Inductive Logic Programming At 30: A New Introduction

The placeholder approach to predicate invention requires generating all possible invented predicates.

“However, this placeholder approach is limited because it requires that a user manually specify the arity and argument types of a symbol (Law et al., 2014), which rather defeats the point, or requires generating all possible invented predicates (Evans & Grefenstette, 2018; Evans et al., 2021), which is computationally expensive.”

• cites Evans & Grefenstette, 2018 sourceDisputed

  Section 3.1 (page unavailable) says the program template specifies the set of invented predicates and the arity of each. Section 3.2 says the system generates clauses satisfying each rule template, subject to restrictions—not all possible invented predicates. Section 7.1 describes exhaustive enumeration of program templates only as a separate, computationally intensive experiment. (LLM, conf=0.84)

  Checked by a reasoning model — may be wrong.

  Experimental — accuracy not yet measured
• cites Evans et al., 2021 sourcePartly supported

  Section 4.1, “Iterating through templates” (page unavailable), says the method enumerates templates that include sets of predicates. But it explicitly says it cannot enumerate all such tuples because there are infinitely many; it uses a bound that increases gradually instead. Section 4.4 discusses computational cost, but does not establish that all possible invented predicates must be generated. (LLM, conf=0.85)

  Experimental — accuracy not yet measured

Because of its symbolic representation, ILP naturally supports lifelong and transfer learning.

“By contrast, because of its symbolic representation, ILP naturally supports lifelong and transfer learning (Torrey et al., 2007; Cropper, 2019), which is considered essential for human-like AI (Lake et al., 2016).”

• cites Torrey et al., 2007Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Cropper, 2019 sourcePartly supported

  The Related Work section (page unavailable) says that ILP meta-learning saves learned programs as background knowledge for reuse on unsolved tasks, supporting learning over time and transfer between tasks. The paper does not attribute that capability specifically to ILP’s symbolic representation or establish it as a natural property of ILP generally; it describes particular approaches that implement it. (LLM, conf=0.83)

  Experimental — accuracy not yet measured

It is impossible to provide a decision tree learner this infinite relation because it would require an infinite feature table.

“For instance, it is impossible to provide a decision tree learner (Quinlan, 1986, 1993) this infinite relation because it would require an infinite feature table.”

• cites Quinlan, 1986 sourceSupported
• cites Quinlan, 1993Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

If learning to recognise events, a user could provide the axioms of the event calculus.

“If learning to recognise events, a user could provide the axioms of the event calculus (Katzouris et al., 2015).”

• cites Katzouris et al., 2015 sourceSupported

Many forms of ML are notorious for their inability to generalise from small numbers of training examples, notably deep learning.

“Many forms of ML are notorious for their inability to generalise from small numbers of training examples, notably deep learning (Marcus, 2018; Chollet, 2019; Bengio et al., 2019).”

• cites Marcus, 2018 sourceSupported
• cites Chollet, 2019 sourceSupported
• cites Bengio et al., 2019 sourcePartly supported

  The Introduction (p. 1) says current machine-learning methods seem weak at generalizing beyond the training distribution. Section 2.2.1 (p. 5) discusses how the number of examples needed for adaptation relates to model complexity, including for neural networks. Neither passage establishes the broader claim that deep learning is generally unable to generalize from small training sets; the paper's few-example discussion concerns adaptation to a related distribution. (LLM, conf=0.78)

  Experimental — accuracy not yet measured

Some ILP systems induce full (unrestricted) clausal theories, such as Claudien and CF-induction.

“Some systems induce full (unrestricted) clausal theories, such as Claudien (De Raedt & Dehaspe, 1997) and CF-induction (Inoue, 2004).”

• cites De Raedt & Dehaspe, 1997 sourceSupported
• cites Inoue, 2004 sourceSupported

It is often difficult and expensive to obtain hand-crafted background knowledge.

“However, it is often difficult and expensive to obtain such BK.”

• cites Evans & Grefenstette, 2018 sourceCould not verify

  Section 5.3.1 describes background knowledge as arithmetic facts supplied for a task, but does not discuss the difficulty or expense of obtaining them. Section 7.1 calls hand-engineering program templates difficult work; those templates are distinct from the background knowledge. Searches for the cost of obtaining background knowledge found no supporting passage. (LLM, conf=0.73)

  Experimental — accuracy not yet measured

• AssumptionRests on an assumption: ILP has traditionally relied on predefined and hand-crafted background knowledge, often designed by domain experts.

There is preliminary work in reasoning about metarules to identify universal sets suitable to learn certain fragments of logic programs.

“For this reason, there is preliminary work in reasoning about metarules to identify universal sets suitable to learn certain fragments of logic programs (Cropper & Muggleton, 2014; Tourret & Cropper, 2019; Cropper & Tourret, 2020).”

• cites Cropper & Muggleton, 2014Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Tourret & Cropper, 2019 sourceSupported
• cites Cropper & Tourret, 2020 sourceSupported

A third new approach has recently emerged called meta-level ILP.

“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).”

• cites Inoue et al., 2013Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Muggleton et al., 2015 sourceSupported
• cites Inoue, 2016Not checked — no full text of the source

Lookahead increases the complexity of learning.

“To avoid this issue, these systems rely on lookahead (Struyf et al., 2006) which increases the complexity of learning.”

• cites Struyf et al., 2006 sourceSupported

These systems rely on lookahead to avoid the issue of getting stuck with suboptimal solutions.

“To avoid this issue, these systems rely on lookahead (Struyf et al., 2006) which increases the complexity of learning.”

• cites Struyf et al., 2006 sourceSupported

Most ILP approaches use an Occamist bias to find the smallest hypothesis, measured in terms of the number of clauses, literals, or description length.

“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).”

• cites Muggleton et al., 2015 sourceSupported
• cites Law et al., 2014 sourceNot checked — no full text of the source
• cites Muggleton, 1995 sourceNot checked — no full text of the source

Most ASP solvers only work on ground programs.

“Moreover, since most ASP solvers only work on ground programs (Gebser et al., 2014), pure ASPbased approaches are inherently restricted to tasks that have a small and finite grounding.”

• cites Gebser et al., 2014 sourceSupported

Pure ASP-based systems struggle to handle infinite domains because most current ASP solvers only work on ground programs and need a finite grounding.

“Pure ASP-based systems (Corapi et al., 2011; Kaminski et al., 2018; Evans et al., 2021) struggle to handle infinite domains because (most) current ASP solvers only work on ground programs, i.e. they need a finite grounding.”

• cites Corapi et al., 2011Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Kaminski et al., 2018Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Evans et al., 2021 sourcePartly supported

  Section 5.2.4 says that both the authors’ system and ILASP ground ASP programs into propositional clauses before solving, and that grounding size affects memory use. It reports that some ILASP programs were unsolvable because they required too much memory. The paper does not establish that most current ASP solvers work only on ground programs or directly assess pure ASP-based systems on infinite domains. (LLM, conf=0.82)

  Experimental — accuracy not yet measured

Some systems, mostly meta-level approaches, cannot handle infinite domains.

“Some systems, mostly meta-level approaches, cannot handle infinite domains (Corapi et al., 2011; Athakravi et al., 2013; Evans & Grefenstette, 2018; Kaminski et al., 2018; Evans et al., 2021).”

• cites Corapi et al., 2011Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Athakravi et al., 2013Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Evans & Grefenstette, 2018 sourcePartly supported

  Section 2.1 says the paper's system disallows function symbols, and Section 6 (Related Work) says it uses Datalog programs with finite Herbrand universes. This supports the limitation on infinite domains for this system, but the paper does not establish that the systems with this limitation are mostly meta-level approaches. (LLM, conf=0.85)

  Experimental — accuracy not yet measured

One approach to mitigate the grounding problem is to use context-dependent examples, where BK can be associated with specific examples so that an ILP system need only ground part of the BK.

“One approach to mitigate this problem is to use context-dependent examples (Law et al., 2016), where BK can be associated with specific examples so that an ILP systems need only ground part of the BK.”

• cites Law et al., 2016 sourceSupported

Knorf substantially reduces the size of the BK program, reducing the number of literals in a program by 50% or more.

“Moreover, the authors also demonstrate that Knorf substantially reduces the size of the BK program, reducing the number of literals in a program by 50% or more.”

• cites Dumancic et al., 2021 sourcePartly supported

  Section 5.1 (p. 6) says refactoring drastically reduces BK size and leaves the number of literals at only a fraction of the original. Section 4.4 (p. 5) defines program size by its number of literals. The text available in our copy does not establish the claimed reduction of 50% or more; Figure 6 is present only as an image placeholder. (LLM, conf=0.86)

  Experimental — accuracy not yet measured

Knorf compresses the learnt program by removing redundancies in it after learning to solve user-supplied tasks in the lifelong learning setting.

“After learning to solve user-supplied tasks in the lifelong learning setting, Knorf compresses the learnt program by removing redundancies in it.”

• cites Dumancic et al., 2021 sourcePartly supported

  Section 4.4 (p. 5) says Knorf seeks a smaller refactored program with less redundancy, and Related Work (p. 2) says it refactors knowledge after learning. But the lifelong-learning description (p. 6) distinguishes earlier learned background tasks from user-provided target tasks, which are solved using that background knowledge. It does not support the claim that Knorf compresses the program after learning to solve the user-provided tasks. (LLM, conf=0.85)

  Experimental — accuracy not yet measured

Generative MLNs learn a probabilistic logic program that explains all predicates in an interpretation, not a single target predicate.

“Generative MLNs learn a (probabilistic) logic program that explains all predicates in an interpretation, not a single target predicate.”

• cites Richardson & Domingos, 2006 sourceSupported

Generating all possible invented predicates is computationally expensive.

“However, this placeholder approach is limited because it requires that a user manually specify the arity and argument types of a symbol (Law et al., 2014), which rather defeats the point, or requires generating all possible invented predicates (Evans & Grefenstette, 2018; Evans et al., 2021), which is computationally expensive.”

• cites Evans & Grefenstette, 2018 sourcePartly supported

  Section 7.1 says that enumerating program templates to avoid hand-engineering required a “massive amount of computation.” Section 3.1 says those templates specify the invented predicates and their arities. The paper therefore supports the computational-cost point, but it does not say its approach generates all possible invented predicates; it describes enumerating templates in a brute-force experiment. (LLM, conf=0.83)

  Experimental — accuracy not yet measured
• cites Evans et al., 2021 sourcePartly supported

  Section 4.1 says the system enumerates templates that include predicate sets, but cannot enumerate all such templates at once because there are infinitely many. Section 4.4 describes substantial computational costs, and the Conclusion says search-space size restricts the system to small-to-medium-size problems. The paper does not specifically establish the cost of generating all possible invented predicates as a separate approach. (LLM, conf=0.80)

  Experimental — accuracy not yet measured

If the learnt program contains invented predicates, Knorf revises them and introduces new ones that would lead to a smaller program.

“If the learnt program contains invented predicates, Knorf revises them and introduces new ones that would lead to a smaller program 37 .”

• cites Dumancic et al., 2021 sourceSupported

Refactoring improves learning performance in lifelong learning.

“The authors experimentally demonstrate that refactoring improves learning performance in lifelong learning.”

• cites Dumancic et al., 2021 sourceSupported

ILP has been applied to identifying and predicting ligands.

“The types of task ILP has been applied to include identifying and predicting ligands (substructures responsible for medical activity) (Finn et al., 1998; Srinivasan et al., 2006; Kaalia et al., 2016)”

• cites Finn et al., 1998 sourcePartly supported

  Section 2.2.3 (page unavailable) says the paper applies ILP to pharmacophore discovery, identifying molecular features associated with biological activity. Section 2.3 (page unavailable) also describes ILP applications to predicting molecular activity. But the paper defines ligands as molecules that bind to a protein, not as activity-related substructures, and does not describe its task as predicting ligands. (LLM, conf=0.87)

  Experimental — accuracy not yet measured
• cites Srinivasan et al., 2006 sourcePartly supported

  The Introduction describes ILP-based structure–activity models that can help suggest new molecules, and the Abstract describes predicting a ligand’s medicinal activity. But the paper defines ligands as small molecules, not the activity-related substructures in the parenthetical; it predicts their activity rather than the ligands themselves. The supplied passages give no page numbers. (LLM, conf=0.86)

  Experimental — accuracy not yet measured
• cites Kaalia et al., 2016 sourcePartly supported

  The Conclusions say ILP-derived pharmacophores can identify potential ligands, and the Introduction reports that screening identified known inhibitors. But the Introduction defines ligands as small molecules, not the activity-related structural features described in the parenthesis; those features are pharmacophores. (LLM, conf=0.87)

  Experimental — accuracy not yet measured

TILDE learns normal logic programs (which include negation) instead of definite logic programs.

“An interesting aspect of TILDE is that it learns normal logic programs (which include negation) instead of definite logic programs.”

• cites Blockeel & De Raedt, 1998 sourcePartly supported

  Section 4 says TILDE induces first-order logical decision trees, not logic programs directly. Section 3 says those trees can always be converted to layered normal logic programs, and the paper defines normal programs as permitting negative literals. The claim captures the role of negation but misstates the form TILDE learns. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

Evans et al. (2021) show that their system can perform human-level performance.

“They show that their system can perform human-level performance.”

• cites Evans et al., 2021 sourceSupported

Another disadvantage of neural approaches is that they often require hand-crafted neural architectures for each domain.

“Another disadvantage of neural approaches is that they often require hand-crafted neural architectures for each domain. For instance, the REPL approach (Ellis et al., 2019) needs a hand-crafted grammar, interpreter, and neural architecture for each domain.”

• cites Ellis et al., 2019 sourceSupported

The authors demonstrate that their framework can learn arithmetic and sorting operations from digits represented as images.

“The authors demonstrate that their framework can learn arithmetic and sorting operations from digits represented as images.”

• cites Dai & Muggleton, 2021 sourceSupported

A handful of approaches have started to tackle the problem of learning from raw data.

“A handful of approaches have started to tackle this problem (Manhaeve et al., 2018; Dai et al., 2019; Evans et al., 2021; Dai & Muggleton, 2021), but developing better ILP techniques that can both perceive sensory input and learn complex relational programs would be a major breakthrough not only for ILP, but the whole of AI.”

• cites Manhaeve et al., 2018 sourceSupported
• cites Dai et al., 2019Not checked — no full text of the source
• cites Evans et al., 2021 sourcePartly supported

  The Introduction describes the Apperception Engine as learning a symbolic program from a sensory sequence, so this paper is an example of an approach tackling the problem. But the Conclusion states that its sensory input must first be discretized by another system; the reported engine does not learn directly from raw continuous data. (LLM, conf=0.85)

  Experimental — accuracy not yet measured

Many systems cannot learn recursive programs, or can only learn it in a limited form.

“As Table 4 shows, many systems cannot learn recursive programs, or can only learn it in a limited form.”

Although larger in clauses and literals, the program learned by Metaopt is more efficient (O(n log n)) than the program learned by Metagol (O(n^2)).

“Although larger, both in terms of clauses and literals, the program learned by Metaopt is more efficient ( O ( n log n )) than the program learned by Metagol ( O ( n 2 )).”

TILDE supports both categorical and numerical data.

“Another advantage of TILDE is that, compared to other ILP systems, it supports both categorical and numerical data.”

• AssumptionRests on an assumption: Other ILP systems usually struggle to handle numerical data.

When learning game rules from observations (Cropper et al., 2020b), ASPAL performs poorly for this reason.

“For instance, when learning game rules from observations (Cropper et al., 2020b), ASPAL performs poorly for this reason.”

• cites Cropper et al., 2020bNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

ILP can learn human-readable hypotheses.

“As this scenario illustrates, ILP can learn human-readable hypotheses.”

ILP can induce hypotheses from small numbers of examples, often from a single example.

“By contrast, ILP can induce hypotheses from small numbers of examples, often from a single example (Lin et al., 2014; Muggleton et al., 2018).”

• cites Lin et al., 2014Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Muggleton et al., 2018Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Relational BK allows us to succinctly represent infinite relations.

“Relational BK allows us to succinctly represent infinite relations.”

If learning causal relations in causal networks, a user can encode constraints about the network.

“For instance, if learning causal relations in causal networks, a user can encode constraints about the network (Inoue et al., 2013).”

• cites Inoue et al., 2013Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

ILP has been widely used for scientific discovery.

“For this reason, ILP has been widely used for scientific discovery.”

• cites Srinivasan et al., 1994Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites King et al., 1992 sourceNot checked — no full text of the source

This data efficiency has made ILP attractive in many real-world applications, especially in drug design, where large numbers of examples are not always easy to obtain.

“This data efficiency has made ILP attractive in many real-world applications, especially in drug design, where large numbers of examples are not always easy to obtain.”

Even if restricted to a finite set of n natural numbers, a table-based approach would still need n^3 features to represent the complete summation relation.

“Even if we restricted ourselves to a finite set of n natural numbers, a table-based approach would still need n 3 features to represent the complete summation relation.”

This knowledge transfer approach leads to a hierarchy of reusable programs, where each program builds on simpler programs.

“Moreover, they show that this knowledge transfer approach leads to a hierarchy of reusable programs, where each program builds on simpler programs.”

• cites Lin et al., 2014Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Gulwani (2011) applies techniques similar to ILP to induce programs from user-provided examples in Microsoft Excel to solve string transformation problems, where it is infeasible to ask a user for thousands of examples.

“For instance, Gulwani (2011) applies techniques similar to ILP to induce programs from user-provided examples in Microsoft Excel to solve string transformation problems, where it is infeasible to ask a user for thousands of examples.”

• cites Gulwani, 2011Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Reasoning about full clausal logic is computationally expensive.

“However, reasoning about full clausal logic is computationally expensive (Nienhuys-Cheng & Wolf, 1997).”

• cites Nienhuys-Cheng & Wolf, 1997 sourceNot checked — no full text of the source

One reason for focusing on Horn theories, rather than full clausal theories, is SLD-resolution, an inference rule that sacrifices expressibility for efficiency.

“One reason for focusing on Horn theories, rather than full clausal theories, is SLD-resolution (Kowalski & Kuehner, 1971), an inference rule that sacrifices expressibility for efficiency.”

• cites Kowalski & Kuehner, 1971 sourceNot checked — no full text of the source

Compared to definite programs, the main advantage of Datalog is decidability.

“Compared to definite programs, the main advantage of Datalog is decidability (Dantsin et al., 2001).”

• cites Dantsin et al., 2001 sourceNot checked — no full text of the source

Horn logic is still Turing complete.

“Horn logic is, however, still Turing complete (T¨ arnlund, 1977).”

• cites T¨ arnlund, 1977Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Checking subsumption between clauses is decidable.

“By contrast, checking subsumption between clauses is decidable (Plotkin, 1971)”

• cites Plotkin, 1971Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Most ILP systems learn fragments of clausal logic, usually definite programs.

“However, reasoning about full clausal theories is computationally expensive, so most systems learn fragments of clausal logic, usually definite programs.”

• AssumptionRests on an assumption: Reasoning about full clausal theories is computationally expensive.

Checking entailment between clauses is undecidable.

“In other words, checking entailment between clauses is undecidable (Church, 1936).”

• cites Church, 1936 sourceNot checked — no full text of the source

In general, deciding subsumption is a NP-complete problem.

“in general, deciding subsumption is a NP-complete problem (Nienhuys-Cheng & Wolf, 1997).”

• cites Nienhuys-Cheng & Wolf, 1997 sourceNot checked — no full text of the source

Inducing higher-order programs can drastically improve learning performance in terms of predictive accuracy, sample complexity, and learning times.

“Cropper et al. (2020) show that inducing higher-order programs can drastically improve learning performance in terms of predictive accuracy, sample complexity, and learning times.”

• cites Cropper et al., 2020Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Some ILP systems can learn unstratified ASP programs.

“By contrast, some systems can learn unstratified ASP programs (Law et al., 2014).”

• cites Law et al., 2014 sourceNot checked — no full text of the source

In the Caesar cipher example, the higher-order program is smaller than the first-order program because the higher-order background relation map abstracts away the need to learn a recursive program.

“The higher-order program is smaller than the first-order program because the higher-order background relation map abstracts away the need to learn a recursive program.”

Using logic programs to represent data allows ILP to learn with complex relational information.

“Using logic programs to represent data allows ILP to learn with complex relational information.”

Most ILP approaches relax the requirement that a hypothesis be complete and consistent, instead seeking a hypothesis that entails as many positive and as few negative examples as possible.

“Therefore, most approaches relax this definition and try to find a hypothesis that entails as many positive and as few negative examples as possible.”

• AssumptionRests on an assumption: Training examples are often noisy.

Most ILP systems learn first-order (or higher-order) programs.

“Although some systems learn propositional programs, such as Duce (Muggleton, 1987), most learn first-order (or higher-order) programs.”

Using too little or insufficient background knowledge may exclude a good hypothesis from the hypothesis space.

“If we use too little or insufficient BK then we may exclude a good hypothesis from the hypothesis space.”

It is still a major challenge to choose appropriate mode declarations.

“Although there is some work on learning mode declarations (McCreath & Sharma, 1995; Ferilli et al., 2004; Picado et al., 2017), it is still a major challenge to choose appropriate ones.”

• cites McCreath & Sharma, 1995Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Ferilli et al., 2004 sourceNot checked — no full text of the source
• cites Picado et al., 2017 sourceNot checked — no full text of the source

If the language bias is too weak, then the search can become intractable.

“If the bias is too weak , then the search can become intractable.”

Choosing an appropriate language bias is essential to make an ILP problem tractable because it defines the hypothesis space.

“Choosing an appropriate language bias is essential to make an ILP problem tractable because it defines the hypothesis space.”

A system can exploit the ordering imposed by a generality relation during the search for a hypothesis.

“A system can exploit this ordering during the search for a hypothesis.”

ILP systems enforce an inductive bias to restrict the hypothesis space.

“To restrict the hypothesis space, systems enforce an inductive bias (Mitchell, 1997).”

• cites Mitchell, 1997Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The trade-off between weak and strong language bias is one of the major problems holding ILP back from being widely used.

“This trade-off is one of the major problems holding ILP back from being widely used 18 .”

If a user provides weak mode declarations (infinite recall, a single type, and no input/output arguments), then the search quickly becomes intractable.

“Moreover, if a user provides weak mode declarations (for instance with infinite recall, a single type, and no input/output arguments), then the search quickly becomes intractable.”

Choosing appropriate background knowledge in ILP is crucial for good learning performance.

“As with choosing appropriate features, choosing appropriate BK in ILP is crucial for good learning performance.”

If the language bias is too strong, then we risk excluding a good solution from the hypothesis space.

“If the bias is too strong then we risk excluding a good solution from the hypothesis space.”

Not providing a recursive mode declaration for the target relation would prevent a mode-based system from finding a good hypothesis.

“Similarly, not providing a recursive mode declaration for the target relation would prevent a mode-based system from finding a good hypothesis.”

Because metarules precisely define the form of hypotheses, they can greatly reduce the hypothesis space, especially if the user knows about the class of programs to be learned.

“Because they precisely define the form of hypotheses, they can greatly reduce the hypothesis space, especially if the user knows about the class of programs to be learned.”

Not providing a recursive metarule would prevent a metarule-based system from inducing a program that generalises to lists of any length, even if all necessary background relations are provided.

“Even if all necessary background relations are provided, not providing a recursive metarule (e.g. R(A,B):P(A,C), R(C,B) ) would prevent a metarule-based system from inducing a program that generalises to lists of any length.”

Bottom-up approaches do not easily support predicate invention.

“However, as Bratko (1999) points out, there are several disadvantages of bottom-up approaches, such as (i) they typically use unnecessarily long hypotheses with many clauses, (ii) it is difficult for them to learn recursive hypotheses and multiple predicates simultaneously, and (iii) they do not easily support predicate invention.”

• cites Bratko, 1999Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

It is difficult for bottom-up approaches to learn recursive hypotheses and multiple predicates simultaneously.

“However, as Bratko (1999) points out, there are several disadvantages of bottom-up approaches, such as (i) they typically use unnecessarily long hypotheses with many clauses, (ii) it is difficult for them to learn recursive hypotheses and multiple predicates simultaneously, and (iii) they do not easily support predicate invention.”

• cites Bratko, 1999Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Top-down approaches rely on iterative improvements.

“Another disadvantage of top-down approaches is their reliance on iterative improvements.”

TILDE can get stuck with suboptimal solutions if the necessary clauses are very long and intermediate specialisations do not improve the score (coverage) of the clause.

“As such, TILDE can get stuck with suboptimal solutions if the necessary clauses are very long and intermediate specialisations do not improve the score (coverage) of the clause.”

Meta-level approaches can often learn optimal and recursive programs.

“Meta-level approaches can often learn optimal and recursive programs.”

• cites Corapi et al., 2011Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Law et al., 2014 sourceNot checked — no full text of the source
• cites Kaminski et al., 2018Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Meta-level approaches often delegate the search for a hypothesis to an off-the-shelf solver.

“Such meta-level approaches often delegate the search for a hypothesis to an off-the-shelf solver (Corapi et al., 2011; Athakravi et al., 2013; Muggleton et al., 2014; Law et al., 2014; Kaminski et al., 2018; Evans et al., 2021; Cropper & Dumanˇ ci´ c, 2020; Cropper & Morel, 2021) after which the meta-level solution is translated back to a standard solution for the ILP problem.”

• cites Corapi et al., 2011Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Athakravi et al., 2013Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Muggleton et al., 2014Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Meta-level approaches formulate the learning problem as a declarative problem, often as an ASP problem.

“In other words, instead of writing a procedure to search in a top-down or bottom-up manner, meta-level approaches formulate the learning problem as a declarative problem, often as an ASP problem (Corapi et al., 2011; Athakravi et al., 2013; Muggleton et al., 2014; Law et al., 2014; Kaminski et al., 2018; Evans et al., 2021; Cropper & Dumanˇ ci´ c, 2020; Cropper & Morel, 2021).”

• cites Corapi et al., 2011Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Athakravi et al., 2013Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Muggleton et al., 2014Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The noise-tolerant version of ILASP uses ASP's optimisation abilities to provably learn the program with the best coverage.

“The noise-tolerant version of ILASP (Law, 2018) uses ASP's optimisation abilities to provably learn the program with the best coverage.”

• cites Law, 2018Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Many meta-level approaches struggle to learn programs with large rules.

“Many approaches also precompute every possible rule in a hypothesis (Corapi et al., 2011; Law et al., 2014), so struggle to learn programs with large rules, although preliminary work tries to address this issue (Cropper & Dumanˇ ci´ c, 2020).”

Aleph offers no guarantees about program size and coverage.

“For instance, Aleph, described in detail in the next section, offers no guarantees about the program size and coverage.”

Newer systems address the limitation of not guaranteeing smallest programs through meta-level reasoning.

“Newer systems address this limitation (Corapi et al., 2011; Law et al., 2014; Cropper & Muggleton, 2016; Kaminski et al., 2018; Cropper & Morel, 2021) through meta-level reasoning (Section 4.5).”

• cites Corapi et al., 2011Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Law et al., 2014 sourceNot checked — no full text of the source
• cites Cropper & Muggleton, 2016 sourceNot checked — no full text of the source

Meta-level approaches can harness state-of-the-art techniques in constraint solving, notably in ASP.

“They can also harness the state-of-the-art techniques in constraint solving, notably in ASP.”

Metaopt (Cropper & Muggleton, 2019) learns efficient programs.

“To address this issue, Metaopt (Cropper & Muggleton, 2019) learns efficient programs.”

• cites Cropper & Muggleton, 2019Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Pure ASP-based approaches are inherently restricted to tasks that have a small and finite grounding.

“Moreover, since most ASP solvers only work on ground programs (Gebser et al., 2014), pure ASPbased approaches are inherently restricted to tasks that have a small and finite grounding.”

Many meta-level approaches struggle to scale to problems with very large domains.

“A key issue is that many approaches encode an ILP problem as a single (often very large) ASP problem (Corapi et al., 2011; Law et al., 2014; Kaminski et al., 2018; Evans et al., 2021), so struggle to scale to problems with very large domains.”

Most ILP systems relax the requirement to entail all positive and none of the negative examples and accept hypotheses that do not cover all positives or cover some negatives.

“Therefore, most systems relax this constraint and accept a hypothesis that does not necessarily cover all positive examples or that covers some negative examples 27 .”

Most ILP systems are not guaranteed to induce the smallest programs.

“Most systems are not, however, guaranteed to induce the smallest programs.”

A key reason most systems are not guaranteed to induce the smallest programs is that many approaches learn a single clause at a time, leading to sub-programs that are sub-optimal in program size and coverage.

“A key reason for this limitation is that many approaches learn a single clause at a time leading to the construction of sub-programs that are sub-optimal in terms of program size and coverage.”

Many meta-level approaches precompute every possible rule in a hypothesis.

“Many approaches also precompute every possible rule in a hypothesis (Corapi et al., 2011; Law et al., 2014), so struggle to learn programs with large rules, although preliminary work tries to address this issue (Cropper & Dumanˇ ci´ c, 2020).”

• cites Corapi et al., 2011Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Law et al., 2014 sourceNot checked — no full text of the source

∂ ILP gives atoms continuous semantics, mapping atoms to the real unit interval [0, 1].

“Rather than an atom being true or false, ∂ ILP gives atoms continuous semantics, which maps atoms to the real unit interval [0, 1].”

Meta-level approaches can learn recursive programs.

“The main advantage of meta-level approaches is that they can learn recursive programs and optimal programs (Corapi et al., 2011; Law et al., 2014; Kaminski et al., 2018; Evans & Grefenstette, 2018; Evans et al., 2021; Cropper & Morel, 2021).”

• cites Corapi et al., 2011Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Law et al., 2014 sourceNot checked — no full text of the source
• cites Kaminski et al., 2018Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Many meta-level approaches encode an ILP problem as a single (often very large) ASP problem.

“A key issue is that many approaches encode an ILP problem as a single (often very large) ASP problem (Corapi et al., 2011; Law et al., 2014; Kaminski et al., 2018; Evans et al., 2021), so struggle to scale to problems with very large domains.”

• cites Corapi et al., 2011Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Law et al., 2014 sourceNot checked — no full text of the source
• cites Kaminski et al., 2018Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Meta-level approaches can learn optimal programs.

“The main advantage of meta-level approaches is that they can learn recursive programs and optimal programs (Corapi et al., 2011; Law et al., 2014; Kaminski et al., 2018; Evans & Grefenstette, 2018; Evans et al., 2021; Cropper & Morel, 2021).”

• cites Corapi et al., 2011Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Law et al., 2014 sourceNot checked — no full text of the source
• cites Kaminski et al., 2018Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Many ILP systems try to learn a textually minimal hypothesis.

“Many systems try to learn a textually minimal hypothesis.”

The assumption that BK is perfect is a major limitation because real-world data, such as images or speech, cannot always be easily translated into a purely noise-free symbolic representation.

“This assumption is a major limitation because real-world data, such as images or speech, cannot always be easily translated into a purely noise-free symbolic representation.”

ASPAL uses ASP's optimisation abilities to provably learn the program with the fewest literals.

“ASPAL uses ASP's optimisation abilities to provably learn the program with the fewest literals.”

• cites Corapi et al., 2011Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Recursion is vital for many program synthesis tasks, such as the quicksort scenario from the introduction.

“Moreover, recursion is vital for many program synthesis tasks, such as the quicksort scenario from the introduction.”

Without recursion it is often difficult for a system to generalise from small numbers of examples.

“Without recursion it is often difficult for a system to generalise from small numbers of examples (Cropper et al., 2015).”

• cites Cropper et al., 2015Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

ASP can work with infinite domains as long as the grounding is finite.

“ASP can work with infinite domains as long as the grounding is finite.”

The grounding problem is essentially the fundamental problem faced by table-based ML approaches.

“This grounding problem is essentially the fundamental problem faced by table-based ML approaches that we discussed in Section 4.3.”

This ASP grounding approach leads to the grounding bottleneck problem, where grounding can be so large that it is intractable.

“This approach leads to the grounding bottleneck problem (Balduccini et al., 2013), where grounding can be so large that it is simply intractable.”

• cites Balduccini et al., 2013 sourceNot checked — no full text of the source

A common limitation is that many systems rely on bottom clause construction.

“A common limitation is that many systems rely on bottom clause construction (Muggleton, 1995), which we discuss in detail in Section 6.1.”

• cites Muggleton, 1995 sourceNot checked — no full text of the source

The grounding issue is especially problematic when reasoning about complex data structures, such as lists.

“This grounding issue is especially problematic when reasoning about complex data structures, such as lists.”

Most systems would need examples of each depth to learn a non-recursive reachability program.

“Moreover, most systems would need examples of each depth to learn such a program.”

There are many negative theoretical results on the learnability of recursive programs.

“Moreover, there are many negative theoretical results on the learnability of recursive programs (Cohen, 1995d).”

• cites Cohen, 1995dNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The non-recursive reachability program does not generalise because it does not define reachability for arbitrary depths.

“This program does not generalise because it does not define reachability for arbitrary depths.”

To learn minimal time complexity logic programs, Metaopt minimises the number of resolution steps.

“To learn minimal time complexity logic programs, Metaopt minimises the number of resolution steps.”

• cites Cropper & Muggleton, 2019Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Metagol can learn mutually recursive programs, such as learning the definition of an even number by also inventing and learning the definition of an odd number.

“Metagol can also learn mutually recursive programs, such as learning the definition of an even number by also inventing and learning the definition of an odd number ( even 1 ):”

Learning recursive programs has long been a difficult problem.

“Despite its importance, learning recursive programs has long been a difficult problem (Muggleton et al., 2012).”

• cites Muggleton et al., 2012Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

A system which supports PI can learn a grandparent program that invents a new predicate symbol inv.

“By contrast, consider the program learned by a system which supports PI:”

A system that supports recursion can learn a recursive reachability program.

“By contrast, a system that supports recursion can learn the program:”

With recursion, systems can generalise from small numbers of examples, often a single example.

“With recursion, systems can generalise from small numbers of examples, often a single example (Lin et al., 2014; Cropper, 2019).”

• cites Lin et al., 2014; Cropper, 2019Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

A system that supports recursion can learn a compact recursive program for finding the last element of a list.

“By contrast, a system that supports recursion can learn the compact program:”

Without recursion, a system would need to learn a separate clause to find the last element for each list of length n.

“As with the reachability example, without recursion, a system would need to learn a separate clause to find the last element for each list of length n , such as:”

Other systems can also learn optimal programs.

“Other systems can also learn optimal programs (Sch¨ uller & Benz, 2018).”

• cites Schüller & Benz, 2018Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

In ILP, recursion is often crucial for generalisation.

“In ILP, recursion is often crucial for generalisation.”

The grounding problem is not specific to ASP-based systems.

“This grounding problem is not specific to ASP-based systems.”

Systems can learn the recursive reachability definition from a small number of examples of arbitrary reachability depth.

“Moreover, systems can learn this definition from a small number of examples of arbitrary reachability depth.”

The recursive reachability program generalises reachability to any depth.

“Although smaller, this program generalises reachability to any depth.”

Metagol induces recursive programs using recursive metarules, such as the tail recursive metarule P(A,B) ← Q(A,C), P(C,B).

“Metagol induces recursive programs using recursive metarules, such as the tail recursive metarule P(A,B) ← Q(A,C), P(C,B) .”

Given suitable examples and no other background relations, a system can learn a non-recursive grandparent program.

“Given suitable examples and no other background relations, a system can learn the program:”

Without recursion, an ILP system would need to learn a separate clause to define reachability of different lengths.

“Without recursion, an ILP system would need to learn a separate clause to define reachability of different lengths.”

Lin et al. (2014) use dependent learning to enable Metagol to learn string transformations programs over time.

“For instance Lin et al. (2014) use a technique called dependent learning to enable Metagol to learn string transformations programs over time.”

• cites Lin et al. (2014)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The predicates invented by ALPs aid the learning of all predicates in the BK.

“The predicates invented by ALPs, therefore, aid the learning of all predicates in the BK.”

Given suitable examples and BK, Metagol ho learns a higher-order droplasts program.

“Given suitable examples and BK, Metagol ho (Cropper et al., 2020) learns the higher-order program:”

• cites Cropper et al., 2020Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

A side-effect of the metarule-driven approach to PI is that problems are forced to be decomposed into smaller problems.

“A side-effect of this metarule-driven approach to PI is that problems are forced to be decomposed into smaller problems.”

The placeholder approach to predicate invention requires a user to manually specify the arity and argument types of a symbol.

“However, this placeholder approach is limited because it requires that a user manually specify the arity and argument types of a symbol (Law et al., 2014), which rather defeats the point, or requires generating all possible invented predicates (Evans & Grefenstette, 2018; Evans et al., 2021), which is computationally expensive.”

• cites Law et al., 2014 sourceNot checked — no full text of the source

The higher-order droplasts program uses map to abstract away the manipulation of the list to avoid the need to learn an explicitly recursive program.

“This higher-order program uses map to abstract away the manipulation of the list to avoid the need to learn an explicitly recursive program (recursion is implicit in map ).”

Knorf optimises the representation of obtained knowledge.

“By doing so, Knorf optimises the representation of obtained knowledge.”

The most important implication of ALPs is that the target programs are easier to express via the invented predicates.

“The most important implication of the approach is that the target programs are easier to express via the invented predicates.”

ALPs change the representation of the problem.

“ALPs, therefore, change the representation of the problem.”

Lin et al.'s multi-task approach performs substantially better than a single-task approach because learned programs are frequently reused.

“The authors experimentally show that their multi-task approach performs substantially better than a single-task approach because learned programs are frequently reused.”

• cites Lin et al. (2014)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

PI can help learn substantially more complex programs.

“This program illustrates the power of PI to help learn substantially more complex programs.”

Aleph is efficient at identifying relevant constant symbols that may appear in a hypothesis.

“Because it uses a bottom clause to bound the search, Aleph is also efficient at identifying relevant constant symbols that may appear in a hypothesis, which is not the case for pure top-down approaches 41 .”

• cites [41]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

ALPs invent predicates by simultaneously learning an encoder and a decoder.

“Auto-encoding logic programs (ALPs) (Dumanˇ ci´ c et al., 2019) invent predicates by simultaneously learning a pair of logic programs: (i) an encoder that maps the examples given as interpretations to new interpretations defined entirely in terms of invented predicates 35 , and (ii) a decoder that reconstructs the original interpretations from the invented ones.”

• cites Dumanˇ ci´ c et al., 2019Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Lin et al.'s approach leads to a hierarchy of BK composed of reusable programs, where each builds on simpler programs.

“Moreover, they show that this approach leads to a hierarchy of BK composed of reusable programs, where each builds on simpler programs.”

• cites Lin et al. (2014)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

ASPAL precomputes every possible rule in a hypothesis.

“It precomputes every possible rule in a hypothesis, which is infeasible on all but trivial problems.”

Aleph struggles to learn optimal programs.

“Because it is based on inverse entailment, and only learns a single clause at a time, Aleph struggles to learn recursive programs and optimal programs and does not support PI.”

ASPAL is infeasible on all but trivial problems.

“It precomputes every possible rule in a hypothesis, which is infeasible on all but trivial problems.”

Because Metagol uses metarules, it can tightly restrict the hypothesis space, making it extremely efficient at finding solutions.

“Because it uses metarules, Metagol can tightly restrict the hypothesis space, which means that it is extremely efficient at finding solutions.”

Lookahead is needed when a single literal is useful only in a conjunction with another literal.

“Lookahead is needed when a single literal is useful only in a conjunction with another literal.”

ILP can make predictions based on the (sub)structured present in biological structures which domain experts can interpret.

“ILP can, therefore, make predictions based on the (sub)structured present in biological structures which domain experts can interpret.”

When one has little knowledge of the solutions, Metagol is unsuitable.

“However, when one has little knowledge of the solutions, then Metagol is unsuitable.”

Aleph struggles to learn recursive programs.

“Because it is based on inverse entailment, and only learns a single clause at a time, Aleph struggles to learn recursive programs and optimal programs and does not support PI.”

The last condition in Metagol's problem setting enforces the strong inductive bias in Metagol.

“It is this condition that enforces the strong inductive bias in Metagol.”

TILDE needs lookahead.

“Another weakness of TILDE is the need for lookahead.”

ILP is especially suitable for bioinformatics and drug design because biological structures, including molecules and protein interaction networks, can easily be expressed as relations.

“ILP is especially suitable for such problems because biological structures, including molecules and protein interaction networks, can easily be expressed as relations: molecular bonds define relations between atoms and interactions define relations between proteins.”

Aleph does not support PI.

“Because it is based on inverse entailment, and only learns a single clause at a time, Aleph struggles to learn recursive programs and optimal programs and does not support PI.”

The main limitation of ASPAL is scalability.

“The main limitation of ASPAL is scalability.”

Pure top-down approaches are not efficient at identifying relevant constant symbols that may appear in a hypothesis.

“Because it uses a bottom clause to bound the search, Aleph is also efficient at identifying relevant constant symbols that may appear in a hypothesis, which is not the case for pure top-down approaches 41 .”

• cites [41]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Metagol supports predicate invention.

“Metagol supports PI, learning recursive programs, and is guaranteed to learn the smallest program.”

PL approaches often aim to find any program that fits the specification, regardless of whether it generalises.

“PL approaches often aim to find any program that fits the specification, regardless of whether it generalises.”

Martínez et al. (2015, 2016) combine LFIT with a reinforcement learning algorithm to learn probabilistic models with exogenous effects from scratch.

“Mart´ ınez et al. (2015, 2016) combine LFIT with a reinforcement learning algorithm to learn probabilistic models with exogenous effects (effects not related to any action) from scratch.”

• cites Mart´ ınez et al., 2015, 2016Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Models of simple dynamical systems can be learned given the knowledge about differential equations.

“and models of simple dynamical systems can be learned given the knowledge about differential equations (Bratko et al., 1991).”

• cites Bratko et al., 1991Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

ILP is well-suited for addressing NLP tasks because linguistics knowledge and principles can be integrated into ILP systems.

“(ii) linguistics knowledge and principles can be integrated into ILP systems”

• AssumptionRests on an assumption: Many natural language processing tasks require an understanding of the syntax and semantics of the language.

Several ILP applications solve problems from first principles: provided physical models of the basic primitives, ILP systems can induce the target hypothesis whose behaviour is derived from the basic primitives.

“Several ILP applications solve problems from fi rst principles : provided physical models of the basic primitives, ILP systems can induce the target hypothesis whose behaviour is derived from the basic primitives.”

ILP is well-suited for addressing NLP tasks because the learnt clauses are understandable to a linguist.

“(iii) the learnt clauses are understandable to a linguist.”

• AssumptionRests on an assumption: Many natural language processing tasks require an understanding of the syntax and semantics of the language.

ILP systems can use a theory of light to understand images.

“For instance, ILP systems can use a theory of light to understand images (Dai et al., 2017; Muggleton et al., 2018).”

• cites Dai et al., 2017Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Muggleton et al., 2018Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Only a few approaches start from core ideas in representation learning, strip them of numerical principles and re-invent them from symbolic principles.

“Only a few approaches (Dumanˇ ci´ c & Blockeel, 2017; Dumanˇ ci´ c et al., 2019; Sourek et al., 2018) start from the core ideas in representation learning, strip them of numerical principles and re-invent them from symbolic principles.”

• cites Dumanˇ ci´ c & Blockeel, 2017Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Dumanˇ ci´ c et al., 2019Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Sourek et al., 2018Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

ILP inherits one of the major limitations of logic programs and logical foundations of knowledge representation: the inability to handle uncertain or incorrect BK.

“As ILP builds upon logic programs and logical foundations of knowledge representation, ILP also inherits one of their major limitations: the inability to handle uncertain or incorrect BK.”

Approaches that force neural networks to invent symbolic constructs, such as ∂ ILP and neural theorem provers, do so by sacrificing the expressivity of logic (they can only learn short Datalog programs).

“Approaches that force neural networks to invent symbolic constructs, such as ∂ ILP and neural theorem provers (Rockt¨ aschel & Riedel, 2017), do so by sacrificing the expressivity of logic (they can only learn short Datalog programs).”

• cites Rockt¨ aschel & Riedel, 2017Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

There have been major advances in many of the directions proposed by Muggleton et al. (2012), including in PI, higher-order logic as representation language and for hypotheses, and applications in learning actions and strategies.

“There have since been major advances in many of these directions, including in PI (Section 5.5), using higher-order logic as a representation language (Section 4.4.2) and for hypotheses (Section 4.2.3), and applications in learning actions and strategies (Section 7).”

• cites Muggleton et al., 2012Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Because of its symbolic representation, a key advantage of ILP is that learned knowledge can be remembered and explicitly stored in the BK.

“Because of its symbolic representation, a key advantage of ILP is that learned knowledge can be remembered and explicitly stored in the BK.”

There is little interaction between PI and representation learning.

“Despite strong connections, there is little interaction between PI and representation learning.”

Neural-ILP techniques make the ILP problem amenable to numerical optimisation by assuming that the entire hypothesis space is the solution program, instead of one member of it.

“Neural-ILP techniques make the ILP problem amenable to numerical optimisation by assuming that the entire hypothesis space is the solution program, instead of one member of it.”

As such a solution program would be able to entail almost anything, neural-ILP techniques relax the notion of entailment.

“As such a solution program would be able to entail almost anything, neural-ILP techniques relax the notion of entailment.”

The latter approaches only apply to propositional learning tasks, not to first-order program induction tasks where infinite domains are impossible to propositionalise.

“A disadvantage of the latter approaches is that they only apply to propositional learning tasks, not to first-order program induction tasks where infinite domains are impossible to propositionalise.”

If a user cannot provide suitable metarules (which is often the case), then Metagol is almost useless.

“However, if a user cannot provide suitable metarules (which is often the case), then Metagol is almost useless.”

The same brittleness applies to ILP systems that employ mode declarations.

“This same brittleness applies to ILP systems that employ mode declarations (Section 4.4.1).”

A more common approach is to transform relational data into a propositional tabular form that can be used as an input to a neural network.

“A more common approach is to transform relational data into a propositional tabular form that can be used as an input to a neural network (Dash et al., 2018; Kaur et al., 2019, 2020).”

• cites Dash et al., 2018 sourceNot checked — no full text of the source
• cites Kaur et al., 2019, 2020Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

In practice, weak mode declarations often lead to very poor performance.

“In practice, however, weak mode declarations often lead to very poor performance.”

New methods for PI have improved the ability of ILP to invent high-level concepts.

“New methods for PI (Section 5.5) have improved the ability of ILP to invent such high-level concepts.”

• AssumptionRests on an assumption: Reaching human-level AI is a desirable goal.

The need for an almost perfect language bias is severely holding back ILP from being widely adopted.

“This need for an almost perfect language bias is severely holding back ILP from being widely adopted.”

• AssumptionRests on an assumption: Wider adoption of ILP both inside and outside academia is desirable.

Because ILP uses logic programming as a uniform representation for examples, BK, and hypotheses, it can easily be applied to arbitrary domains.

“By contrast, because ILP uses logic programming as a uniform representation for examples, BK, and hypotheses, it can easily be applied to arbitrary domains.”

StarAI formalisms allow a user to explicitly quantify the confidence in the correctness of the BK by annotating parts of BK with probabilities.

“StarAI formalisms allow a user to explicitly quantify the confidence in the correctness of the BK by annotating parts of BK with probabilities.”

The majority of existing neural-ILP systems reframe the ILP problem as structure learning by parameter learning.

“The majority of existing neural-ILP systems reframe the ILP problem as structure learning by parameter learning .”

Most ILP approaches assume noiseless BK, i.e. a relation is true or false without any room for uncertainty.

“This issue is especially problematic because, as mentioned in Section 5.1, most ILP approaches assume noiseless BK, i.e. a relation is true or false without any room for uncertainty.”

This situation leads to the problem of catastrophic remembering: the inability for a learner to forget knowledge.

“This situation leads to the problem of catastrophic remembering (Cropper, 2020): the inability for a learner to forget knowledge.”

• cites Cropper, 2020Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionRests on an assumption: Reaching human-level AI is a desirable goal.

The empirical efficacy of neural relevance scoring for BK has yet to be clearly demonstrated.

“However, the empirical efficacy of this approach has yet to be clearly demonstrated.”

Handling the complexity of thousands of tasks is challenging because ILP systems struggle to handle large amounts of BK.

“But handling the complexity of thousands of tasks is challenging because, as we explained in Section 4.3, ILP systems struggle to handle large amounts of BK.”

• AssumptionRests on an assumption: Reaching human-level AI is a desirable goal.

Muggleton et al. (2018) empirically demonstrate improved human understanding directly through learned hypotheses.

“Muggleton et al. (2018) empirically demonstrate improved human understanding directly through learned hypotheses.”

• cites Muggleton et al., 2018Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.