Cut elimination for a simple formulation of epsilon calculus

AbstractA simple cut elimination proof for arithmetic with the epsilon symbol is used to establish the termination of a modified epsilon substitution process. This opens a possibility of extension to much stronger systems

A termination proof for epsilon substitution using partial derivations

AbstractEpsilon substitution method introduced by Hilbert is a successive approximation process providing numerical realizations from proofs of existential formulas. Most convergence (termination) proofs for it use assignments of decreasing ordinals to stages of the process and work only for predicative systems. We describe a new ordinal assignment for the case of first-order arithmetic admitting extension to impredicative systems. It is based on an interpretation of individual epsilon substitutions forming the substitution process as incomplete finite proofs, each encoding a complete but infinite proof

A Mathematical Model of Quantum Computer by Both Arithmetic and Set Theory

A practical viewpoint links reality, representation, and language to calculation by the concept of Turing (1936) machine being the mathematical model of our computers. After the Gödel incompleteness theorems (1931) or the insolvability of the so-called halting problem (Turing 1936; Church 1936) as to a classical machine of Turing, one of the simplest hypotheses is completeness to be suggested for two ones. That is consistent with the provability of completeness by means of two independent Peano arithmetics discussed in Section I. Many modifications of Turing machines cum quantum ones are researched in Section II for the Halting problem and completeness, and the model of two independent Turing machines seems to generalize them. Then, that pair can be postulated as the formal definition of reality therefore being complete unlike any of them standalone, remaining incomplete without its complementary counterpart. Representation is formal defined as a one-to-one mapping between the two Turing machines, and the set of all those mappings can be considered as “language” therefore including metaphors as mappings different than representation. Section III investigates that formal relation of “reality”, “representation”, and “language” modeled by (at least two) Turing machines. The independence of (two) Turing machines is interpreted by means of game theory and especially of the Nash equilibrium in Section IV. Choice and information as the quantity of choices are involved. That approach seems to be equivalent to that based on set theory and the concept of actual infinity in mathematics and allowing of practical implementations

Representation and Reality by Language: How to make a home quantum computer?

A set theory model of reality, representation and language based on the relation of completeness and incompleteness is explored. The problem of completeness of mathematics is linked to its counterpart in quantum mechanics. That model includes two Peano arithmetics or Turing machines independent of each other. The complex Hilbert space underlying quantum mechanics as the base of its mathematical formalism is interpreted as a generalization of Peano arithmetic: It is a doubled infinite set of doubled Peano arithmetics having a remarkable symmetry to the axiom of choice. The quantity of information is interpreted as the number of elementary choices (bits). Quantum information is seen as the generalization of information to infinite sets or series. The equivalence of that model to a quantum computer is demonstrated. The condition for the Turing machines to be independent of each other is reduced to the state of Nash equilibrium between them. Two relative models of language as game in the sense of game theory and as ontology of metaphors (all mappings, which are not one-to-one, i.e. not representations of reality in a formal sense) are deduced

Unprovability and phase transitions in Ramsey theory

The first mathematically interesting, first-order arithmetical example of incompleteness was given in the late seventies and is know as the Paris-Harrington principle. It is a strengthened form of the finite Ramsey theorem which can not be proved, nor refuted in Peano Arithmetic. In this dissertation we investigate several other unprovable statements of Ramseyan nature and determine the threshold functions for the related phase transitions. Chapter 1 sketches out the historical development of unprovability and phase transitions, and offers a little information on Ramsey theory. In addition, it introduces the necessary mathematical background by giving definitions and some useful lemmas. Chapter 2 deals with the pigeonhole principle, presumably the most well-known, finite instance of the Ramsey theorem. Although straightforward in itself, the principle gives rise to unprovable statements. We investigate the related phase transitions and determine the threshold functions. Chapter 3 explores a phase transition related to the so-called infinite subsequence principle, which is another instance of Ramsey’s theorem. Chapter 4 considers the Ramsey theorem without restrictions on the dimensions and colours. First, generalisations of results on partitioning α-large sets are proved, as they are needed later. Second, we show that an iteration of a finite version of the Ramsey theorem leads to unprovability. Chapter 5 investigates the template “thin implies Ramsey”, of which one of the theorems of Nash-Williams is an example. After proving a more universal instance, we study the strength of the original Nash-Williams theorem. We conclude this chapter by presenting an unprovable statement related to Schreier families. Chapter 6 is intended as a vast introduction to the Atlas of prefixed polynomial equations. We begin with the necessary definitions, present some specific members of the Atlas, discuss several issues and give technical details

Investigation, Development, and Evaluation of Performance Proving for Fault-tolerant Computers

A number of methodologies for verifying systems and computer based tools that assist users in verifying their systems were developed. These tools were applied to verify in part the SIFT ultrareliable aircraft computer. Topics covered included: STP theorem prover; design verification of SIFT; high level language code verification; assembly language level verification; numerical algorithm verification; verification of flight control programs; and verification of hardware logic

Hybrid semantic-document models

This thesis presents the concept of hybrid semantic-document models to aid information management when using standards for complex technical domains such as military data communication. These standards are traditionally text based documents for human interpretation, but prose sections can often be ambiguous and can lead to discrepancies and subsequent implementation problems. Many organisations produce semantic representations of the material to ensure common understanding and to exploit computer aided development. In developing these semantic representations, no relationship is maintained to the original prose. Maintaining relationships between the original prose and the semantic model has key benefits, including assessing conformance at a semantic level, and enabling original content authors to explicitly define their intentions, thus reducing ambiguity and facilitating computer aided functionality. Through the use of a case study method based on the military standard MIL-STD-6016C, a framework of relationships is proposed. These relationships can integrate with common document modelling techniques and provide the necessary functionality to allow semantic content to be mapped into document views. These relationships are then generalised for applicability to a wider context. Additionally, this framework is coupled with a templating approach which, for repeating sections, can improve consistency and further enhance quality. A reflective approach to model driven web rendering is presented and evaluated. This reflective approach uses self-inspection at runtime to read directly from the model, thus eliminating the need for any generative processes which result in data duplication across source used for different purpose

Clustering Smart Metering Data for Energy Efficiency

Nowadays, huge quantities of metering data of each consumer are being taken from the electric distribution network through smart meters and stored in databases. These metering data are consumption readings useful for many data analysis applications as, for example, fraud detection. However, a large number of possible analysis on this data are still unexplored. In this dissertation, we explore smart meters as a way of improving Energy Efficiency. To do that, we need to understand more about the way clients consume and find behavioural patterns on their consumption. The identification of all these profiles of consumption is essential since it will allow EDP Distribuição (EDPD) to know more about its types of clients, providing focused feedback and consumption advice. Clustering algorithms are useful to understand the distribution of patterns in large data sets. By creating several groups/clusters, we will be able to understand the profile of a specific client by the characteristics of its cluster. It makes it possible to categorise a customer from its group behaviour rather than expecting that each customer as it own specific profile. This allows to save a lot of time analysing the types of consumers. In this dissertation, we intend to analyse clients from several perspectives in order to capture different types of behaviours. For example, we may want to analyse the clients based on their absolute consumption values, in order to compare their scales or compare them from the consumption regularity point of view. So, we use the clustering algorithms with the appropriate features as a data mining approach to find structure in our data. The results that we present highlight the strengths and weaknesses of each clustering algorithm and validate their applicability to the EDP Distribuição (EDPD) use case. So, this dissertation will bring added knowledge about clustering techniques and analysis over smart metering data