Dualized Simple Type Theory

We propose a new bi-intuitionistic type theory called Dualized Type Theory (DTT). It is a simple type theory with perfect intuitionistic duality, and corresponds to a single-sided polarized sequent calculus. We prove DTT strongly normalizing, and prove type preservation. DTT is based on a new propositional bi-intuitionistic logic called Dualized Intuitionistic Logic (DIL) that builds on Pinto and Uustalu's logic L. DIL is a simplification of L by removing several admissible inference rules while maintaining consistency and completeness. Furthermore, DIL is defined using a dualized syntax by labeling formulas and logical connectives with polarities thus reducing the number of inference rules needed to define the logic. We give a direct proof of consistency, but prove completeness by reduction to L.Comment: 47 pages, 10 figure

Completeness of Interacting Particles

This thesis concerns the completeness of scattering states of n _-interacting particles in one dimension. Only the repulsive case is treated, where thereare no bound states and the spectrum is entirely absolutely continuous, so the scattering Hilbert space is the whole of L2(Rn). The thesis consists of 4 chapters: The first chapter describes the model, the scattering states as given by the Bethe Ansatz, and the main completeness problem. The second chapter contains the proof of the completeness relation in the case of two particles: n = 2. This case had in fact already been treated by B. Smit (1997), [17], but it is useful to include this case as it clarifies the more general case. In particular, the more algebraic approach used for the n-particle case is illustrated in this simple example. In Chapter 3 the case n = 3 is examined. This is useful for illustrative purposes as the scattering states can still be written explicitly term by term and it is not yet necessary to introduce the complicated notation used in the general case. On the other hand, this case shows up certain technical difficulties to do with the non-commutativity of the permutation group (S3) which do not occur in the 2-particle case. Finally, Chapter 4 contains the proof of the completeness relation in the general n-particle case. The method used is the same as in the 3-particle case, but the algebra is much more complicated. In particular a number of interesting lemmas and one theorem is proved. The first lemma for 3-particle case and its generalisation - theorem for n-particle case essentially concerns the Yang-Baxter relation for this model, as first written by Yang. Indeed, Yang proposed his version of these relations as a consistency condition for the Bethe Ansatz solution of the model but never actually gave a complete proof of the consistency given these relations. Here a complete inductive proof is given. Some algebraic manipulation reduces the left-hand side of the completeness relation to a simpler form. Another lemma, which seems to be new, then shows that this expression does not contain divergent terms and consists of a sum of integrals similar to those encountered in the 2- and 3-particle cases. Evaluation of these integrals then leads to the required _-relation

Ratio coordinates for higher Teichm\"uller spaces

We define new coordinates for Fock-Goncharov's higher Teichm\"uller spaces for a surface with holes, which are the moduli spaces of representations of the fundamental group into a reductive Lie group $G$. Some additional data on the boundary leads to two closely related moduli spaces, the $\mathscr{X}$-space and the $\mathscr{A}$-space, forming a cluster ensemble. Fock and Goncharov gave nice descriptions of the coordinates of these spaces in the cases of $G = PGL_m$ and $G=SL_m$, together with Poisson structures. We consider new coordinates for higher Teichm\"uller spaces given as ratios of the coordinates of the $\mathscr{A}$-space for $G=SL_m$, which are generalizations of Kashaev's ratio coordinates in the case $m=2$. Using Kashaev's quantization for $m=2$, we suggest a quantization of the system of these new ratio coordinates, which may lead to a new family of projective representations of mapping class groups. These ratio coordinates depend on the choice of an ideal triangulation decorated with a distinguished corner at each triangle, and the key point of the quantization is to guarantee certain consistency under a change of such choices. We prove this consistency for $m=3$, and for completeness we also give a full proof of the presentation of Kashaev's groupoid of decorated ideal triangulations.Comment: 42 pages, 6 figure

Graphical Classification of Global SO(n) Invariants and Independent General Invariants

This paper treats some basic points in general relativity and in its perturbative analysis. Firstly a systematic classification of global SO(n) invariants, which appear in the weak-field expansion of n-dimensional gravitational theories, is presented. Through the analysis, we explain the following points: a) a graphical representation is introduced to express invariants clearly; b) every graph of invariants is specified by a set of indices; c) a number, called weight, is assigned to each invariant. It expresses the symmetry with respect to the suffix-permutation within an invariant. Interesting relations among the weights of invariants are given. Those relations show the consistency and the completeness of the present classification; d) some reduction procedures are introduced in graphs for the purpose of classifying them. Secondly the above result is applied to the proof of the independence of general invariants with the mass-dimension $M^6$ for the general geometry in a general space dimension. We take a graphical representation for general invariants too. Finally all relations depending on each space-dimension are systematically obtained for 2, 4 and 6 dimensions.Comment: LaTex, epsf, 60 pages, many figure

A different approach to logic: absolute logic

The paper is about 'absolute logic': an approach to logic that differs from the standard first-order logic and other known approaches. It should be a new approach the author has created proposing to obtain a general and unifying approach to logic and a faithful model of human mathematical deductive process. In first-order logic there exist two different concepts of term and formula, in place of these two concepts in our approach we have just one notion of expression. In our system the set-builder notation is an expression-building pattern. In our system we can easily express second-order, third order and any-order conditions. The meaning of a sentence will depend solely on the meaning of the symbols it contains, it will not depend on external 'structures'. Our deductive system is based on a very simple definition of proof and provides a good model of human mathematical deductive process. The soundness and consistency of the system are proved. We discuss on the completeness of our deductive systems. We also discuss how our system relates to the most know types of paradoxes, from the discussion no specific vulnerability to paradoxes comes out. The paper provides both the theoretical material and a fully documented example of deduction.Comment: 105 pages. Moved the discussions on consistency and completeness from chapter 7 to chapter 4, with additions to the discussion. Updated chapter 7 because of this change. Other minor change

Avoiding Unnecessary Information Loss: Correct and Efficient Model Synchronization Based on Triple Graph Grammars

Model synchronization, i.e., the task of restoring consistency between two interrelated models after a model change, is a challenging task. Triple Graph Grammars (TGGs) specify model consistency by means of rules that describe how to create consistent pairs of models. These rules can be used to automatically derive further rules, which describe how to propagate changes from one model to the other or how to change one model in such a way that propagation is guaranteed to be possible. Restricting model synchronization to these derived rules, however, may lead to unnecessary deletion and recreation of model elements during change propagation. This is inefficient and may cause unnecessary information loss, i.e., when deleted elements contain information that is not represented in the second model, this information cannot be recovered easily. Short-cut rules have recently been developed to avoid unnecessary information loss by reusing existing model elements. In this paper, we show how to automatically derive (short-cut) repair rules from short-cut rules to propagate changes such that information loss is avoided and model synchronization is accelerated. The key ingredients of our rule-based model synchronization process are these repair rules and an incremental pattern matcher informing about suitable applications of them. We prove the termination and the correctness of this synchronization process and discuss its completeness. As a proof of concept, we have implemented this synchronization process in eMoflon, a state-of-the-art model transformation tool with inherent support of bidirectionality. Our evaluation shows that repair processes based on (short-cut) repair rules have considerably decreased information loss and improved performance compared to former model synchronization processes based on TGGs.Comment: 33 pages, 20 figures, 3 table

Proving inductive equalities algorithms and implementation

The aim of this paper is first to describe an algorithm for testing sufficient completeness and second to present concepts necessary to understand the behavior of an implementation of an automatic prover of inductive properties of functional programs or specifications of abstract data types. These programs or specifications are rewriting systems and relations between constructors are allowed. The method is essentially based on a proof by consistency implemented through a Knuth-Bendix completion, extending the Huet-Hullot approach in many respects. This requires to prove the inductive completeness of the set of relations among the constructors, and the relative (or sufficient) completeness of the definitions of the function. After introducing the concept of inductive completeness, inductively complete theories are presented. On the other hand a test of relative completeness is implemented by an extension of an algorithm that worked only when no relation among constructors existed

Architectural Refinement in HETS

The main objective of this work is to bring a number of improvements to the Heterogeneous Tool Set HETS, both from a theoretical and an implementation point of view. In the first part of the thesis we present a number of recent extensions of the tool, among which declarative specifications of logics, generalized theoroidal comorphisms, heterogeneous colimits and integration of the logic of the term rewriting system Maude. In the second part we concentrate on the CASL architectural refinement language, that we equip with a notion of refinement tree and with calculi for checking correctness and consistency of refinements. Soundness and completeness of these calculi is also investigated. Finally, we present the integration of the VSE refinement method in HETS as an institution comorphism. Thus, the proof manangement component of HETS remains unmodified