General-Specific Ordering

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A General-Specific Ordering is a distance metric based on generalization and Specialization measures.

References

2018a

  • (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Anti-unification_(computer_science)#Generalization,_specialization Retrieved:2018-4-8.
    • If a term [math]\displaystyle{ t }[/math] has an instance equivalent to a term [math]\displaystyle{ u }[/math], that is, if [math]\displaystyle{ t \sigma \equiv u }[/math] for some substitution [math]\displaystyle{ \sigma }[/math], then [math]\displaystyle{ t }[/math] is called more general than [math]\displaystyle{ u }[/math] , and [math]\displaystyle{ u }[/math] is called more special than, or subsumed by, [math]\displaystyle{ t }[/math] . For example, [math]\displaystyle{ x \oplus a }[/math] is more general than [math]\displaystyle{ a \oplus b }[/math] if [math]\displaystyle{ \oplus }[/math] is commutative, since then [math]\displaystyle{ (x \oplus a)\{x \mapsto b\} = b \oplus a \equiv a \oplus b }[/math] .

      If [math]\displaystyle{ \equiv }[/math] is literal (syntactic) identity of terms, a term may be both more general and more special than another one only if both terms differ just in their variable names, not in their syntactic structure; such terms are called variants, or renamings of each other.

      For example, [math]\displaystyle{ f(x_1,a,g(z_1),y_1) }[/math] is a variant of [math]\displaystyle{ f(x_2,a,g(z_2),y_2) }[/math] , since [math]\displaystyle{ f(x_1,a,g(z_1),y_1) \{ x_1 \mapsto x_2, y_1 \mapsto y_2, z_1 \mapsto z_2\} = f(x_2,a,g(z_2),y_2) }[/math] and [math]\displaystyle{ f(x_2,a,g(z_2),y_2) \{x_2 \mapsto x_1, y_2 \mapsto y_1, z_2 \mapsto z_1\} = f(x_1,a,g(z_1),y_1) }[/math] .

      However, [math]\displaystyle{ f(x_1,a,g(z_1),y_1) }[/math] is not a variant of [math]\displaystyle{ f(x_2,a,g(x_2),x_2) }[/math] , since no substitution can transform the latter term into the former one, although [math]\displaystyle{ \{x_1 \mapsto x_2, z_1 \mapsto x_2, y_1 \mapsto x_2 \} }[/math] achieves the reverse direction.

      The latter term is hence properly more special than the former one.

      A substitution [math]\displaystyle{ \sigma }[/math] is more special than, or subsumed by, a substitution [math]\displaystyle{ \tau }[/math] if [math]\displaystyle{ x \sigma }[/math] is more special than [math]\displaystyle{ x \tau }[/math] for each variable [math]\displaystyle{ x }[/math] .

      For example, [math]\displaystyle{ \{ x \mapsto f(u), y \mapsto f(f(u)) \} }[/math] is more special than [math]\displaystyle{ \{ x \mapsto z, y \mapsto f(z) \} }[/math] , since [math]\displaystyle{ f(u) }[/math] and [math]\displaystyle{ f(f(u)) }[/math] is more special than [math]\displaystyle{ z }[/math] and [math]\displaystyle{ f(z) }[/math] , respectively.

2018b

  • (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Anti-unification_(computer_science)#Anti-unification_problem Retrieved:2018-4-8.
    • An anti-unification problem is a pair [math]\displaystyle{ \langle t_1, t_2 \rangle }[/math] of terms.

      A term [math]\displaystyle{ t }[/math] is a common generalization, or anti-unifier, of [math]\displaystyle{ t_1 }[/math] and [math]\displaystyle{ t_2 }[/math] if [math]\displaystyle{ t \sigma_1 \equiv t_1 }[/math] and [math]\displaystyle{ t \sigma_2 \equiv t_2 }[/math] for some substitutions [math]\displaystyle{ \sigma_1, \sigma_2 }[/math] .

      For a given anti-unification problem, a set [math]\displaystyle{ S }[/math] of anti-unifiers is called complete if each generalization subsumes some term [math]\displaystyle{ t \in S }[/math] ; the set [math]\displaystyle{ S }[/math] is called minimal if none of its members subsumes another one.

2017a

  • (Sammut, 2017) ⇒ Sammut, C. (2017) https://link.springer.com/referenceworkentry/10.1007/978-1-4899-7687-1_327 "Generalization". . In: Sammut, C., Webb, G.I. (eds) [Encyclopedia of Machine Learning and Data Mining]. Springer, Boston, MA
    • QUOTE: A hypothesis, h, is a predicate that maps an instance to true or false. That is, if [math]\displaystyle{ h(x) }[/math] is true, then [math]\displaystyle{ x }[/math] is hypothesized to belong to the concept being learned, the target. Hypothesis, [math]\displaystyle{ h_1 }[/math], is more general than or equal to [math]\displaystyle{ h_2 }[/math], if [math]\displaystyle{ h_1 }[/math] covers at least as many examples as [math]\displaystyle{ h_2 }[/math] (Mitchell, 1997 [1]). That is, [math]\displaystyle{ h_1 \geq h_2 }[/math] if and only if

      [math]\displaystyle{ (\forall x)[h_1(x) \rightarrow h_2(x)] }[/math]

2017b

2017c

2008

1997

  1. ↑ Mitchell, T., Buchanan, B., DeJong, G., Dietterich, T., Rosenbloom, P., & Waibel, A. (1990). Machine learning. Annual Review of Computer Science, 4(1), 417-433.
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