L-Fuzzy Relations in Coq

Heyting categories, a variant of Dedekind categories, and Arrow categories provide a convenient framework for expressing and reasoning about fuzzy relations and programs based on those methods. In this thesis we present an implementation of Heyting and arrow categories suitable for reasoning and program execution using Coq, an interactive theorem prover based on Higher-Order Logic (HOL) with dependent types. This implementation can be used to specify and develop correct software based on L-fuzzy relations such as fuzzy controllers. We give an overview of lattices, L-fuzzy relations, category theory and dependent type theory before describing our implementation. In addition, we provide examples of program executions based on our framework.

Moral reasoning and ethics in educational leadership /

School leaders face difficult decisions regarding discipline matters. Often, such decisions play an important role in determining the moral tone of the school and the health of the school community. Many stakeholders are affected by the outcome of such decisions. Codes of conduct, board and school policies, and discipline meetings are often shrouded under secrecy, making the discipline process mysterious. .; In this study I examined the process of moral reasoning. I sought to determine the extent to which school leaders were aware that they were involved in a process of moral reasoning, and ftirthermore, what kind of moral reasoning they practiced. As well, I investigated the ethical grounds and foundations underlying moral reasoning. Thus, in this study I probed the awareness of the process of moral reasoning and sought to find the ethical grounding of decision making. This qualitative study featured short field research. The process involved individual interviews with three different participants: school leaders of a public. Catholic, and an independent school. It found that each school leader practiced moral reasoning to varying degrees through the discipline process. It also explored the possible democratization of moral reasoning by linking to concepts such as fairness, due process, public accountability, and greater participation in the administering of discipline. This study has implications for practice, theory, and future research. The examination of school leaders as the primary focus for discipline matters opens the door to future research that could explore differences between the school systems and possibly other parties affected by moral reasoning in discipline cases.

Violent Video Game Playing, Moral Reasoning, and Attitudes Towards Violence in Adolescents: Is There a Connection?

In this study of 109 adolescents from the eighth grade of seven public elementary schools in Ontario, the relationship among adolescents’ violent video game playing patterns, habits and attitudes, their levels of moral reasoning, and their attitudes towards violence in real life was investigated. In addition, gender differences were addressed. The mixed-methodology was employed combining qualitative and quantitative data. The research results confirmed that playing video games in general is a very popular activity among those adolescents. Significant negative relationship was found between adolescents’ amount of time playing violent video games during the day and their scores on The Sociomoral Reflection Measure. Significant difference was also found between adolescents who play violent video games and those who do not play violent video games on their scores on The Attitudes Towards Violence Scale. Boys and girls significantly differed in the amount of playing video games during the day, the reasons for playing video games, their favourite video game choices, and their favourite video game character choices. Boys and girls also significantly differed on their choices of personality traits of selected video game characters, the identification with video game characters, and their mood experiences while playing video games. The findings are put into the educational context and the context of normal development, and suggestions are given for parents, for educators, and for future violent video game research.

Equational Reasoning about Object-Oriented Programs

Formal verification of software can be an enormous task. This fact brought some software engineers to claim that formal verification is not feasible in practice. One possible method of supporting the verification process is a programming language that provides powerful abstraction mechanisms combined with intensive reuse of code. In this thesis we present a strongly typed functional object-oriented programming language. This language features type operators of arbitrary kind corresponding to so-called type protocols. Sub classing and inheritance is based on higher-order matching, i.e., utilizes type protocols as basic tool for reuse of code. We define the operational and axiomatic semantics of this language formally. The latter is the basis of the interactive proof assistant VOOP (Verified Object-Oriented Programs) that allows the user to prove equational properties of programs interactively.

Generating Relation Algebras for Qualitative Spatial Reasoning

Basic relationships between certain regions of space are formulated in natural language in everyday situations. For example, a customer specifies the outline of his future home to the architect by indicating which rooms should be close to each other. Qualitative spatial reasoning as an area of artificial intelligence tries to develop a theory of space based on similar notions. In formal ontology and in ontological computer science, mereotopology is a first-order theory, embodying mereological and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries between parts. We shall introduce abstract relation algebras and present their structural properties as well as their connection to algebras of binary relations. This will be followed by details of the expressiveness of algebras of relations for region based models. Mereotopology has been the main basis for most region based theories of space. Since its earliest inception many theories have been proposed for mereotopology in artificial intelligence among which Region Connection Calculus is most prominent. The expressiveness of the region connection calculus in relational logic is far greater than its original eight base relations might suggest. In the thesis we formulate ways to automatically generate representable relation algebras using spatial data based on region connection calculus. The generation of new algebras is a two pronged approach involving splitting of existing relations to form new algebras and refinement of such newly generated algebras. We present an implementation of a system for automating aforementioned steps and provide an effective and convenient interface to define new spatial relations and generate representable relational algebras.

An Abstract Algebraic Theory of L-Fuzzy Relations for Relational Databases

Classical relational databases lack proper ways to manage certain real-world situations including imprecise or uncertain data. Fuzzy databases overcome this limitation by allowing each entry in the table to be a fuzzy set where each element of the corresponding domain is assigned a membership degree from the real interval [0…1]. But this fuzzy mechanism becomes inappropriate in modelling scenarios where data might be incomparable. Therefore, we become interested in further generalization of fuzzy database into L-fuzzy database. In such a database, the characteristic function for a fuzzy set maps to an arbitrary complete Brouwerian lattice L. From the query language perspectives, the language of fuzzy database, FSQL extends the regular Structured Query Language (SQL) by adding fuzzy specific constructions. In addition to that, L-fuzzy query language LFSQL introduces appropriate linguistic operations to define and manipulate inexact data in an L-fuzzy database. This research mainly focuses on defining the semantics of LFSQL. However, it requires an abstract algebraic theory which can be used to prove all the properties of, and operations on, L-fuzzy relations. In our study, we show that the theory of arrow categories forms a suitable framework for that. Therefore, we define the semantics of LFSQL in the abstract notion of an arrow category. In addition, we implement the operations of L-fuzzy relations in Haskell and develop a parser that translates algebraic expressions into our implementation.

L-Fuzzy Structured Query Language

Lattice valued fuzziness is more general than crispness or fuzziness based on the unit interval. In this work, we present a query language for a lattice based fuzzy database. We define a Lattice Fuzzy Structured Query Language (LFSQL) taking its membership values from an arbitrary lattice L. LFSQL can handle, manage and represent crisp values, linear ordered membership degrees and also allows membership degrees from lattices with non-comparable values. This gives richer membership degrees, and hence makes LFSQL more flexible than FSQL or SQL. In order to handle vagueness or imprecise information, every entry into an L-fuzzy database is an L-fuzzy set instead of crisp values. All of this makes LFSQL an ideal query language to handle imprecise data where some factors are non-comparable. After defining the syntax of the language formally, we provide its semantics using L-fuzzy sets and relations. The semantics can be used in future work to investigate concepts such as functional dependencies. Last but not least, we present a parser for LFSQL implemented in Haskell.