## Automatic Methods of Inductive Inference

##### Abstract

This thesis is concerned with algorithms for generating
generalisations-from experience. These algorithms are viewed as
examples of the general concept of a hypothesis discovery system
which, in its turn, is placed in a framework in which it is seen as
one component in a multi-stage process which includes stages of
hypothesis criticism or justification, data gathering and analysis
and prediction. Formal and informal criteria, which should be
satisfied by the discovered hypotheses are given. In particular,
they should explain experience and be simple. The formal work uses
the first-order predicate calculus.
These criteria are applied to the case of hypotheses which are
generalisations from experience. A formal definition of generalisation
from experience, relative to a body of knowledge is developed and
several syntactical simplicity measures are defined. This work uses
many concepts taken from resolution theory (Robinson, 1965). We
develop a set of formal criteria that must be satisfied by any hypothesis
generated by an algorithm for producing generalisation from experience.
The mathematics of generalisation is developed. In particular,
in the case when there is no body of knowledge, it is shown that there is
always a least general generalisation of any two clauses, in the
generalisation ordering. (In resolution theory, a clause is an
abbreviation for a disjunction of literals.) This least general
generalisation is effectively obtainable. Some lattices induced by the generalisation ordering, in the case
where there is no body of knowledge, are investigated.
The formal set of criteria is investigated. It is shown that for
a certain simplicity measure, and under the assumption that there is no
body of knowledge, there always exist hypotheses which satisfy them.
Generally, however, there is no algorithm which, given the sentences
describing experience, will produce as output a hypothesis satisfying
the formal criteria. These results persist for a wide range of other
simplicity measures. However several useful cases for which algorithms
are available are described, as are some general properties of the set of
hypotheses which satisfy the criteria.
Some connections with philosophy are discussed. It is shown that,
with sufficiently large experience, in some cases, any hypothesis which
satisfies the formal criteria is acceptable in the sense of Hintikka and
Hilpinen (1966). The role of simplicity is further discussed. Some
practical difficulties which arise because of Goodman's (1965) "grue"
paradox of confirmation theory are presented.
A variant of the formal criteria suggested by the work of Meltzer
(1970) is discussed. This allows an effective method to be developed
when this was not possible before. However, the possibility is
countenanced that inconsistent hypotheses might be proposed by the
discovery algorithm.
The positive results on the existence of hypotheses satisfying the formal criteria are extended to include some simple types of knowledge.
It is shown that they cannot be extended much further without changing
the underlying simplicity ordering.
A program which implements one of the decidable cases is described.
It is used to find definitions in the game of noughts and crosses and in
family relationships.
An abstract study is made of the progression of hypothesis discovery
methods through time.
Some possible and some impossible behaviours of such methods are
demonstrated. This work is an extension of that of Gold (1967) and
Feldman (1970). The results are applied to the case of machines that
discover generalisations. They are found to be markedly sensitive to
the underlying simplicity ordering employed.