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(3) When learning executes, some instances are extracted from instance space.
The set constructed by these instances is referred to as positive instance set. Then,
other instances are extracted from instance space, which is referred to as negative
instance set.
(4) If a concept A, which includes positive instance set completely and whose
intersection with negative instance set is empty set, can be found in limited steps,
A is the individual concept for learning and learning is success, otherwise fail.
(5) If a definite algorithm exists so that learning is always successful for any
given positive and negative instance set, the instance space is called acquisitive in
the form of the language.
The representative approaches of inductive learning include version space,
AQ11 algorithm, decision tree, etc. which will be discussed in this chapter
respectively.
7.2 Logic Foundation of Inductive Learning
7.2.1
Inductive general paradigm
In order to depict conceptual inductive learning concretely, the general paradigm
of inductive learning is given here (Michalski, 1983).
Given:
(1) premise statements (facts), F, it is knowledge related to individual object in
an object category or partial features of an object.
(2) a tentative inductive asertion (which may be empty), it is generalization item
or generalization description about objects.
(3) background knowledge that defines assumptions and constraints on observing
statements and candidate inductive hypotheses, assertions generated by them,
including any related general or domain specific knowledge.
Find:
an inductive assertion (hypothesis),
, that is tautology or weak implication
observing statements which should meet background knowledge.
An hypothesis
H
H
tautological implicit fact
F
which means that
F
is the logic
reasoning of
H
i.e.
H
¼
F
holds. That is, if expression
H
¼
F
is always true in
H
|
any explanation, it can be represented as follows:
F
,
H is
specialized to
F
or
F
H.
Here, the procedure reasoning
F
from
H
is a tautological procedure. Since
H
¼
|<
H
.
F is
summed up or generalized to
F
must hold according to patterns mentioned above, so if
H
is true,
F
must be
true. On the contrary, the procedure reasoning
H
from fact
F
is a non-tautological
procedure. That is, if fact
F
is false,
H
must be false.
