Inductive logic programming (ILP) is a subfield of symbolic artificial intelligence which uses logic programming as a uniform representation for examples, background knowledge and hypotheses. The term "inductive" here refers to philosophical (i.e. suggesting a theory to explain observed facts) rather than mathematical (i.e. proving a property for all members of a well-ordered set) induction. Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesised logic program which entails all the positive and none of the negative examples. Schema: positive examples + negative examples + background knowledge ⇒ hypothesis.
Bioinformatics and drug design have been highlighted as a principal application area of inductive logic programming techniques.
History Building on earlier work on Inductive inference, Gordon Plotkin was the first to formalise induction in a clausal setting around 1970, adopting an approach of generalising from examples. In 1981, Ehud Shapiro introduced several ideas that would shape the field in his new approach of model inference, an algorithm employing refinement and backtracing to search for a complete axiomatisation of given examples. Reprinted in His first implementation was the Model Inference System in 1981: a Prolog program that inductively inferred Horn clause logic programs from positive and negative examples. The term Inductive Logic Programming was first introduced in a paper by Stephen Muggleton in 1990, defined as the intersection of machine learning and logic programming. Muggleton and Wray Buntine introduced predicate invention and inverse resolution in 1988.
Several inductive logic programming systems that proved influential appeared in the early 1990s. FOIL, introduced by Ross Quinlan in 1990 was based on upgrading propositional learning algorithms AQ and ID3. Golem, introduced by Muggleton and Feng in 1990, went back to a restricted form of Plotkin's least generalisation algorithm. See also The Progol system, introduced by Muggleton in 1995, first implemented inverse entailment, and inspired many later systems. Aleph, a descendant of Progol introduced by Ashwin Srinivasan in 2001, is still one of the most widely used systems .
At around the same time, the first practical applications emerged, particularly in bioinformatics, where by 2000 inductive logic programming had been successfully applied to drug design, carcinogenicity and mutagenicity prediction, and elucidation of the structure and function of proteins. Unlike the focus on automatic programming inherent in the early work, these fields used inductive logic programming techniques from a viewpoint of relational data mining. The success of those initial applications and the lack of progress in recovering larger traditional logic programs shaped the focus of the field.
Recently, classical tasks from automated programming have moved back into focus, as the introduction of meta-interpretative learning makes predicate invention and learning recursive programs more feasible. This technique was pioneered with the Metagol system introduced by Muggleton, Dianhuan Lin, Niels Pahlavi and Alireza Tamaddoni-Nezhad in 2014. This allows ILP systems to work with fewer examples, and brought successes in learning string transformation programs, answer set grammars and general algorithms.
Setting Inductive logic programming has adopted several different learning settings, the most common of which are learning from entailment and learning from interpretations. In both cases, the input is provided in the form of background knowledge , a logical theory (commonly in the form of clauses used in logic programming), as well as positive and negative examples, denoted and respectively. The output is given as a hypothesis , itself a logical theory that typically consists of one or more clauses.
The two settings differ in the format of examples presented.
Learning from entailment , learning from entailment is by far the most popular setting for inductive logic programming. In this setting, the positive and negative examples are given as finite sets and of positive and negated ground literals, respectively. A correct hypothesis is a set of clauses satisfying the following requirements, where the turnstile symbol stands for logical entailment:
Completeness requires any generated hypothesis to explain all positive examples , and consistency forbids generation of any hypothesis that is inconsistent with the negative examples , both given the background knowledge .
In Muggleton's setting of concept learning,; here: Sect.2.1 "completeness" is referred to as "sufficiency", and "consistency" as "strong consistency". Two further conditions are added: "Necessity", which postulates that does not entail , does not impose a restriction on , but forbids any generation of a hypothesis as long as the positive facts are explainable without it. "Weak consistency", which states that no contradiction can be derived from , forbids generation of any hypothesis that contradicts the background knowledge . Weak consistency is implied by strong consistency; if no negative examples are given, both requirements coincide. Weak consistency is particularly important in the case of noisy data, where completeness and strong consistency cannot be guaranteed.
Learning from interpretations In learning from interpretations, the positive and negative examples are given as a set of complete or partial Herbrand structures, each of which are themselves a finite set of ground literals. Such a structure is said to be a model of the set of clauses if for any substitution and any clause in such that , also holds. The goal is then to output a hypothesis that is complete, meaning every positive example is a model of , and consistent, meaning that no negative example is a model of .
Approaches to ILP An inductive logic programming system is a program that takes as an input logic theories and outputs a correct hypothesis with respect to theories . A system is complete if and only if for any input logic theories any correct hypothesis with respect to these input theories can be found with its hypothesis search procedure. Inductive logic programming systems can be roughly divided into two classes, search-based and meta-interpretative systems.
Search-based systems exploit that the space of possible clauses forms a complete lattice under the subsumption relation, where one clause subsumes another clause if there is a substitution such that , the result of applying to , is a subset of . This lattice can be traversed either bottom-up or top-down.
Bottom-up search Bottom-up methods to search the subsumption lattice have been investigated since Plotkin's first work on formalising induction in clausal logic in 1970. Techniques used include least general generalisation, based on anti-unification, and inverse resolution, based on inverting the resolution inference rule.
Least general generalisation A least general generalisation algorithm takes as input two clauses and and outputs the least general generalisation of and , that is, a clause that subsumes and , and that is subsumed by every other clause that subsumes and . The least general generalisation can be computed by first computing all selections from and , which are pairs of literals sharing the same predicate symbol and negated/unnegated status. Then, the least general generalisation is obtained as the disjunction of the least general generalisations of the individual selections, which can be obtained by first-order syntactical anti-unification.
To account for background knowledge, inductive logic programming systems employ relative least general generalisations, which are defined in terms of subsumption relative to a background theory. In general, such relative least general generalisations are not guaranteed to exist; however, if the background theory is a finite set of ground literals, then the negation of is itself a clause. In this case, a relative least general generalisation can be computed by disjoining the negation of with both and and then computing their least general generalisation as before.
Relative least general generalisations are the foundation of the bottom-up system Golem.
Inverse resolution Inverse resolution is an inductive reasoning technique that involves inverting the resolution operator.
Inverse resolution takes information about the resolvent of a resolution step to compute possible resolving clauses. Two types of inverse resolution operator are in use in inductive logic programming: V-operators and W-operators. A V-operator takes clauses and as input and returns a clause such that is the resolvent of and . A W-operator takes two clauses and and returns three clauses , and such that is the resolvent of and and is the resolvent of and .
Inverse resolution was first introduced by Stephen Muggleton and Wray Buntine in 1988 for use in the inductive logic programming system Cigol. By 1993, this spawned a surge of research into inverse resolution operators and their properties.
Top-down search The ILP systems Progol, Hail and Imparo find a hypothesis using the principle of the inverse entailment for theories , , : . First they construct an intermediate theory called a bridge theory satisfying the conditions and . Then as , they generalize the negation of the bridge theory with anti-entailment. However, the operation of anti-entailment is computationally more expensive since it is highly nondeterministic. Therefore, an alternative hypothesis search can be conducted using the inverse subsumption (anti-subsumption) operation instead, which is less non-deterministic than anti-entailment.
Questions of completeness of a hypothesis search procedure of specific inductive logic programming system arise. For example, the Progol hypothesis search procedure based on the inverse entailment inference rule is not complete by Yamamoto's example. On the other hand, Imparo is complete by both anti-entailment procedure and its extended inverse subsumption procedure.
Metainterpretive learning Rather than explicitly searching the hypothesis graph, metainterpretive or meta-level systems encode the inductive logic programming program as a meta-level logic program which is then solved to obtain an optimal hypothesis. Formalisms used to express the problem specification include Prolog and answer set programming, with existing Prolog systems and answer set solvers used for solving the constraints.
And example of a Prolog-based system is Metagol, which is based on a meta-interpreter in Prolog, while ASPAL and ILASP are based on an encoding of the inductive logic programming problem in answer set programming.
Evolutionary learning Evolutionary algorithms in ILP use a population-based approach to evolve hypotheses, refining them through selection, crossover, and mutation. Methods like EvoLearner have been shown to outperform traditional approaches on structured machine learning benchmarks.
List of implementations 1BC and 1BC2: first-order naive Bayesian classifiers: ACE (A Combined Engine) Aleph Atom
DL-Learner DMax FastLAS (Fast Learning from Answer Sets) FOIL (First Order Inductive Learner) Golem ILASP (Inductive Learning of Answer Set Programs) Imparo Inthelex (INcremental THEory Learner from EXamples) Lime Metagol
MIS (Model Inference System) by Ehud Shapiro Ontolearn Popper PROGOL RSD Warmr (now included in ACE) ProGolem
Probabilistic inductive logic programming Probabilistic inductive logic programming adapts the setting of inductive logic programming to learning probabilistic logic programs. It can be considered as a form of statistical relational learning within the formalism of probabilistic logic programming.
Given
background knowledge as a probabilistic logic program , and a set of positive and negative examples and