragments of metarules can be logically reduced
to minimal finite subsets. We consider two traditional forms of logical
reduction: subsumption and entailment. We also consider a new reduction
technique called \emph{derivation reduction}, which is based on SLD-resolution.
We compute reduced sets of metarules for fragments relevant to ILP and
theoretically show whether these reduced sets are reductions for more general
infinite fragments. We experimentally compare learning with reduced sets of
metarules on three domains: Michalski trains, string transformations, and game
rules. In general, derivation reduced sets of metarules outperforms subsumption
and entailment reduced sets, both in terms of predictive accuracies and
learning times.
