Generalization

• See: Classification; Specialization; Subsumption; Logic of Generality; Generalization Operation; Inductive Argument; Predictive Classifier; Performance; Generalization Error, Clinical Generalizability Measure.

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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 ). 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

• (De Raedt, 2017b) ⇒ De Raedt L. (2017) Logic of Generality. In: Sammut, C., Webb, G.I. (eds) Encyclopedia of Machine Learning and Data Mining. Springer, Boston, MA
  • QUOTE: One hypothesis is more general than another one if it covers all instances that are also covered by the latter one. The former hypothesis is called a generalization of the latter one, and the latter a specialization of the former. When using logical formulae as hypotheses, the generality relation coincides with the notion of logical entailment, which implies that the generality relation can be analyzed from a logical perspective. The logical analysis of generality, which is pursued in this chapter, leads to the perspective of induction as the inverse of deduction. This forms the basis for an analysis of various logical frameworks for reasoning about generality and for traversing the space of possible hypotheses. Many of these frameworks (such as for instance, θ-subsumption) are employed in the field of inductive logic programming and are introduced below.

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.

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