The DOI matches its Crossref record; 69 of the 70 references it cites were found in a registry or an index, and 1 are not checked yet; 8 of the 31 sources its claims cite were checked for support.
Meta-interpretive learning of higher-order dyadic datalog: predicate invention revisited
Open accessVerified 2026-10-19 · 1 reference not checked yet
Overview
AI snapshot
Written from the paper’s full text.
TL;DR
The paper generalises meta-interpretive learning (MIL) to higher-order dyadic datalog programs in the class H₂², showing that H₂² is Turing expressive with an infinite signature but decidable with a finite signature. It implements MetagolD and demonstrates efficient learning with predicate invention and recursion on robot strategies, East-West trains, and NELL data.
Key findings
- H₂² has universal Turing expressivity when the constant signature is infinite, but satisfiability is decidable when the predicate and constant signatures are finite; the MIL setting is decidable when meta-rules, background knowledge, and examples are Datalog and signatures are finite.
- MetagolD, using Knuth-Bendix ordering and logarithmic clause bounding, PAC-learns minimal cardinality H₂² definitions; Theorem 3 gives polynomial sample complexity, and experiments show rapid error reduction with roughly linear learning-time growth under the logarithmic bound.
- On robot wall-building, MetagolD reaches predictive accuracy above 90% at training size 2 with one positive and one negative example; a dramatic time increase occurs at training size 10 because an additional negative example forces a larger hypothesis.
- On East-West trains, Metagol learns a recursive theory that correctly classifies all 20 trains in about 45 s on a 2.5 GHz MacBook laptop, equivalent to the winning competition entry, which took about 1 day on a 40 MHz Sun Sparc 10; on NELL, a greedy variant runs in about 0.004 s, hundreds of times faster, and MetagolD learns higher-order concepts such as symmetry.
Methods
- MIL uses higher-order meta-rules and a modified Prolog meta-interpreter that proves examples and saves metalogical substitutions for existentially quantified higher-order variables; applying these substitutions yields a first-order definite program.
- MetagolD is implemented in Yap Prolog and constrains search with total orderings over the Herbrand Base (lexicographic/Knuth-Bendix and interval inclusion), iterative deepening for minimal cardinality hypotheses, logarithmic clause bounding, and episodes for multi-predicate incremental learning.
- Experiments cover robot wall-building (training sizes 2–80, test size 200, half positive/half negative, averaged over five resamples), East-West trains (20 training trains, 100 test trains, averaged over 10 random samples), and NELL athletehomestadium (tenfold cross-validation, 120 examples, 919 background facts, training sizes up to 108).
Limitations
- The paper states that the NELL evaluation can lead to excessive runtimes; the logarithmic bound was dropped and a 2 min time limit used, causing MetagolD’s learning curve to drop significantly at training size 80.
- The paper states that the effect of incompleteness of meta-rules on predictive accuracy and robustness was not explored, and that the user is expected to provide the total orderings ⪰P and ⪰C over initial predicates and constants.
- The paper states that the greedy search variant can lead to overly specific, non-minimal programs and has unresolved overfitting issues; it was not applied to the other experiments. Evident from the text, the robot model is a simplification of a real-world robotics application, and the NELL data is incomplete, with 17% of examples missing teamhomestadium information.
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Claims the paper makes
46 Own interpretation · 25 Own result · 23 No basis given · 9 No full text of the sources · 8 Own method · 5 Sources not found · 3 Partly supported · 3 Not checked yet · 2 Supported · 1 Mixed support · accuracy not yet measured
Every source the paper cites for this claim was checked and supports it.
In the paper
“Since the late 1990s predicate invention has been under-explored within inductive logic programming due to difficulties in formulating efficient search mechanisms.”
- Also stated in: § 1 Introduction · p. 2 · plain
“Although the topic of Predicate Invention was investigated in early Inductive Logic Programming (ILP) research (Muggleton and Buntine 1988; Stahl 1992) it is still seen as hard and under-explored (Muggleton et al. 2011).”
What the sources say
- Supportedcites Muggleton et al. 2011 source
In “Whatever happened to predicate invention?” (section 4.2; page not supplied), the paper says research over the preceding 15 years had not kept pace with activity around 1990. It attributes the difficulty to an explosive number of possible new predicates and the challenge of recognizing their benefits during search. (LLM, conf=0.92)
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Also stated in: § 1 Introduction · p. 2 · plain
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“A key difference is that inductive generalisation and predicate invention are not achieved in existing EBG paradigms, which assume a complete first-order domain theory.”
What the sources say
- Not matchedcites De Jong 1981
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not checked — no full text of the sourcecites Kedar-Cabelli and McCarty 1987 source
- Partly supportedcites Mitchell et al. 1986 sourceExperimental — accuracy not yet measured
Section 4.2.1 says EBG assumes a domain theory sufficient to prove the training example belongs to the goal concept, so its generalizations follow deductively from existing knowledge. The EBG as Reformulating/Operationalizing/Deducing section likewise says the method compiles deductive consequences of its domain theory. These passages support the distinction from inductive generalisation, but they do not establish that EBG requires a complete first-order domain theory or address predicate invention. (LLM, conf=0.85)
plain- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
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- Not matched
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This result is consistent with our experiments which indicate that Metagol D efficiently learns compact H 2 2 definitions involving predicate invention for learning robotic strategies, the East-West train challenge and NELL.”
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“We show that with an infinite signature the higher-order dyadic datalog class H 2 2 has universal Turing expressivity though H 2 2 is decidable given a finite signature.”
- Also stated in: § 3.4 Language classes and expressivity · p. 8 · plain
“Theorem 2 (Decidable fragment of H 2 2 ) The satisfiability of G , P is decidable when P , C is finite.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Also stated in: § 3.4 Language classes and expressivity · p. 8 · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Additionally higher-order concepts were learned in the NELL language learning domain.”
- Also stated in: § 2 Related work · p. 4 · boosted
“The NELL language application (Sect. 5.3) demonstrates that higher-order concepts can be readily and naturally expressed in H 2 2 and learned within the MIL framework.”
boostedSingle case- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Also stated in: § 2 Related work · p. 4 · boosted
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Additionally we show that Knuth-Bendix ordering of the hypothesis space together with logarithmic clause bounding allows our MIL implementation Metagol D to PAC-learn minimal cardinality H 2 2 definitions.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“We show that with an infinite signature the higher-order dyadic datalog class H 2 2 has universal Turing expressivity though H 2 2 is decidable given a finite signature.”
- Also stated in: § 3.4 Language classes and expressivity · p. 7 · boosted
“This fragment also has UTM expressivity, as demonstrated by the following H 2 2 encoding of a UTM in which S , S 1 , T represent Turing machine tapes.”
boosteddirect- Asserted: passes — The authors put it forward as true.
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- The authors state this as an assumption: The UTM has a suitably designed set of machine instructions representing functions of the form used in the encoding.
- Also stated in: § 3.4 Language classes and expressivity · p. 7 · boosted
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“At training size 2 with only one positive and one negative example, Metagol D already reaches a predictive accuracy in excess of 90%.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The minimal logic program found at depth 4 is shown in Fig. 8. In this case predicates are invented which correspond to both grandparent and parent .”
plaindirectSingle case- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Theorem 2 (Decidable fragment of H 2 2 ) The satisfiability of G , P is decidable when P , C is finite.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
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- Probably not a substantial claim. 46
- It may state two separate claims as one. 26
- The quote only partly states this claim. 1
Some cited sources support this claim and some don’t — read each source’s verdict below.
In the paper
“However, unlike the approach for learning recursive robot strategies described in Sect. 5.1 the Relational Reinforcement approaches are based on the use of ground instances and do not involve predicate invention to extend the relational vocabulary provided by the user.”
What the sources say
- Partly supportedcites Džeroski et al. 2001 sourceExperimental — accuracy not yet measured
The Introduction describes states as sets of ground facts, and §5.2 specifies relational states built from basic facts. Section 4.4 says the user can influence learning through supplied background predicates and declarative bias. The paper studies blocks-world planning, however, and the passages do not explicitly establish that the approach never invents predicates or that it learns robot strategies. (LLM, conf=0.84)
- Disputedcites Katz et al. 2008 sourceExperimental — accuracy not yet measured
The section “Representation of Q-Value Function” (page unavailable) explicitly says that states and actions are relational and stored uninstantiated, contrary to the claim that the approach uses ground instances. The paper describes a defined set of predicates, but does not establish the broader claim about user-provided vocabularies or predicate invention across relational reinforcement approaches. (LLM, conf=0.88)
Checked by a reasoning model — may be wrong.
- Partly supportedcites Driessens and Ramon 2003Experimental — accuracy not yet measured
Section 3 (page unavailable) says RRL builds examples from encountered states and chosen actions, represented in datalog or Herbrand interpretations; this supports the claim that it uses ground instances. The paper does not state that RRL avoids predicate invention or limits itself to a user-provided relational vocabulary, and it does not establish the claim specifically for robot strategies. (LLM, conf=0.82)
plain- Asserted: passes — The authors put it forward as true.
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
- Partly supported
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“ILP systems such as ALEPH (Srinivasan 2001) and FOIL (Quinlan 1990) have no predicate invention and limited recursion learning and therefore cannot learn recursive grammars from example sequences.”
What the sources say
- Not matchedcites Srinivasan 2001
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Partly supportedcites Quinlan 1990 sourceExperimental — accuracy not yet measured
The Discussion (page not available in our copy) says FOIL cannot invent new predicates, and §5.7 says its handling of recursion is incomplete. But the paper also says FOIL can find recursive definitions; it does not establish that FOIL cannot learn recursive grammars from example sequences. It does not address ALEPH. (LLM, conf=0.88)
plaindirect- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Not matched
Every source the paper cites for this claim was checked and supports it.
In the paper
“Failure to address these issues has led to limited progress being made in this important topic over a protracted period (Muggleton et al. 2011).”
What the sources say
- Supportedcites Muggleton et al. 2011 source
In the predicate-invention discussion (section 4.2; page unavailable), the paper says research over the preceding 15 years did not keep pace with the early period and describes the problem’s high combinatorial complexity. The Conclusion (section 5; page unavailable) adds that predicate invention has not been well addressed recently, supporting the claim of limited progress over a prolonged period. (LLM, conf=0.86)
plaindirect- Asserted: passes — The authors put it forward as true.
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Supported
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“Blumer-bound arguments are provided which indicate that search constrained in this way achieves not only speed improvements but also reduction in out-ofsample error.”
What the sources say
- Partly supportedcites Blumer et al., 1989 sourceExperimental — accuracy not yet measured
The Introduction and Section 2 give VC-dimension bounds on the out-of-sample error of consistent hypotheses, supporting the use of a Blumer bound in a generalization argument. Our copy does not show that the particular search constraint described by the citing paper improves speed or reduces out-of-sample error; those appear to be conclusions of the citing authors' arguments. (LLM, conf=0.84)
hedgeddirect- Asserted: passes — The authors put it forward as true.
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- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Partly supported
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Metagol 's accuracy and learning time plots shown in Fig. 12 indicate that, consistent with the analysis in Sect. 4.3, Metagol D , given increasing number of randomly chosen examples, produces rapid error reduction while learning time increases roughly linearly.”
hedgeddirect- Asserted: passes — The authors put it forward as true.
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- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Note that H 2 2 is sufficient for the kinship example in Sect. 1.”
plain- Asserted: passes — The authors put it forward as true.
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“There are seven clauses in this hypothesis.”
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In our initial experiment Metagol inductively inferred the clause above from NELL data and used it to abduce the following facts, not found in the database.”
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In the NELL experiment described in Sect. 5.3.2 we found that a greedy modification of the Metagol D search strategy leads to considerable speed increases.”
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“However, 1 and 3 are erroneous.”
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Therefore, the running time with the greedy search strategy is around 0.004s, which is hundreds of times faster than the original MetagolD without being greedy.”
plaindirectSingle case- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“However, this data is incomplete. Specifically, 17% of the examples have incomplete information on teamhomestadium , requiring abduction of ground facts.”
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“When provided with all 20 trains, Progol can quickly find a partial solution shown in Fig. 18.”
plainSingle case- Asserted: passes — The authors put it forward as true.
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- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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The quote only partly states this claim.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In an evaluation experiment it was found that learning on the NELL dataset can lead to excessive runtimes, which are improved using a greedy version of the search Metagol mechanism.”
plaindirect- Asserted: passes — The authors put it forward as true.
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A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Using this Metagol D abduced the following hypothesis.”
plaindirectSingle case- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“This theory correctly classifies all 20 trains and is equivalent to the winning entry for the first competition submitted by Bernhard Pfahringer (Turney 1995).”
What the sources say
- Not matchedcites Turney 1995
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
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- Not matched
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The background knowledge contained 919 facts about athleteplays f orteam and teamhomestadium .”
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This fragment is shown to be Turing expressive in the case of an infinite signature, but decidable otherwise.”
plain- Asserted: passes — The authors put it forward as true.
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What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“As shown in Fig. 3 each meta-rule has an associated Order constraint, which is designed to ensure termination of the proof (see Sect. 4.1).”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
In the paper
“By contrast, in (Muggleton et al. 2014) definite clause grammars were learned with predicate invention using Meta-Interpretive Learning (MIL).”
What the sources say
- Not checked — no full text of the sourcecites Muggleton et al. 2014 source
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A claim is checked only against the cited paper’s full text, never its abstract, and none is available for its sources.
In the paper
“MIL (Muggleton and Lin 2013; Muggleton et al. 2014; Lin et al. 2014) is a technique which supports efficient predicate invention and learning of recursive logic programs built as a set of metalogical substitutions by a modified Prolog meta-interpreter (see Fig. 2) which acts as the central part of the ILP learning engine.”
What the sources say
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
In the paper
“However, a recent paper demonstrated that both predicate invention and the learning of recursion can be efficiently implemented for regular and context-free grammars, by way of metalogical substitutions with respect to a modified Prolog meta-interpreter which acts as the learning engine.”
What the sources say
- Not checked — no full text of the sourcecites Muggleton et al. 2014 source
boosteddirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“New predicate symbols are introduced as constants representing existentially quantified higher-order variables.”
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“The Metagol code and datasets described in this paper have been made publicly available on a website to allow reproduction of results in this paper.”
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We show that a fragment of this class is Turing equivalent, allowing the learning of complex recursive programs such as robot strategies.”
plaindirect- Asserted: passes — The authors put it forward as true.
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What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“Termination is guaranteed because atoms higher in the Herbrand Base are always proved by ones lower in the ordering. Also the ordering is finite and so cannot be infinitely descending.”
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“Early approaches were based on the use of W operators within the inverting resolution framework (Muggleton and Buntine 1988; Rouveirol and Puget 1989). However, apart from the inherent problems in controlling the search, the completeness of these approaches was never demonstrated, partly because of the lack of a declarative bias to delimit the hypothesis space.”
What the sources say
- Not checked — no full text of the sourcecites Muggleton and Buntine 1988 source
- Not matchedcites Rouveirol and Puget 1989
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Owing to the existentially quantified variables in the meta-rules, the resulting first-order theories are strictly logical generalisation of the meta-rules.”
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
None of the sources cited for this claim have been checked yet.
In the paper
“Predicate Invention has been viewed as an important problem since the early days of ILP (e.g. Muggleton and Buntine 1988; Rouveirol and Puget 1989; Stahl 1992) since it is essential for automating the introduction of auxiliary predicates within top-down programming.”
plaindirect- Asserted: passes — The authors put it forward as true.
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- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“One important feature of MIL, which distinguishes it from other existing approaches, is that it introduces new predicate symbols which represent relations rather than new objects or propositions.”
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This restriction limits the number of possible meta-rules and forces more prolific predicate invention, leading to more opportunities for internal re-use of part of the hypothesised program.”
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“New predicate names are introduced as higher-order skolem constants, a finite number of which are introduced during every iterative deepening of the search.”
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
In the paper
“The approach of Pasula and Lang (Pasula et al. 2004) for learning STRIPS-like operators in a relational language is closer to the approach described in Sect. 5.1, but is restricted to learning non-recursive operators and does not involve predicate invention.”
What the sources say
- Not matchedcites Pasula et al. 2004
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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It may state two separate claims as one.
- Not matched
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This phenomenon can be explained by considering that time taken for searching the hypothesis space for the joint definition is a function of the product of the hypothesis spaces of the individual predicates.”
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper may give no reason why it holds.
None of the sources cited for this claim have been checked yet.
In the paper
“Various authors (Feng and Muggleton 1992; Lloyd 2003) have advocated higher-order logic learning frameworks. However, to date these approaches have difficulties in incorporation of background knowledge and compatibility with more developed logic programming frameworks.”
What the sources say
- Not matchedcites Feng and Muggleton 1992
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not checked — no full text of the sourcecites Lloyd 2003 source
plain- Asserted: passes — The authors put it forward as true.
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- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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It may state two separate claims as one.
- Not matched
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“Moreover, unlike many ILP systems, only hypotheses consistent with all examples are considered.”
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
In the paper
“λ -Prolog (Miller 1991), is a notable exception which achieves efficient unification through assumptions on the flexibility of terms.”
What the sources say
- Not checked — no full text of the sourcecites Miller 1991 source
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“For instance, in Cropper and Muggleton (2014) it is demonstrated that sets of irreducible, or minimal sets of meta-rules can be found automatically by applying Plotkin's clausal theory reduction algorithm to an enumeration of all meta-rules in a given finite hypothesis language, resulting in meta-rules which exhibit lower runtimes and higher predictive accuracies.”
What the sources say
- Not matchedcites Cropper and Muggleton 2014
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
boosteddirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
In the paper
“Within ILP, search efficiency depends on the partial order of θ -subsumption (NienhuysCheng and de Wolf 1997).”
What the sources say
- Not matchedcites NienhuysCheng and de Wolf 1997
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“Metagol D uses iterative deepening to ensure the first hypothesis returned contains the minimal number of clauses.”
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Moreover logical equivalence with the larger set ensures completeness of the reduced set.”
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“By comparison with other forms of declarative bias in ILP, such as modes (Muggleton 1995; Srinivasan 2001) or grammars (Cohen 1994), meta-rules are logical statements. This provides the potential for reasoning about them and manipulating them alongside normal first-order background knowledge.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
In the paper
“Although McCarthy (1999) long advocated the use of higher-order logic for representing common sense reasoning, most knowledge representation languages avoid higher-order quantification owing to problems with decidability (Huet 1975) and theorem-proving efficiency.”
What the sources say
- Not checked — no full text of the sourcecites Huet 1975 source
plaindirect- Asserted: passes — The authors put it forward as true.
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The use of the meta-rules clarifies the declarative bias being employed.”
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
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Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Since the user has no ascribed meaning for P ′ and C ′ , it would not be possible to provide examples containing full interpretations, and therefore the Learning from Interpretations setting is inappropriate.”
boosteddirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- The authors state this as an assumption: P' ⊆ P and C' ⊆ C represent uninterpreted predicates and constants whose interpretation is assigned by the Meta-interpreter.
- Rests on an assumption: The user has no ascribed meaning for P' and C'.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As a knowledge representation, higher-order datalog (see Sect. 3), first introduced in Pahlavi and Muggleton (2012), has advantages in being both expressive and decidable.”
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. It may state two separate claims as one.
In the paper
“Owing to the efficiency of Prolog backtracking MIL implementations have been demonstrated to search the hypothesis space 100-1,000 times faster than state-of-the-art ILP systems (Muggleton et al. 2014) in the task of learning recursive grammars.”
What the sources say
- Not checked — no full text of the sourcecites Muggleton et al. 2014 source
boosteddirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In the lexicographic ordering predicates which are higher in the ordering, such as grandparent/2 , are defined in terms of ones which are lower, such as parent/2 .”
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In the MIL framework described in Muggleton et al. (2014) and in this paper, predicate invention is conducted via construction of substitutions for meta-rules employed by a meta-interpreter.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“By contrast, induction, abduction and predicate invention are all achieved in MIL by way of the meta-rules.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In the NELL task abduction with respect to inductively inferred rules uncovered a systematic error in the existing NELL data, indicating that Metagol D shows promise in helping debug large-scale knowledge bases.”
hedgeddirectSingle case- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Finally it is worth noting that a Universal Turing Machine can be considered as simply a meta-interpreter incorporated within hardware.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
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Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This example demonstrates the potential for the MIL framework to use and infer higher-order concepts.”
boostedSingle case- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“However, the greedy approach can lead to overly specific results.”
plaindirectSingle case- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“This provides an interesting topic for further work since the use of a non-greedy complete search for each episode leads to search time increasing exponentially in the maximum number of clauses considered [see search bound in Lin et al. (2014)].”
What the sources say
- Not checked — no full text of the sourcecites Lin et al. (2014) source
plaindirect- Asserted: passes — The authors put it forward as true.
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“However, this greedy search strategy has not been applied to other experiments.”
plainSingle case- Asserted: passes — The authors put it forward as true.
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“However, the approach can also be implemented within more sophisticated solvers such as ASP (see Muggleton et al. 2014).”
What the sources say
- Not checked — no full text of the sourcecites Muggleton et al. 2014 source
plain- Asserted: passes — The authors put it forward as true.
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The NELL experiments are distinct from those on robot strategies and east-west trains in providing an initial indication of the power of the technique to reveal new and unexpected insights in large-scale real-world data.”
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The experiment also indicates the potential for learning higher-order concepts like symmetric .”
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In the Prolog implementation, Metagol D , efficiency of search is achieved by constraining the backtracking search in several ways.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“NELL presently incorporates manual annotation on concepts being symmetric or transitive.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“A related issue is that the user is expected to provide the total orderings ⪰ P and ⪰ C over the initial predicate symbols and constants.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This leads to a large hypothesis space.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For instance, a Knuth-Bendix style total ordering can be imposed over the Herbrand base which requires predicates which are higher in the ordering to be defined in terms of lower ones.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“When learning up to seven clauses, not only is the hypothesis space excessively large but learning requires more training examples than is consistent with the logarithmic bound in Theorem 3.”
plaindirectSingle case- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The approach seems promising though there are unresolved issues relating to potential overfitting.”
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Alternatively, an interval inclusion ordering ensures finite termination of (mutual) recursion in the case that the Herbrand Base is infinitely ascending but finitely descending.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
The paper relies on this without support.
SLD resolution is complete, so all hypotheses consistent with the examples can be constructed.
“Completeness of SLD resolution ensures that all hypotheses consistent with the examples can be constructed.”
The authors state this as an assumption.
P' ⊆ P and C' ⊆ C represent uninterpreted predicates and constants whose interpretation is assigned by the Meta-interpreter.
“To account for predicate invention we assume that P ′ ⊆ P and C ′ ⊆ C represent uninterpreted predicates and constants respectively, whose interpretation is assigned by the Meta-interpreter.”
The authors state this as an assumption.
The UTM has a suitably designed set of machine instructions representing functions of the form used in the encoding.
“We assume the UTM has a suitably designed set of machine instructions representing functions of the form”
The paper relies on this without support.
The user has no ascribed meaning for P' and C'.
“Since the user has no ascribed meaning for P ′ and C ′ , it would not be possible to provide examples containing full interpretations, and therefore the Learning from Interpretations setting is inappropriate.”
The authors state this as an assumption.
The big data assumption holds.
“under the big data assumption (Sect. 4.3)”
The paper relies on this without support.
The simplified model of wall building is adequate for evaluating Metagol D's robot strategy learning.
“This model is a simplification of a real-world robotics application.”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“However, the expressiveness of Metagol is bounded by the metarules provided, requiring domain expertise to design appropriate templates.”
From ADVENT: LLM-Driven Automatic Predicate Invention for ILP (2026) · cites this paper as [13]
Experimental — accuracy not yet measuredThe cited source says what the claim attributes to it.
In the citing paper
“Recent meta-level ILP approaches often learn optimal solutions, such as programs with the fewest clauses (Muggleton et al. 2015; Cropper and Muggleton 2016; Kaminski et al. 2018) or literals (Corapi et al. 2011; Law et al. 2014).”
From Learning programs by learning from failures (2021) · cites this paper as Muggleton et al. 2015
The cited source says what the claim attributes to it.
In the citing paper
“Metagol performs iterative deepening over the number of clauses allowed in a solution (Muggleton et al. 2015).”
From Learning programs by learning from failures (2021) · cites this paper as Muggleton et al. 2015
The cited source says what the claim attributes to it.
In the citing paper
“A third approach has recently emerged called meta-level ILP (Inoue et al., 2013; Muggleton et al., 2015; Inoue, 2016; Law et al., 2020; Cropper & Morel, 2021).”
From Inductive Logic Programming At 30: A New Introduction (2022) · cites this paper as Muggleton et al., 2015
The cited source says what the claim attributes to it.
In the citing paper
“For instance, Metagol (Muggleton et al., 2015; Cropper & Muggleton, 2016) uses a Prolog meta-interpreter to search for a proof of a meta-level Prolog program.”
From Inductive Logic Programming At 30: A New Introduction (2022) · cites this paper as Muggleton et al., 2015
The cited source says what the claim attributes to it.
In the citing paper
“A third new approach has recently emerged called meta-level ILP (Inoue et al., 2013; Muggleton et al., 2015; Inoue, 2016; Law et al., 2020; Cropper & Morel, 2021).”
From Inductive Logic Programming At 30: A New Introduction (2022) · cites this paper as Muggleton et al., 2015
The cited source says what the claim attributes to it.
In the citing paper
“Most approaches use an Occamist bias to find the smallest hypothesis, measured in terms of the number of clauses (Muggleton et al., 2015), literals (Law et al., 2014), or description length (Muggleton, 1995).”
From Inductive Logic Programming At 30: A New Introduction (2022) · cites this paper as Muggleton et al., 2015
The cited source says what the claim attributes to it.
In the citing paper
“Although inverse resolution approaches could support PI, they never demonstrated completeness, partly because of the lack of a declarative bias to delimit the hypothesis space (Muggleton et al., 2015).”
From Inductive Logic Programming At 30: A New Introduction (2022) · cites this paper as Muggleton et al., 2015
The cited source says what the claim attributes to it.
In the citing paper
“MetaInterpretive Learning (MIL) systems such as Metagol [13] employ metarules as templates to guide the invention of new predicates, enabling the construction of recursive and compositional hypotheses.”
From ADVENT: LLM-Driven Automatic Predicate Invention for ILP (2026) · cites this paper as [13]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ 5.3.2 Evaluation experiment: learning athletehomestadium · p. 19
“When the training size was smaller than 108, it was derived from the first two examples of the corresponding set of 108 examples.”
§ 5.3.2 Evaluation experiment: learning athletehomestadium · p. 19
“We considered different training sizes of [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 12 , 14 , 16 , 18 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , 100 , 108 ] .”
Citations
Cites
Reinforcement Learning: An Introduction
Sutton, R. S., & Barto, A. G. (1998) · IEEE Transactions on Neural Networks · doi:10.1109/tnn.1998.712192 · Cited by 27,225
Reinforcement Learning: An Introduction
Sutton, R. S., & Barto, A. (2005) · IEEE Transactions on Neural Networks · doi:10.1109/tnn.2004.842673 · Cited by 25,268
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Neurosymbolic AI: the 3rd wave
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Next-Generation Global Biomonitoring: Large-scale, Automated Reconstruction of Ecological Networks
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Self-citation
Metadata
- Authors
- Muggleton, S. H., Lin, D., & Tamaddoni-Nezhad, A.
- Affiliations
- Imperial College London (GB)
- Corresponding author
- Muggleton, Stephen H.
- Venue
- Machine Learning
- Published
- 2015-03-12
- Volume
- 100
- Issue
- 1
- Pages
- 49-73
- ISSN
- 0885-6125 (ISSN-L)
- 1573-0565
- OpenAlex
- W1769664091
- Cited by
- 184
- References
- 66
- Citations by year
- 2026: 3 · 2025: 8 · 2024: 13 · 2023: 18 · 2022: 8 · 2021: 23 · 2020: 20 · 2019: 25 · 2018: 24 · 2017: 12 · 2016: 9 · 2015: 10
- FWCI
- 17.10
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- www.springer.com/tdm (tdm)
- Locations
- Machine Learning (publishedVersion)
- Spiral (Imperial College London) (submittedVersion)
- Spiral (Imperial College London) (submittedVersion, cc-by-nc-nd)
- Funders
- Royal Academy of Engineering
- Imperial College London
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Logic, Reasoning, and Knowledge
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Computability, Logic, AI Algorithms
- Physical Sciences › Computer Science › Artificial Intelligence › AI-based Problem Solving and Planning
- Keywords
- predicate invention, higher-order logic, PAC learnability
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-10-19, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-19 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Cited sources are checked with a language model’s help (gpt-6-sol).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
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