Generalization

From GM-RKB
(Redirected from generalisation)
Jump to navigation Jump to search


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] , resp

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

2009

  • http://www.uky.edu/~rosdatte/phi120/glossary.htm
    • generalization: an argument in which a conclusion is drawn about a group on the basis of characteristics of a sample of the group.
  • http://www.philosophy.uncc.edu/mleldrid/logic/logiglos.html
    • Empirical Generalization: Empirical (or inductive) generalizations are general statements based upon experience. Most student desks in older classroom buildings at UNC Charlotte have gum stuck underneath the desk tops. A good generalization will be developed from a large number of varied experiences. For instance, one could offer as a justification for the previous generalization: I've looked underneath several desks in several classrooms. Generalizations drawn from a small number of instances or from anecdotal evidence are said to be hasty generalizations.
  • http://clopinet.com/isabelle/Projects/ETH/Exam_Questions.html
    • generalization: The capability the a predictive system f(x) has to make "good" predictions on examples that were not used for training.

  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.