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In this work we show that second-order metarules for MIL can be learned by MIL. We define a generality ordering of metarules by\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\theta$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03b8<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -subsumption and show that user-defined\n                    <jats:italic>sort metarules<\/jats:italic>\n                    are derivable by specialisation of the most-general\n                    <jats:italic>matrix metarules<\/jats:italic>\n                    in a language class; and that these matrix metarules are in turn derivable by specialisation of third-order\n                    <jats:italic>punch metarules<\/jats:italic>\n                    with variables quantified over the set of atoms and for which only an upper bound on their number of literals need be user-defined. We show that the cardinality of a metarule language is polynomial in the number of literals in punch metarules. We re-frame MIL as metarule specialisation by resolution. We modify the MIL metarule specialisation operator to return new metarules rather than first-order clauses and prove the correctness of the new operator. We implement the new operator as TOIL, a sub-system of the MIL system Louise. Our experiments show that as user-defined sort metarules are progressively replaced by sort metarules learned by TOIL, Louise\u2019s predictive accuracy and training times are maintained. We conclude that automatically derived metarules can replace user-defined metarules.\n                  <\/jats:p>","DOI":"10.1007\/s10994-022-06156-1","type":"journal-article","created":{"date-parts":[[2022,4,15]],"date-time":"2022-04-15T11:03:11Z","timestamp":1650020591000},"page":"3703-3731","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Meta-interpretive learning as metarule specialisation"],"prefix":"10.1007","volume":"111","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2266-4663","authenticated-orcid":false,"given":"S.","family":"Patsantzis","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S. 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Author 2 provided feedback and corrections on all sections of the paper. The authors have no conflicts of interest to disclose. Ethics approval, consent to participate and consent for publication were not required. Code and data have been made available in Sect.\u00a0\n                      \n                      .","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of interest"}}]}}