537 research outputs found
Recursion does not always help
We show that adding recursion does not increase the total functions definable
in the typed -calculus or the partial functions definable in
the -calculus. As a consequence, adding recursion does not
increase the class of partial or total definable functions on free algebras and
so, in particular, on the natural numbers.Comment: Improved presentation a littl
A Model of Cooperative Threads
We develop a model of concurrent imperative programming with threads. We
focus on a small imperative language with cooperative threads which execute
without interruption until they terminate or explicitly yield control. We
define and study a trace-based denotational semantics for this language; this
semantics is fully abstract but mathematically elementary. We also give an
equational theory for the computational effects that underlie the language,
including thread spawning. We then analyze threads in terms of the free algebra
monad for this theory.Comment: 39 pages, 5 figure
Smart Choices and the Selection Monad
Describing systems in terms of choices and their resulting costs and rewards
offers the promise of freeing algorithm designers and programmers from
specifying how those choices should be made; in implementations, the choices
can be realized by optimization techniques and, increasingly, by machine
learning methods. We study this approach from a programming-language
perspective. We define two small languages that support decision-making
abstractions: one with choices and rewards, and the other additionally with
probabilities. We give both operational and denotational semantics.
In the case of the second language we consider three denotational semantics,
with varying degrees of correlation between possible program values and
expected rewards. The operational semantics combine the usual semantics of
standard constructs with optimization over spaces of possible execution
strategies.
The denotational semantics, which are compositional and can also be viewed as
an implementation by translation to a simpler language, rely on the selection
monad, to handle choice, combined with an auxiliary monad, to handle other
effects such as rewards or probability.
We establish adequacy theorems that the two semantics coincide in all cases.
We also prove full abstraction at ground types, with varying notions of
observation in the probabilistic case corresponding to the various degrees of
correlation. We present axioms for choice combined with rewards and
probability, establishing completeness at ground types for the case of rewards
without probability
Three Inadequate Models
The connection between operational and denotational semantics is of longstanding
interest in the study of programming languages. One naturally seeks positive results.
For example in [FP94, Sim99] adequacy results are given for models in a variety of
categories. Again, the failure of full abstraction in the standard models constructed
using complete partial orders and continuous functions [Plo77, Mil77] prompted the
exploration of other categories (see, e.g., [BCL85, FJM96, AM98, AC98]) with varying
degrees of success.
In this paper we interest ourselves in counterexamples in order to make a case that
these natural avenues of research had a degree of necessity. To this end, we construct
inadequate models and investigate whether one can do better than the standard
model, but still stay in the category of complete partial orders. (In contrast, an
inadequate standard model of PCF is given in [Sim99]—but in a specially constructed
category.
Automatic Methods of Inductive Inference
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
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