Английская Википедия:Golem (ILP)

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Golem is an inductive logic programming algorithm developed by Stephen Muggleton and Cao Feng in 1990. It uses the technique of relative least general generalisation proposed by Gordon Plotkin, leading to a bottom-up search through the subsumption lattice. In 1992, shortly after its introduction, Golem was considered the only inductive logic programming system capable of scaling to tens of thousands of examples.

Description

Golem takes as input a definite program Шаблон:Mvar as background knowledge together with sets of positive and negative examples, denoted <math display="inline">E^{+}</math> and <math display="inline">E^{-}</math> respectively. The overall idea is to construct the least general generalisation of <math display="inline">E^{+}</math> with respect to the background knowledge. However, if Шаблон:Mvar is not merely a finite set of ground atoms, then this relative least general generalisation may not exist. Therefore, rather than using Шаблон:Mvar directly, Golem uses the set <math display="inline">B^{h}</math> of all ground atoms that can be resolved from Шаблон:Mvar in at most Шаблон:Mvar resolution steps. An additional difficulty is that if <math display="inline">E^{-}</math> is non-empty, the least general generalisation of <math display="inline">E^{+}</math> may entail a negative example. In this case, Golem generalises different subsets of <math display="inline">E^{+}</math> separately to obtain a program of several clauses. Golem also employs some restrictions on the hypothesis space, ensuring that relative least general generalisations are polynomial in the number of training examples. Golem demands that all variables in the head of a clause also appears in a literal of the clause body; that the number of substitutions needed to instantiate existentially quantified variables introduced in a literal is bounded; and that the depth of the chain of substitutions needed to instantiate such a variable is also bounded.

Example

Файл:Family relations example for inductive logic programming article.gif

Assumed family relations

The following example about learning definitions of family relations uses the abbreviations

Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math, and Шаблон:Math.

It starts from the background knowledge (cf. picture)

<math>\textit{par}(h,m) \land \textit{par}(h,t) \land \textit{par}(g,m) \land \textit{par}(t,e) \land \textit{par}(n,e) \land \textit{fem}(h) \land \textit{fem}(m) \land \textit{fem}(n) \land \textit{fem}(e)</math>,

the positive examples

<math>\textit{dau}(m,h) \land \textit{dau}(e,t)</math>,

and the trivial proposition Шаблон:Mvar to denote the absence of negative examples.

The relative least general generalisation is now computed as follows to obtain a definition of the daughter relation.

• Relativise each positive example literal with the complete background knowledge:

  <math>\begin{align}

\textit{dau}(m,h) \leftarrow \textit{par}(h,m) \land \textit{par}(h,t) \land \textit{par}(g,m) \land \textit{par}(t,e) \land \textit{par}(n,e) \land \textit{fem}(h) \land \textit{fem}(m) \land \textit{fem}(n) \land \textit{fem}(e) \\ \textit{dau}(e,t) \leftarrow \textit{par}(h,m) \land \textit{par}(h,t) \land \textit{par}(g,m) \land \textit{par}(t,e) \land \textit{par}(n,e) \land \textit{fem}(h) \land \textit{fem}(m) \land \textit{fem}(n) \land \textit{fem}(e) \end{align}</math>,

• Convert into clause normal form:

  <math>\begin{align}

\textit{dau}(m,h) \lor \lnot \textit{par}(h,m) \lor \lnot \textit{par}(h,t) \lor \lnot \textit{par}(g,m) \lor \lnot \textit{par}(t,e) \lor \lnot \textit{par}(n,e) \lor \lnot \textit{fem}(h) \lor \lnot \textit{fem}(m) \lor \lnot \textit{fem}(n) \lor \lnot \textit{fem}(e) \\ \textit{dau}(e,t) \lor \lnot \textit{par}(h,m) \lor \lnot \textit{par}(h,t) \lor \lnot \textit{par}(g,m) \lor \lnot \textit{par}(t,e) \lor \lnot \textit{par}(n,e) \lor \lnot \textit{fem}(h) \lor \lnot \textit{fem}(m) \lor \lnot \textit{fem}(n) \lor \lnot \textit{fem}(e) \end{align}</math>,

• Anti-unify each compatible pair of literals:
  • <math>\textit{dau}(x_{me},x_{ht})</math> from <math>\textit{dau}(m,h)</math> and <math>\textit{dau}(e,t)</math>,
  • <math>\lnot \textit{par}(x_{ht},x_{me})</math> from <math>\lnot \textit{par}(h,m)</math> and <math>\lnot \textit{par}(t,e)</math>,
  • <math>\lnot \textit{fem}(x_{me})</math> from <math>\lnot \textit{fem}(m)</math> and <math>\lnot \textit{fem}(e)</math>,
  • <math>\lnot \textit{par}(g,m)</math> from <math>\lnot \textit{par}(g,m)</math> and <math>\lnot \textit{par}(g,m)</math>, similar for all other background-knowledge literals
  • <math>\lnot \textit{par}(x_{gt},x_{me})</math> from <math>\lnot \textit{par}(g,m)</math> and <math>\lnot \textit{par}(t,e)</math>, and many more negated literals
• Delete all negated literals containing variables that don't occur in a positive literal:
  • after deleting all negated literals containing other variables than <math>x_{me},x_{ht}</math>, only <math>\textit{dau}(x_{me},x_{ht}) \lor \lnot \textit{par}(x_{ht},x_{me}) \lor \lnot \textit{fem}(x_{me})</math> remains, together with all ground literals from the background knowledge
• Convert clauses back to Horn form:
  • <math>\textit{dau}(x_{me},x_{ht}) \leftarrow \textit{par}(x_{ht},x_{me}) \land \textit{fem}(x_{me}) \land (\text{all background knowledge facts})</math>

The resulting Horn clause is the hypothesis Шаблон:Mvar obtained by Golem. Informally, the clause reads "<math>x_{me}</math> is called a daughter of <math>x_{ht}</math> if <math>x_{ht}</math> is the parent of <math>x_{me}</math> and <math>x_{me}</math> is female", which is a commonly accepted definition.