6 research outputs found
Introducing Quantified Cuts in Logic with Equality
Cut-introduction is a technique for structuring and compressing formal
proofs. In this paper we generalize our cut-introduction method for the
introduction of quantified lemmas of the form (for
quantifier-free ) to a method generating lemmas of the form . Moreover, we extend the original method to predicate
logic with equality. The new method was implemented and applied to the TSTP
proof database. It is shown that the extension of the method to handle equality
and quantifier-blocks leads to a substantial improvement of the old algorithm
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 Baars’ 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
Automatic Construction and Verification of Isotopy Invariants
We extend our previous study of the automatic construction of isomorphic classification theorems for algebraic domains by considering the isotopy equivalence relation. Isotopism is an important generalisation of isomorphism, and is studied by mathematicians in domains such as loop theory. This extension was not straightforward, and we had to solve two major technical problems, namely, generating and verifying isotopy invariants. Concentrating on the domain of loop theory, we have developed three novel techniques for generating isotopic invariants, by using the notion of universal identities and by using constructions based on subblocks. In addition, given the complexity of the theorems that verify that a conjunction of the invariants form an isotopy class, we have developed ways of simplifying the problem of proving these theorems. Our techniques employ an interplay of computer algebra, model generation, theorem proving, and satisfiability-solving methods. To demonstrate the power of the approach, we generate isotopic classification theorems for loops of size 6 and 7, which extend the previously known enumeration results. This work was previously beyond the capabilities of automated reasoning techniques
Automatic construction and verification of isotopy invariants
We extend our previous study of the automatic construction of isomorphic classification theorems for algebraic domains by considering the isotopy equivalence relation, which is of more importance than isomorphism in certain domains. This extension was not straightforward, and we had to solve two major technical problems, namely generating and verifying isotopy invariants. Concentrating on the domain of loop theory, we have developed three novel techniques for generating isotopic invariants, by using the notion of universal identities and by using constructions based on substructures. In addition, given the complexity of the theorems which verify that a conjunction of the invariants form an isotopy class, we have developed ways of simplifying the problem of proving these theorems. Our techniques employ an intricate interplay of computer algebra, model generation, theorem proving and satisfiability solving methods. To demonstrate the power of the approach, we generate an isotopic classification theorem for loops of size 6, which extends the previously known result that there are 22. This result was previously beyond the capabilities of automated reasoning techniques
Automatic construction and verification of isotopy invariants
Abstract. We extend our previous study of the automatic construction of isomorphic classification theorems for algebraic domains by considering the isotopy equivalence relation. Isotopism is an important generalisation of isomorphism, and is studied by mathematicians in domains such as loop theory. This extension was not straightforward, and we had to solve two major technical problems, namely generating and verifying isotopy invariants. Concentrating on the domain of loop theory, we have developed three novel techniques for generating isotopic invariants, by using the notion of universal identities and by using constructions based on sub-blocks. In addition, given the complexity of the theorems which verify that a conjunction of the invariants form an isotopy class, we have developed ways of simplifying the problem of proving these theorems. Our techniques employ an interplay of computer algebra, model generation, theorem proving and satisfiability solving methods. To demonstrate the power of the approach, we generate isotopic classification theorems for loops of size 6 and 7, which extends the previously known enumeration results. This result was previously beyond the capabilities of automated reasoning techniques