Abstract
We present a method to prune hypothesis spaces in the context of inductive logic programming. The main strategy of our method consists in removing hypotheses that are equivalent to already considered hypotheses. The distinguishing feature of our method is that we use learned domain theories to check for equivalence, in contrast to existing approaches which only prune isomorphic hypotheses. Specifically, we use such learned domain theories to saturate hypotheses and then check if these saturations are isomorphic. While conceptually simple, we experimentally show that the resulting pruning strategy can be surprisingly effective in reducing both computation time and memory consumption when searching for long clauses, compared to approaches that only consider isomorphism.
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Notes
- 1.
As we show later in the paper, the completeness requirement disqualifies relative subsumption [15] as a candidate for such a pruning method.
- 2.
We use invariants based on a generalized version of Weisfeiler-Lehman procedure [22].
- 3.
This is true for the \(\theta \)-subsumption solver based on [8] which we use in our implementation.
- 4.
The formulation of Hebrand’s theorem used here is taken from notes by Cook and Pitassi: http://www.cs.toronto.edu/~toni/Courses/438/Mynotes/page39.pdf.
- 5.
Indeed, if \(\mathcal {T} \wedge \lnot C_{Sk}\) is unsatisfiable in \(\mathcal {L}\), then there is a set of corresponding \(\mathcal {L}\)-ground instances of clauses that are unsatisfiable. If we replace each constant appearing in these ground clauses which does not appear in \(C_{sk}\) by an arbitrary constant that does appear in \(C_{sk}\), then the resulting set of ground clauses must still be inconsistent, since T does not contain any constants and there is no equality in the language, meaning that \(\mathcal {T} \wedge \lnot C_{Sk}\) cannot be satisfiable in \(\mathcal {L}_{Sk}\).
- 6.
In the physical world, bonds do not necessarily have to be symmetric, e.g. there is an obvious asymmetry in polar bonds. However, it is a common simplification in data mining on molecular datasets to assume that bonds are symmetric.
- 7.
Note that we are slightly abusing notation here, as \(\theta ^{-1}\) is not a substitution.
- 8.
Note that we only use OI-subsumption to partially order the constructed hypotheses, not to check the entailment relation.
- 9.
What we call refinement operator in this paper is often called downward refinement operator. Since we only consider downward refinement operators in this paper, we omit the word downward.
- 10.
If we ordered the set of clauses by \(\theta \)-subsumption instead of OI-subsumption then there would not have to exist a maximal clause with this property.
- 11.
- 12.
A clause is said to be connected if it cannot be written as disjunction of two non-empty clauses. For instance \(\forall X,Y : p_1(X) \vee p_2(Y)\) is not connected because it can be written also as \((\forall X : p_1(X)) \vee (\forall Y : p_2(Y))\) but \(\forall X,Y : p_1(X) \vee p_2(Y) \vee p_3(X,Y)\) is connected. If a clause is connected then its saturation is also connected.
- 13.
Frequent conjunctive pattern mining can be emulated in our setting. It is enough to notice that the clauses that we construct are just negations of conjunctive patterns.
- 14.
Available from https://github.com/martinsvat.
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Acknowledgements
MS, GŠ and FŽ acknowledge support by project no. 17-26999S granted by the Czech Science Foundation. This work was done while OK was with Cardiff University and supported by a grant from the Leverhulme Trust (RPG-2014-164). SS is supported by ERC Starting Grant 637277. Computational resources were provided by the CESNET LM2015042 and the CERIT Scientific Cloud LM2015085, provided under the programme “Projects of Large Research, Development, and Innovations Infrastructures”.
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Svatoš, M., Šourek, G., Železný, F., Schockaert, S., Kuželka, O. (2018). Pruning Hypothesis Spaces Using Learned Domain Theories. In: Lachiche, N., Vrain, C. (eds) Inductive Logic Programming. ILP 2017. Lecture Notes in Computer Science(), vol 10759. Springer, Cham. https://doi.org/10.1007/978-3-319-78090-0_11
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