Logic of Generality
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.
- 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] .
2017a
- (Sammut & Webb, 2017) ⇒ (2017) Specialization. In: Sammut, C., Webb, G.I. (eds) Encyclopedia of Machine Learning and Data Mining. Springer, Boston, MA
- QUOTE: Specialization is the converse of generalization. Thus, if [math]\displaystyle{ h_1 }[/math] is a generalization of [math]\displaystyle{ h_2 }[/math] then [math]\displaystyle{ h_2 }[/math] is a specialization of [math]\displaystyle{ h_1 }[/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.