41 research outputs found
Mathematical applications of inductive logic programming
Accepted versio
Automated theory formation in pure mathematics
The automation of specific mathematical tasks such as theorem proving and algebraic
manipulation have been much researched. However, there have only been a few isolated
attempts to automate the whole theory formation process. Such a process involves
forming new concepts, performing calculations, making conjectures, proving theorems
and finding counterexamples. Previous programs which perform theory formation are
limited in their functionality and their generality. We introduce the HR program
which implements a new model for theory formation. This model involves a cycle of
mathematical activity, whereby concepts are formed, conjectures about the concepts
are made and attempts to settle the conjectures are undertaken.HR has seven general production rules for producing a new concept from old ones and
employs a best first search by building new concepts from the most interesting old
ones. To enable this, HR has various measures which estimate the interestingness of a
concept. During concept formation, HR uses empirical evidence to suggest conjectures
and employs the Otter theorem prover to attempt to prove a given conjecture. If this
fails, HR will invoke the MACE model generator to attempt to disprove the conjecture
by finding a counterexample. Information and new knowledge arising from the attempt
to settle a conjecture is used to assess the concepts involved in the conjecture, which
fuels the heuristic search and closes the cycle.The main aim of the project has been to develop our model of theory formation and
to implement this in HR. To describe the project in the thesis, we first motivate
the problem of automated theory formation and survey the literature in this area.
We then discuss how HR invents concepts, makes and settles conjectures and how
it assesses the concepts and conjectures to facilitate a heuristic search. We present
results to evaluate HR in terms of the quality of the theories it produces and the
effectiveness of its techniques. A secondary aim of the project has been to apply HR to
mathematical discovery and we discuss how HR has successfully invented new concepts
and conjectures in number theory
A global workspace framework for combined reasoning
Artificial Intelligence research has produced
many effective techniques for solving a wide range
of problems. Practitioners tend to concentrate their efforts in one particular problem solving
paradigm and, in the main, AI research describes new methods for solving particular types of
problems or improvements in existing approaches. By contrast, much less research has considered
how to fruitfully combine different problem solving techniques. Numerous studies have
demonstrated how a combination of reasoning approaches can improve the effectiveness of one of
those methods. Others have demonstrated how, by using several different reasoning techniques,
a system or method can be developed to accomplish a novel task, that none of the individual
techniques could perform. Combined reasoning systems, i.e., systems which apply disparate
reasoning techniques in concert, can be more than the sum of their parts. In addition, they
gain leverage from advances in the individual methods they encompass. However, the benefits
of combined reasoning systems are not easily accessible, and systems have been hand-crafted
to very specific tasks in certain domains. This approach means those systems often suffer from
a lack of clarity of design and are inflexible to extension. In order for the field of combined reasoning
to advance, we need to determine best practice and identify effective general approaches.
By developing useful frameworks, we can empower researchers to explore the potential of combined
reasoning, and AI in general. We present here a framework for developing combined
reasoning systems, based upon Baarsâ Global Workspace Theory. The architecture describes a
collection of processes, embodying individual reasoning techniques, which communicate via a
global workspace. We present, also, a software toolkit which allows users to implement systems
according to the framework. We describe how, despite the restrictions of the framework, we
have used it to create systems to perform a number of combined reasoning tasks. As well
as being as effective as previous implementations, the simplicity of the underlying framework
means they are structured in a straightforward and comprehensible manner. It also makes the
systems easy to extend to new capabilities, which we demonstrate in a number of case studies.
Furthermore, the framework and toolkit we describe allow developers to harness the parallel
nature of the underlying theory by enabling them to readily convert their implementations into
distributed systems. We have experimented with the framework in a number of application domains
and, through these applications, we have contributed to constraint satisfaction problem
solving and automated theory formation
Automated conjecturing III : property-relations conjectures
Discovery in mathematics is a prototypical intelligent behavior, and an early and continuing goal of artificial intelligence research. We present a heuristic for producing mathematical conjectures of a certain typical form and demonstrate its utility. Our program conjectures relations that hold between properties of objects (property-relation conjectures). These objects can be of a wide variety of types. The statements are true for all objects known to the program, and are the simplest statements which are true of all these objects. The examples here include new conjectures for the hamiltonicity of a graph, a well-studied property of graphs. While our motivation and experiments have been to produce mathematical conjectures-and to contribute to mathematical research-other kinds of interesting property-relation conjectures can be imagined, and this research may be more generally applicable to the development of intelligent machinery
A Computational Model of Lakatos-style Reasoning
Institute for Computing Systems ArchitectureLakatos outlined a theory of mathematical discovery and justification,
which suggests ways in which concepts, conjectures and proofs
gradually evolve via interaction between mathematicians. Different
mathematicians may have different interpretations of a conjecture,
examples or counterexamples of it, and beliefs regarding its value or
theoremhood. Through discussion, concepts are refined and conjectures
and proofs modified. We hypothesise that: (i) it is possible to
computationally represent Lakatos's theory, and (ii) it is
useful to do so. In order to test our hypotheses we have developed a
computational model of his theory.
Our model is a multiagent dialogue system. Each agent has a copy of a
pre-existing theory formation system, which can form concepts and make
conjectures which empirically hold for the objects of interest
supplied. Distributing the objects of interest between agents means
that they form different theories, which they communicate to each
other. Agents then find counterexamples and use methods identified by
Lakatos to suggest modifications to conjectures, concept definitions
and proofs.
Our main aim is to provide a computational reading of Lakatos's
theory, by interpreting it as a series of algorithms and implementing
these algorithms as a computer program.
This is the first systematic automated realisation of Lakatos's
theory. We contribute to the computational philosophy of science by
interpreting, clarifying and extending his theory. We also contribute
by evaluating his theory, using our model to test hypotheses about it,
and evaluating our extended computational theory on the basis of
criteria proposed by several theorists. A further contribution is to
automated theory formation and automated theorem proving. The process
of refining conjectures, proofs and concept definitions requires a
flexibility which is inherently useful in fields which handle
ill-specified problems, such as theory formation. Similarly, the
ability to automatically modify an open conjecture into one which can
be proved, is a valuable contribution to automated theorem proving
A creativity support system based on causal mapping.
Theory development is a very complex process that requires creativity and highly specialized analytical skills. This article presents a new algorithm, based on causal mapping, for assisting in the creation of qualitative theories. This algorithm is able to conjecture and prove new theorems, to test for consistency and completeness of the theory, and to derive meta-theorems comparing the different concepts in it. The use of the algorithm is exemplified in developing a theory to explain structural inertia in organizations