Proof diagrams and term rewriting with applications to computational algebra

In this thesis lessons learned from the use of computer algebra systems and machine assisted theorem provers are developed in order to give an insight into both the problems and their solutions. Many algorithms in computational algebra and automated deduction (for example Grobner basis computations and Knuth-Bendix completion) tend to produce redundant facts and can contain more than one proof of any particular fact. This thesis introduces proof diagrams in order to compare and contrast the proofs of facts which such procedures generate. Proof diagrams make it possible to analyse the effect of heuristics which can be used to guide implementations of such algorithms. An extended version of an inference system for Knuth-Bendix completion is introduced. It is possible to see that this extension characterises the applicability of critical pair criteria, which are heuristics used in completion. We investigate a number of executions of a completion procedure by analysing the associated proof diagrams. This leads to a better understanding of the heuristics used to control these examples. Derived rales of inference are also investigated in this thesis. This is done in the formalism of proof diagrams. Rewrite rules for proof diagrams are defined: this is motivated by the notion of a transformation tactic in the Nuprl proof development system. A method to automatically extract 'useful' derived inference rales is also discussed. 'Off the shelf' theorem provers, such as the Larch Prover and Otter, are compared to specialised programs from computational group theory. This analysis makes it possible to see where methods from automated deduction can improve on the tools which group theorists currently use. Problems which can be attacked with theorem provers but not with currently used specialised programs are also indicated. Tietze transformations, from group theory, are discussed. This makes it possible to link ideas used in Knuth-Bendix completion programs and group presentation simplification programs. Tietze transformations provide heuristics for more efficient and effective implementations of these programs

Emergent Design

Explorations in Systems Phenomenology in Relation to Ontology, Hermeneutics and the Meta-dialectics of Design SYNOPSIS A Phenomenological Analysis of Emergent Design is performed based on the foundations of General Schemas Theory. The concept of Sign Engineering is explored in terms of Hermeneutics, Dialectics, and Ontology in order to define Emergent Systems and Metasystems Engineering based on the concept of Meta-dialectics. ABSTRACT Phenomenology, Ontology, Hermeneutics, and Dialectics will dominate our inquiry into the nature of the Emergent Design of the System and its inverse dual, the Meta-system. This is an speculative dissertation that attempts to produce a philosophical, mathematical, and theoretical view of the nature of Systems Engineering Design. Emergent System Design, i.e., the design of yet unheard of and/or hitherto non-existent Systems and Metasystems is the focus. This study is a frontal assault on the hard problem of explaining how Engineering produces new things, rather than a repetition or reordering of concepts that already exist. In this work the philosophies of E. Husserl, A. Gurwitsch, M. Heidegger, J. Derrida, G. Deleuze, A. Badiou, G. Hegel, I. Kant and other Continental Philosophers are brought to bear on different aspects of how new technological systems come into existence through the midwifery of Systems Engineering. Sign Engineering is singled out as the most important aspect of Systems Engineering. We will build on the work of Pieter Wisse and extend his theory of Sign Engineering to define Meta-dialectics in the form of Quadralectics and then Pentalectics. Along the way the various ontological levels of Being are explored in conjunction with the discovery that the Quadralectic is related to the possibility of design primarily at the Third Meta-level of Being, called Hyper Being. Design Process is dependent upon the emergent possibilities that appear in Hyper Being. Hyper Being, termed by Heidegger as Being (Being crossed-out) and termed by Derrida as Differance, also appears as the widest space within the Design Field at the third meta-level of Being and therefore provides the most leverage that is needed to produce emergent effects. Hyper Being is where possibilities appear within our worldview. Possibility is necessary for emergent events to occur. Hyper Being possibilities are extended by Wild Being propensities to allow the embodiment of new things. We discuss how this philosophical background relates to meta-methods such as the Gurevich Abstract State Machine and the Wisse Metapattern methods, as well as real-time architectural design methods as described in the Integral Software Engineering Methodology. One aim of this research is to find the foundation for extending the ISEM methodology to become a general purpose Systems Design Methodology. Our purpose is also to bring these philosophical considerations into the practical realm by examining P. Bourdieu’s ideas on the relationship between theoretical and practical reason and M. de Certeau’s ideas on practice. The relationship between design and implementation is seen in terms of the Set/Mass conceptual opposition. General Schemas Theory is used as a way of critiquing the dependence of Set based mathematics as a basis for Design. The dissertation delineates a new foundation for Systems Engineering as Emergent Engineering based on General Schemas Theory, and provides an advanced theory of Design based on the understanding of the meta-levels of Being, particularly focusing upon the relationship between Hyper Being and Wild Being in the context of Pure and Process Being

Hybrid simulation techniques in the structural analysis and testing of architectural heritage

L'abstract è presente nell'allegato / the abstract is in the attachmen

Kant, Heidegger and spacing

Kant's metaphysics of space periods is explored via his in both the pre-critical and critical relation to Leibniz, the incongruent counterparts argument, and the distinctive arguments of the Transcendental Aesthetic. Sequentially, Heidegger's phenomenology of space from the period of Being and Time is explicated in terms of concepts like deseverance, directionality, region and equipmentality. The two analyses are found to overlap on several key points. These include: the priority of the whole over the parts, openness, and exteriority and thus non-discursivity. The points of overlap we call 'spacing' . Through further analyses, it is discovered that the concepts of spacing are precisely the concepts required by these two philosophers even when they treat of subjects not normally considered essentially spacial. These subjects include the nature of temporal relations, of selfhood and self-constancy, and of the experience and significance of art. The importance of spacing for these subjects is individually discussed, as are possible reasons why the language of space should be required

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

Person-based Prominence in Ojibwe

This dissertation develops a formal and psycholinguistic theory of person-based prominence effects, the finding that certain categories of person such as first and second (the local persons) are privileged by the grammar. The thesis takes on three questions: (i) What are the possible categories related to person? (ii) What are the possible prominence relationships between these categories? And (iii) how is prominence information used to parse and interpret linguistic input in real time? The empirical through-line is understanding obviation — a “spotlighting” system, found most prominently in the Algonquian family of languages, that splits the (ani- mate) third persons into two categories: proximate, the person who is in the spotlight, and obviative, the persons who are introduced into the discourse, but are not in the spotlight. I provide a semantics for the feature [proximate], and detail a lattice-based theory of feature composition to derive the categories related to obviation in Border Lakes Ojibwe and beyond. This leads to insights about the syntactic and semantic relationships between person, animacy-based noun classification, number, and obviation. The novel contribution to the theory of person-based prominence effects is to de- compose person features into sets of primitives. This proposal allows the stipulated entailment relationships between categories and features, as encoded in prominence hierarchies and feature geometries, to be derived from the first principles of set theory. I further motivate the account by showing that it has increased empirical coverage, and apply it to capture patterns of agreement and word order in Border Lakes Ojibwe. Finally, I present a psycholinguistic study on how obviation is used to process filler- gap dependencies in Border Lakes Ojibwe. I show that obviation, and by extension, prominence information more generally, is used immediately to predictively encode movement chains, prior to bottom-up information from voice marking about the argument structure of the clause. I argue for a modular and syntax-first model of parsing, revising the Active Filler Strategy to be guided by pressures to minimize syntactic distance and maximize the expected well-formedness of each link in the chain. These pressures compete, accounting for effects of prediction, integration, and reanalysis in long-distance dependency formation