Inductive logic programming
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. 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. Inductive logic programming is particularly useful in bioinformatics and natural language processing. Gordon Plotkin and Ehud Shapiro laid the initial theoretical foundation for inductive machine learning in a logical setting. Shapiro built  their first implementation (Model Inference System) in 1981: a Prolog program that inductively inferred logic programs from positive and negative examples. The term "Inductive Logic Programming" was first introduced in a paper by Stephen Muggleton in 1991. Muggleton also founded the annual international conference on Inductive Logic Programming, introduced the theoretical ideas of Predicate Invention, Inverse resolution, and Inverse entailment. Muggleton implemented Inverse entailment first in the PROGOL system. 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 . Formal definition. The "background knowledge" is given as a logic theory , commonly in the form of Horn clauses used in logic programming. The "positive" and "negative" examples are given as a conjunction formula_1 and formula_2 of unnegated and negated ground literals, respectively. A "correct hypothesis" is a logic proposition satisfying the following requirements. ""Necessity"" does not impose a restriction on , but forbids any generation of a hypothesis as long as the positive facts are explainable without it. ""Sufficiency"" requires any generated hypothesis to explain all positive examples formula_1. ""Weak consistency"" forbids generation of any hypothesis that contradicts the background knowledge  . ""Strong consistency"" also forbids generation of any hypothesis that is inconsistent with the negative examples formula_2, given the background knowledge ; it implies ""Weak consistency""; if no negative examples are given, both requirements coincide. Džeroski requires only ""Sufficiency"" (called "Completeness" there) and ""Strong consistency"". Example. The following well-known example about learning definitions of family relations uses the abbreviations It starts from the background knowledge (cf. picture) the positive examples and the trivial proposition to denote the absence of negative examples. Plotkin's ""relative least general generalization (rlgg)"" approach to "inductive logic programming" shall be used to obtain a suggestion about  how to formally define the daughter relation . This approach uses the following steps. The resulting Horn clause is the hypothesis obtained by the rlgg approach. Ignoring the background knowledge facts, the clause informally reads ""formula_28 is called a daughter of formula_29 if formula_29 is the parent of formula_28 and formula_28 is female"", which is a commonly accepted definition. Concerning the above requirements, ""Necessity"" was satisfied because the predicate doesn't appear in the background knowledge, which hence cannot imply any property containing this predicate, such as the positive examples are. ""Sufficiency"" is satisfied by the computed hypothesis , since it , together with formula_33 from the background knowledge, implies the first positive example formula_11, and similarly and formula_35 from the background knowledge implies the second positive example formula_12. ""Weak consistency"" is satisfied by , since holds in the (finite) Herbrand structure described by the background knowledge; similar for ""Strong consistency"". The common definition of the grandmother relation, viz. formula_37, cannot be learned using the above approach, since the variable occurs in the clause body only; the corresponding literals would have been deleted in the 4th step of the approach. To overcome this flaw, that step has to be modified such that  it can be parametrized with different "literal post-selection heuristics". Historically, the GOLEM implementation is based on the rlgg approach. Inductive Logic Programming system. Inductive Logic Programming system is a program that takes as an input logic theories formula_38 and outputs a correct hypothesis wrt theories formula_38 An algorithm of an ILP system consists of two parts: hypothesis search and hypothesis selection. First a hypothesis is searched with an inductive logic programming procedure, then a subset of the found hypotheses (in most systems one hypothesis) is chosen by a selection algorithm. A selection algorithm scores each of the found hypotheses  and returns the ones with the highest score. An example of score function include minimal compression length where a hypothesis with a lowest Kolmogorov complexity has the highest score and is returned. An ILP system is complete iff for any input logic theories formula_38 any correct hypothesis wrt to these input theories can be found with its hypothesis search procedure. Hypothesis search. Modern ILP systems like Progol, Hail and Imparo find a hypothesis using the principle of the inverse entailment for theories , , : formula_41. First they construct an intermediate theory called a bridge theory satisfying the conditions formula_42 and formula_43. Then  as formula_44, they generalize the negation of the bridge theory with the anti-entailment. However, the operation of the anti-entailment since being highly non-deterministic is computationally more expensive. Therefore, an alternative hypothesis search can be conducted using the operation of the inverse subsumption (anti-subsumption) instead which is less non-deterministic than anti-entailment. Questions of completeness of a hypothesis search procedure of specific ILP system arise. For example, Progol's 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.