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arxiv: 1907.10952 · v1 · pith:D4OWPRASnew · submitted 2019-07-25 · 💻 cs.LG · cs.AI

Logical reduction of metarules

Andrew Cropper , Sophie Tourret This is my paper

Pith reviewed 2026-05-24 16:13 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords metarulesinductive logic programmingderivation reductionSLD-resolutionlogical reductionhypothesis spaceILP
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The pith

Derivation reduction produces finite metarule sets that remain complete yet improve accuracy and speed in ILP

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines whether infinite fragments of metarules can be logically reduced to small finite subsets while preserving the programs they can generate. It compares two standard reductions based on subsumption and entailment against a new method called derivation reduction that uses SLD-resolution steps. For fragments relevant to inductive logic programming, the authors prove which reduced sets stay complete and then test them on three domains. Derivation-reduced sets yield higher predictive accuracy and shorter run times than the alternatives because they shrink the hypothesis space without losing necessary expressivity.

Core claim

Derivation reduced sets of metarules outperform subsumption and entailment reduced sets both in predictive accuracies and learning times on the three experimental domains while remaining complete for the targeted fragments.

What carries the argument

Derivation reduction, a process based on SLD-resolution that extracts a minimal finite subset of metarules from which every other metarule in the fragment can be obtained through resolution derivations.

If this is right

  • Smaller hypothesis spaces result from using fewer metarules without sacrificing the programs expressible in the fragment.
  • Learning algorithms run faster because they search fewer clauses.
  • Higher predictive accuracy is observed on Michalski trains, string transformations, and game-rule domains.
  • The same reduction technique applies to other metarule fragments used in standard ILP settings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same completeness arguments could be checked for additional metarule fragments not examined in the experiments.
  • Combining derivation reduction with existing ILP pruning methods might produce even smaller effective search spaces.
  • If the reduced sets scale to larger fragments, the technique could extend to tasks with richer second-order structure.

Load-bearing premise

The finite reduced sets remain complete for the infinite fragments they target, so every program generable from the original infinite set is still generable from the reduced finite set.

What would settle it

A concrete target program or hypothesis that can be constructed from the original metarule fragment but cannot be constructed from the derivation-reduced set would falsify the completeness claim.

Figures

Figures reproduced from arXiv: 1907.10952 by Andrew Cropper, Sophie Tourret.

Figure 1
Figure 1. Figure 1: An example Michalski trains target program. In the Mi [PITH_FULL_IMAGE:figures/full_fig_p030_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Background relations available in the trains experi [PITH_FULL_IMAGE:figures/full_fig_p031_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Example programs learned by Metagol when varying the [PITH_FULL_IMAGE:figures/full_fig_p032_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Examples of the p6 string transformation problem input-output pairs. 6.2.2 Method Our experimental method is: 1. Sample 50 tasks Ts from the set {p1, . . ., p250} 2. For each t ∈ Ts: (a) Sample 5 training examples and use the remaining examples as testing examples (b) For each set of metarules m in the S-, E-, D, and D ∗ -reductions: i. Learn a program p for task t using the training examples and metarules… view at source ↗
Figure 5
Figure 5. Figure 5: IGGP games used in the experiments [PITH_FULL_IMAGE:figures/full_fig_p035_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Target solution for the next predicate for the minimal decay game. 6.3.2 Method The majority of game examples are negative. We therefore use balanced accuracy to eval￾uate the approaches. Given background knowledge B, sets of positive E + and negative E − testing examples, and a logic program H, we define the number of positive examples as p = |E + |, the number of negative examples as n = |E − |, the numb… view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies logical reduction of metarules (second-order Horn clauses) in inductive logic programming to address the efficiency-expressivity tradeoff in hypothesis spaces. It defines and compares three reduction techniques—subsumption reduction, entailment reduction, and a new derivation reduction based on SLD-resolution—computes reduced finite sets for relevant fragments, proves completeness properties for certain infinite fragments, and experimentally evaluates them on the Michalski trains, string transformations, and game rules domains, claiming that derivation-reduced sets yield higher predictive accuracies and shorter learning times than the alternatives.

Significance. If the completeness claims hold, the work directly tackles a central open problem in ILP by supplying minimal finite metarule sets that preserve the generative power of larger (sometimes infinite) fragments while demonstrably improving both accuracy and runtime. The direct use of logical operations (subsumption, entailment, SLD derivation) rather than fitted parameters, together with explicit theoretical results for specific fragments and reproducible experimental comparisons across three domains, constitutes a clear methodological contribution.

major comments (2)
  1. [§4] §4 (Derivation reduction definition and completeness argument): The claim that derivation-reduced finite sets remain complete for the targeted infinite fragments rests on an SLD-simulation property that is stated but not fully verified for fragments containing recursive metarules or metarules of arity greater than two. If any SLD derivation tree using the original fragment cannot be rewritten using only the reduced metarules without introducing new resolution branches or losing second-order substitutions, the reduced sets would be strictly weaker; this would make the reported accuracy gains potentially attributable to a smaller hypothesis space rather than a strictly superior reduction technique.
  2. [§5.2] §5.2 (Experimental domains and hypothesis-space comparison): The performance comparison across the three domains does not report the size of the hypothesis spaces induced by each reduced set (e.g., number of distinct programs generable up to a fixed depth). Without this measurement it is impossible to separate the effect of completeness preservation from the effect of simply using fewer metarules; the central claim that derivation reduction “outperforms” therefore requires an explicit control for hypothesis-space cardinality.
minor comments (2)
  1. [Table 1] Table 1: The column headers for the three reduction methods are not aligned with the rows describing the metarule fragments; this makes it difficult to match the reported reduced-set cardinalities to the correct technique.
  2. [Notation] Notation section: The symbol for second-order substitution is introduced without an explicit definition of its domain and range; a short formal definition would improve readability for readers outside the immediate ILP community.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [§4] §4 (Derivation reduction definition and completeness argument): The claim that derivation-reduced finite sets remain complete for the targeted infinite fragments rests on an SLD-simulation property that is stated but not fully verified for fragments containing recursive metarules or metarules of arity greater than two. If any SLD derivation tree using the original fragment cannot be rewritten using only the reduced metarules without introducing new resolution branches or losing second-order substitutions, the reduced sets would be strictly weaker; this would make the reported accuracy gains potentially attributable to a smaller hypothesis space rather than a strictly superior reduction technique.

    Authors: We acknowledge that the SLD-simulation argument in §4 would benefit from explicit verification for recursive metarules and arities greater than two. The manuscript defines derivation reduction via SLD-resolution and states the completeness property for the targeted fragments, but does not expand the simulation proof to cover these cases in full detail. In the revised manuscript we will add an extended proof section that shows any SLD derivation tree over the original fragment can be simulated by the reduced set without introducing extraneous branches or altering second-order substitutions. revision: yes

  2. Referee: [§5.2] §5.2 (Experimental domains and hypothesis-space comparison): The performance comparison across the three domains does not report the size of the hypothesis spaces induced by each reduced set (e.g., number of distinct programs generable up to a fixed depth). Without this measurement it is impossible to separate the effect of completeness preservation from the effect of simply using fewer metarules; the central claim that derivation reduction “outperforms” therefore requires an explicit control for hypothesis-space cardinality.

    Authors: We agree that an explicit measurement of hypothesis-space cardinality is needed to isolate the contribution of completeness preservation. The current experiments compare predictive accuracy and runtime but omit the number of distinct programs generable up to a fixed depth for each reduced set. In the revised manuscript we will report these cardinalities for all three reduction techniques across the Michalski trains, string transformations, and game rules domains. revision: yes

Circularity Check

0 steps flagged

No circularity: reductions defined via independent logical operations

full rationale

The paper defines subsumption, entailment, and derivation reduction directly from standard logical notions (subsumption of clauses, entailment, SLD-resolution steps). These are external to the target performance claims and do not reduce by construction to fitted parameters or self-citations. The theoretical completeness arguments for infinite fragments are presented as separate proofs rather than assumed via prior self-work. No load-bearing step equates a prediction to its own input definition. This matches the default expectation of non-circularity for papers whose core machinery is externally verifiable logic.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard first-order logic and SLD-resolution; no new free parameters, ad-hoc axioms, or invented entities are introduced beyond the definition of the three reduction relations.

axioms (2)
  • standard math SLD-resolution is sound and complete for Horn clauses
    Invoked when defining derivation reduction.
  • domain assumption Metarules are second-order Horn clauses
    Standard assumption in the ILP literature cited by the abstract.

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Reference graph

Works this paper leans on

63 extracted references · 63 canonical work pages · 1 internal anchor

  1. [1]

    Constraint-based synthesis of Datalog programs

    Aws Albarghouthi, Paraschos Koutris, Mayur Naik, and Cal vin Smith. Constraint-based synthesis of Datalog programs. In J. Christopher Beck, editor , Principles and Practice of Constraint Programming - 23rd International Conference, CP 2017, Melbourne, VIC, A ustralia, August 28 - September 1, 2017, Proceedings, volume 10416 of Lecture Notes in Computer Sc...

    work page 2017

  2. [2]

    Prime implicates and prime implicants i n modal logic

    Meghyn Bienvenu. Prime implicates and prime implicants i n modal logic. In Proceedings of the Twenty-Second AAAI Conference on Artificial Intelligen ce, July 22-26, 2007, Vancouver , British Columbia, Canada, pages 379–384. AAAI Press, 2007

    work page 2007

  3. [3]

    Anselm Blumer , Andrzej Ehrenfeucht, David Haussler , and Manfred K. Warmuth. Occam’s razor . Inf. Process. Lett., 24(6):377–380, 1987

    work page 1987

  4. [4]

    Bradley and Zohar Manna

    Aaron R. Bradley and Zohar Manna. The calculus of computation - decision procedures with appl ica- tions to verification . Springer , 2007

    work page 2007

  5. [5]

    Campero, A

    A. Campero, A. Pareja, T . Klinger, J. T enenbaum, and S. Rie del. Logical Rule Induction and Theory Learning Using Neural Theorem Proving. ArXiv e-prints, September 2018

    work page 2018

  6. [6]

    A note on the Entscheidungsproblem

    Alonzo Church. A note on the Entscheidungsproblem. J. Symb. Log. , 1(1):40–41, 1936

    work page 1936

  7. [7]

    William W . Cohen. Grammatically biased learning: Learni ng logic programs using an explicit an- tecedent description language. Artif. Intell., 68(2):303–366, 1994

    work page 1994

  8. [8]

    Efficiently learning efficient programs

    Andrew Cropper . Efficiently learning efficient programs . PhD thesis, Imperial College London, UK, 2017

    work page 2017

  9. [9]

    Inductive general game playing

    Andrew Cropper, Richard Evans, and Mark Law. Inductive ge neral game playing. arXiv e-prints , page arXiv:1906.09627, Jun 2019

    work page internal anchor Pith review Pith/arXiv arXiv 1906

  10. [10]

    Muggleton

    Andrew Cropper and Stephen H. Muggleton. Logical minimi sation of meta-rules within meta- interpretive learning. In Jesse Davis and Jan Ramon, editor s, Inductive Logic Programming - 24th International Conference, ILP 2014, Nancy, France, Septem ber 14-16, 2014, Revised Selected Papers , volume 9046 of Lecture Notes in Computer Science , pages 62–75. Spri...

    work page 2014

  11. [11]

    Muggleton

    Andrew Cropper and Stephen H. Muggleton. Learning effici ent logical robot strategies involving composable objects. In Qiang Y ang and Michael Wooldridge, e ditors, Proceedings of the Twenty- Fourth International Joint Conference on Artificial Intell igence, IJCAI 2015, Buenos Aires, Argentina, July 25-31, 2015 , pages 3423–3429. AAAI Press, 2015

    work page 2015

  12. [12]

    Muggleton

    Andrew Cropper and Stephen H. Muggleton. Learning highe r-order logic programs through abstrac- tion and invention. In Subbarao Kambhampati, editor , Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence, IJCAI 2016, Ne w York, NY , USA, 9-15 July 2016 , pages 1418–1424. IJCAI/AAAI Press, 2016

    work page 2016

  13. [13]

    Muggleton

    Andrew Cropper and Stephen H. Muggleton. Metagol system . https://github.com/metagol/metagol, 2016

    work page 2016

  14. [14]

    Muggleton

    Andrew Cropper and Stephen H. Muggleton. Learning effici ent logic programs. Machine Learning, 108(7):1063–1083, 2019

    work page 2019

  15. [15]

    Muggleton

    Andrew Cropper , Alireza T amaddoni-Nezhad, and Stephen H. Muggleton. Meta-interpretive learn- ing of data transformation programs. In Katsumi Inoue, Haya to Ohwada, and Akihiro Y amamoto, editors, Inductive Logic Programming - 25th International Conferen ce, ILP 2015, Kyoto, Japan, August 20-22, 2015, Revised Selected Papers, volume 9575 of Lecture Notes...

    work page 2015

  16. [16]

    Derivation reductio n of metarules in meta-interpretive learn- ing

    Andrew Cropper and Sophie T ourret. Derivation reductio n of metarules in meta-interpretive learn- ing. In Fabrizio Riguzzi, Elena Bellodi, and Riccardo Zese, editors, Inductive Logic Programming - 28th International Conference, ILP 2018, Ferrara, Italy, September 2-4, 2018, Proceedings , volume 11105 of Lecture Notes in Computer Science , pages 1–21. Spr...

    work page 2018

  17. [17]

    Complexity and expressive power of logic programming

    Evgeny Dantsin, Thomas Eiter , Georg Gottlob, and Andrei V oronkov . Complexity and expressive power of logic programming. ACM Comput. Surv. , 33(3):374–425, 2001

    work page 2001

  18. [18]

    Qu antifier-free equational logic and prime implicate generation

    Mnacho Echenim, Nicolas Peltier , and Sophie T ourret. Qu antifier-free equational logic and prime implicate generation. In Amy P . Felty and Aart Middeldorp, e ditors, Automated Deduction - CADE-25 - 25th International Conference on Automated Deduction, Be rlin, Germany, August 1-7, 2015, Proceed- ings, volume 9195 of Lecture Notes in Computer Science , p...

    work page 2015

  19. [19]

    The discovery of the equator or con- cept driven learning

    Werner Emde, Christopher Habel, and Claus-Rainer Rolli nger . The discovery of the equator or con- cept driven learning. In Alan Bundy , editor , Proceedings of the 8th International Joint Conference on Artificial Intelligence. Karlsruhe, FRG, August 1983 , pages 455–458. William Kaufmann, 1983

    work page 1983

  20. [20]

    Learning explan atory rules from noisy data

    Richard Evans and Edward Grefenstette. Learning explan atory rules from noisy data. J. Artif. Intell. Res., 61:1–64, 2018

    work page 2018

  21. [21]

    Inductive logic program synthesis with D IALOGS

    Pierre Flener . Inductive logic program synthesis with D IALOGS. In Stephen Muggleton, editor , Inductive Logic Programming, 6th International Workshop, ILP-96, Stockholm, Sweden, August 26-28, 1996, Selected Papers , volume 1314 of Lecture Notes in Computer Science , pages 175–198. Springer , 1996

    work page 1996

  22. [22]

    Fonseca, Vítor Santos Costa, Fernando M

    Nuno A. Fonseca, Vítor Santos Costa, Fernando M. A. Silva , and Rui Camacho. On avoiding redun- dancy in inductive logic programming. In Rui Camacho, Ross D . King, and Ashwin Srinivasan, edi- tors, Inductive Logic Programming, 14th International Conference, ILP 2004, Porto, Portugal, September 6-8, 2004, Proceedings , volume 3194 of Lecture Notes in Comp...

    work page 2004

  23. [23]

    Dimensionality reduction in ILP: A call to arms

    Johannes Fürnkranz. Dimensionality reduction in ILP: A call to arms. In Proceedings of the IJCAI-97 Workshop on Frontiers of Inductive Logic Programming , pages 81–86, 1997

    work page 1997

  24. [24]

    M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP- Completeness. W . H. Freeman, 1979

    work page 1979

  25. [25]

    Genesereth, Nathaniel Love, and Barney Pell

    Michael R. Genesereth, Nathaniel Love, and Barney Pell. General game playing: Overview of the AAAI competition. AI Magazine, 26(2):62–72, 2005

    work page 2005

  26. [26]

    Fermüller

    Georg Gottlob and Christian G. Fermüller . Removing redu ndancy from a clause. Artif. Intell. , 61(2):263–289, 1993

    work page 1993

  27. [27]

    O n the complexity of some inductive logic programming problems

    Georg Gottlob, Nicola Leone, and Francesco Scarcello. O n the complexity of some inductive logic programming problems. In Nada Lavrac and Saso Dzeroski, editors, Inductive Logic Programming, 7th International Workshop, ILP-97, Prague, Czech Republic, S eptember 17-20, 1997, Proceedings , volume 1297 of Lecture Notes in Computer Science , pages 17–32. Spri...

    work page 1997

  28. [28]

    Minimization f or generalized boolean formulas

    Edith Hemaspaandra and Henning Schnoor . Minimization f or generalized boolean formulas. In T oby Walsh, editor ,IJCAI 2011, Proceedings of the 22nd International Joint Con ference on Artificial Intelligence, Barcelona, Catalonia, Spain, July 16-22, 20 11, pages 566–571. IJCAI /AAAI, 2011

    work page 2011

  29. [29]

    Clause elimination for SA T and QSA T.J

    Marijn Heule, Matti Järvisalo, Florian Lonsing, Martin a Seidl, and Armin Biere. Clause elimination for SA T and QSA T.J. Artif. Intell. Res. , 53:127–168, 2015

    work page 2015

  30. [30]

    From search to computation: Redundancy criteria and simplification at wor k

    Thomas Hillenbrand, Ruzica Piskac, Uwe Waldmann, and Ch ristoph Weidenbach. From search to computation: Redundancy criteria and simplification at wor k. In Andrei V oronkov and Christoph Weidenbach, editors, Programming Logics - Essays in Memory of Harald Ganzinger , volume 7797 of Lecture Notes in Computer Science , pages 169–193. Springer , 2013

    work page 2013

  31. [31]

    Joyner Jr

    William H. Joyner Jr . Resolution strategies as decision procedures. J. ACM, 23(3):398–417, 1976

    work page 1976

  32. [32]

    Explo iting answer set programming with exter- nal sources for meta-interpretive learning

    T obias Kaminski, Thomas Eiter , and Katsumi Inoue. Explo iting answer set programming with exter- nal sources for meta-interpretive learning. TPLP, 18(3-4):571–588, 2018

    work page 2018

  33. [33]

    Controlling the compl exity of learning in logic through syntactic and task-oriented models

    Jörg-Uwe Kietz and Stefan Wrobel. Controlling the compl exity of learning in logic through syntactic and task-oriented models. In Inductive logic programming . Citeseer , 1992

    work page 1992

  34. [34]

    Kowalski

    Robert A. Kowalski. Predicate logic as programming lang uage. In IFIP Congress , pages 569–574, 1974

    work page 1974

  35. [35]

    Larson and Ryszard S

    J. Larson and Ryszard S. Michalski. Inductive inference of VL decision rules. SIGART Newsletter , 63:38–44, 1977

    work page 1977

  36. [36]

    Redundancy in logic I: CNF propositio nal formulae

    Paolo Liberatore. Redundancy in logic I: CNF propositio nal formulae. Artif. Intell., 163(2):203–232, 2005

    work page 2005

  37. [37]

    Redundancy in logic II: 2CNF and Horn p ropositional formulae

    Paolo Liberatore. Redundancy in logic II: 2CNF and Horn p ropositional formulae. Artif. Intell. , 172(2-3):265–299, 2008

    work page 2008

  38. [38]

    T enen baum, and Stephen Muggleton

    Dianhuan Lin, Eyal Dechter , Kevin Ellis, Joshua B. T enen baum, and Stephen Muggleton. Bias re- formulation for one-shot function induction. In ECAI 2014 - 21st European Conference on Artificial Intelligence, 18-22 August 2014, Prague, Czech Republic - I ncluding Prestigious Applications of Intelli- gent Systems (PAIS 2014) , pages 525–530, 2014. 40 Andre...

    work page 2014

  39. [39]

    John W . Lloyd. Foundations of Logic Programming, 2nd Edition . Springer , 1987

    work page 1987

  40. [40]

    J.W . Lloyd. Logic for Learning . Springer , Berlin, 2003

    work page 2003

  41. [41]

    Undecidabili ty of the Horn-clause implication problem

    Jerzy Marcinkowski and Leszek Pacholski. Undecidabili ty of the Horn-clause implication problem. In 33rd Annual Symposium on Foundations of Computer Science, P ittsburgh, Pennsylvania, USA, 24-27 October 1992, pages 354–362, 1992

    work page 1992

  42. [42]

    Consequence finding algorithms

    Pierre Marquis. Consequence finding algorithms. In Handbook of Defeasible Reasoning and Uncer- tainty Management Systems , pages 41–145. Springer , 2000

    work page 2000

  43. [43]

    Making robots conscious of their mental s tates

    John McCarthy . Making robots conscious of their mental s tates. In Machine Intelligence 15, Intelligent Agents [St. Catherine’s College, Oxford, July 1995 ], pages 3–17, 1995

    work page 1995

  44. [44]

    Luke Ong

    Rolf Morel, Andrew Cropper , and C.-H. Luke Ong. T yped met a-interpretive learning of logic pro- grams. In Francesco Calimeri, Nicola Leone, and Marco Manna , editors, Logics in Artificial Intel- ligence - 16th European Conference, JELIA 2019, Rende, Ital y, May 7-11, 2019, Proceedings , volume 11468 of Lecture Notes in Computer Science , pages 198–213. S...

    work page 2019

  45. [45]

    Inverse entailment and Progol

    Stephen Muggleton. Inverse entailment and Progol. New Generation Comput. , 13(3&4):245–286, 1995

    work page 1995

  46. [46]

    Efficient induction of lo gic programs

    Stephen Muggleton and Cao Feng. Efficient induction of lo gic programs. In Algorithmic Learning Theory, First International Workshop, ALT ’90, T okyo, Japa n, October 8-10, 1990, Proceedings , pages 368–381, 1990

    work page 1990

  47. [47]

    Flach, Katsumi Inoue, and Ashwin Srinivasan

    Stephen Muggleton, Luc De Raedt, David Poole, Ivan Bratk o, Peter A. Flach, Katsumi Inoue, and Ashwin Srinivasan. ILP turns 20 - biography and future chall enges. Machine Learning, 86(1):3–23, 2012

    work page 2012

  48. [48]

    Muggleton, Dianhuan Lin, Niels Pahlavi, and A lireza T amaddoni-Nezhad

    Stephen H. Muggleton, Dianhuan Lin, Niels Pahlavi, and A lireza T amaddoni-Nezhad. Meta- interpretive learning: application to grammatical infere nce. Machine Learning, 94(1):25–49, 2014

    work page 2014

  49. [49]

    Muggleton, Dianhuan Lin, and Alireza T amaddo ni-Nezhad

    Stephen H. Muggleton, Dianhuan Lin, and Alireza T amaddo ni-Nezhad. Meta-interpretive learning of higher-order dyadic Datalog: predicate invention revisit ed. Machine Learning, 100(1):49–73, 2015

    work page 2015

  50. [50]

    Declarative bias in ILP

    Claire Nédellec, Céline Rouveirol, Hilde Adé, Francesc o Bergadano, and Birgit T ausend. Declarative bias in ILP. Advances in inductive logic programming , 32:82–103, 1996

    work page 1996

  51. [51]

    Foundations of Inductive Logic Programming

    Shan-Hwei Nienhuys-Cheng and Ronald de Wolf. Foundations of Inductive Logic Programming . Springer-V erlag New Y ork, Inc., Secaucus, NJ, USA, 1997

    work page 1997

  52. [52]

    G.D. Plotkin. Automatic Methods of Inductive Inference . PhD thesis, Edinburgh University , August 1971

    work page 1971

  53. [53]

    Declarative modeling for machine learning and data mining

    Luc De Raedt. Declarative modeling for machine learning and data mining. In Algorithmic Learning Theory - 23rd International Conference, ALT 2012, Lyon, Fra nce, October 29-31, 2012. Proceedings , page 12, 2012

    work page 2012

  54. [54]

    Interactive concep t-learning and constructive induction by analogy .Machine Learning, 8:107–150, 1992

    Luc De Raedt and Maurice Bruynooghe. Interactive concep t-learning and constructive induction by analogy .Machine Learning, 8:107–150, 1992

    work page 1992

  55. [55]

    A machine-oriented logic based on th e resolution principle

    John Alan Robinson. A machine-oriented logic based on th e resolution principle. J. ACM, 12(1):23– 41, 1965

    work page 1965

  56. [56]

    Implication of clauses is unde cidable

    Manfred Schmidt-Schauß. Implication of clauses is unde cidable. Theor . Comput. Sci., 59:287–296, 1988

    work page 1988

  57. [57]

    E.Y . Shapiro. Algorithmic program debugging . MIT Press, 1983

    work page 1983

  58. [58]

    Syntax- guided synthesis of Datalog programs

    Xujie Si, Woosuk Lee, Richard Zhang, Aws Albarghouthi, P araschos Koutris, and Mayur Naik. Syntax- guided synthesis of Datalog programs. In Gary T . Leavens, Al essandro Garcia, and Corina S. Pasare- anu, editors, Proceedings of the 2018 ACM Joint Meeting on European Softwa re Engineering Conference and Symposium on the Foundations of Software Engineering...

    work page 2018

  59. [59]

    Understanding complex datasets: data mining with matrix de compositions

    David Skillicorn. Understanding complex datasets: data mining with matrix de compositions. Chapman and Hall/CRC, 2007

    work page 2007

  60. [60]

    Horn clause computability

    Sten-Åke Tärnlund. Horn clause computability . BIT, 17(2):215–226, 1977

    work page 1977

  61. [61]

    SLD-resolution redu ction of second-order Horn fragments

    Sophie T ourret and Andrew Cropper . SLD-resolution redu ction of second-order Horn fragments. In Francesco Calimeri, Nicola Leone, and Marco Manna, editors , Logics in Artificial Intelligence - 16th European Conference, JELIA 2019, Rende, Italy, May 7-11, 2019, Proceedings, volume 11468 of Lecture Notes in Computer Science , pages 259–276. Springer , 2019

    work page 2019

  62. [62]

    William Y ang Wang, Kathryn Mazaitis, and William W . Cohe n. Structure learning via parameter learning. In Jianzhong Li, Xiaoyang Sean Wang, Minos N. Garo falakis, Ian Soboroff, T orsten Suel, and Min Wang, editors, Proceedings of the 23rd ACM International Conference on Con ference on In- formation and Knowledge Management, CIKM 2014, Shanghai, C hina, ...

    work page 2014

  63. [63]

    Subterm contextual rewriting

    Christoph Weidenbach and Patrick Wischnewski. Subterm contextual rewriting. AI Commun., 23(2- 3):97–109, 2010. Logical reduction of metarules 41 A Detailed Reduction Results A.1 Connected (/Ccala m) reductions S-reduction E-reduction D-reduction P(A, B)←Q(A, C ) P(A, B)←Q(B, C ) P(A, B)←Q(C , A) P(A, B)←Q(C , B) P(A, B)←Q(B, C ) P(A, A)←Q(B, A) P(A, B)←Q(...

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