The use of data-mining for the automatic formation of tactics

This paper discusses the usse of data-mining for the automatic formation of tactics. It was presented at the Workshop on Computer-Supported Mathematical Theory Development held at IJCAR in 2004. The aim of this project is to evaluate the applicability of data-mining techniques to the automatic formation of tactics from large corpuses of proofs. We data-mine information from large proof corpuses to find commonly occurring patterns. These patterns are then evolved into tactics using genetic programming techniques

Validating reasoning heuristics using next generation theorem provers

The specification of enterprise information systems using formal specification languages enables the formal verification of these systems. Reasoning about the properties of a formal specification is a tedious task that can be facilitated much through the use of an automated reasoner. However, set theory is a corner stone of many formal specification languages and poses demanding challenges to automated reasoners. To this end a number of heuristics has been developed to aid the Otter theorem prover in finding short proofs for set-theoretic problems. This dissertation investigates the applicability of these heuristics to next generation theorem provers.ComputingM.Sc. (Computer Science

Formal methods and digital systems validation for airborne systems

This report has been prepared to supplement a forthcoming chapter on formal methods in the FAA Digital Systems Validation Handbook. Its purpose is as follows: to outline the technical basis for formal methods in computer science; to explain the use of formal methods in the specification and verification of software and hardware requirements, designs, and implementations; to identify the benefits, weaknesses, and difficulties in applying these methods to digital systems used on board aircraft; and to suggest factors for consideration when formal methods are offered in support of certification. These latter factors assume the context for software development and assurance described in RTCA document DO-178B, 'Software Considerations in Airborne Systems and Equipment Certification,' Dec. 1992

Hacia un autómata geómetra

Este artículo tiene como objetivo ilustrar al lector, muy particularmente a los colegas de la comunidad matemática española, de la RSME, de la evolución de nuestro grupo de investigación y de su trabajo a lo largo de los últimos 30 años, en pos de la creación de un autómata geómetra. Para ello, en las secciones siguientes se trazan unas pinceladas sobre la historia de las contribuciones del equipo de investigación creado en torno a los autores de este artículo, incluyendo, en particular, una descripción informal de los conceptos, problemas y métodos que fueron desarrollando. No pretende ser, no puede ser, una historia universal de la demostración automática de teoremas geométricos, pero creemos que el lector interesado encontrará datos relevantes a este respecto en las referencias que se incluyen a lo largo de este trabajo. Finalmente se presenta el resultado más «visible» de nuestro trabajo, el autómata geómetra AG, a través de diversos ejemplos, concluyendo con unas someras reflexiones acerca de su potencial impacto en el mundo educativo

Deductive Systems in Traditional and Modern Logic

The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic

Rethinking inconsistent mathematics

This dissertation has two main goals. The first is to provide a practice-based analysis of the field of inconsistent mathematics: what motivates it? what role does logic have in it? what distinguishes it from classical mathematics? is it alternative or revolutionary? The second goal is to introduce and defend a new conception of inconsistent mathematics - queer incomaths - as a particularly effective answer to feminist critiques of classical logic and mathematics. This sets the stage for a genuine revolution in mathematics, insofar as it suggests the need for a shift in mainstream attitudes about the rolee of logic and ethics in the practice of mathematics

The theory of inconsistency: inconsistant mathematics and paraconsistent logic

Each volume includes author's previously published papers.Bibliography: leaves 147-151 (v. 1).3 v. :Thesis (D.Sc.)--University of Adelaide, School of Mathematical Sciences, 200

Three Dogmas of First-Order Logic and some Evidence-based Consequences for Constructive Mathematics of differentiating between Hilbertian Theism, Brouwerian Atheism and Finitary Agnosticism

We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'. We then adopt what may be labelled a finitary, evidence-based, `agnostic' perspective and argue that Brouwerian atheism is merely a restricted perspective within the finitary agnostic perspective, whilst Hilbertian theism contradicts the finitary agnostic perspective. We then consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary, evidence-based, definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways. We show that the two definitions correspond to two distinctly different---not necessarily evidence-based but complementary---assignments of satisfaction and truth to the compound formulas of PA over N. We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both the complementary interpretations; and conclude some unsuspected constructive consequences of such complementarity for the foundations of mathematics, logic, philosophy, and the physical sciences

The Significance of Evidence-based Reasoning in Mathematics, Mathematics Education, Philosophy, and the Natural Sciences

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines

The development of 'for experts systems' as heuristic reasoning platforms in risk decision support: a consideration of tool design, technology transfer and compatability with Bayesian decision analysis

This work considers the creation of two risk and decision support systems, one for the National Air Traffic Services of the UK and one for Unilever, a multi-national. Their development contributes to risk decision science in the area of decision support in particular. This contribution is based on the development real-life systems, it has three key elements. One, it addresses the fact that, for practical environments like these, the science of risk and decisions is insufficiently resolved to be accepted and easily used. Two, the systems share an arena with subjective Bayesian decision analysis. The benefits of a hybrid form of the two approaches to generate higher levels of user acceptance and organisational transfer is discussed. Three, they take the unique approach of being 'for experts' systems rather than 'expert systems'. This approach offers a number of benefits to applied user communities. These include: a decision support system which remains grounded within the reasoning world view of the decision makers; an expansion and refinement of the existing 'natural heuristics' that decision makers use currently; a scoring and visualisation environment which is both fast and flexible but allows for, previously unavailable, levels of reasoning transparency and comparison. Taken in total the combination of the tool design, the heuristic artefacts within them and their influence on the hosts organisations, the two systems have proven they can provide an effective and valued 'heuristic reasoning platform' for risks and issues. A future research direction is to explore ways in which the highly transferable heuristic artefacts in these systems, particularly measurement and data manipulation, might be strengthened via hybridisation with more powerful, but less transferred, formal systems like Bayes decision analysis