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.

Do Neuro-Muscular Adaptations Occur in Endurance-Trained Boys and Men?

Most research on the effects of endurance training has focused on endurance training's health-related benefits and metabolic effects in both children and adults. The purpose of this study was to examine the neuromuscular effects of endurance training and to investigate whether they differ in children (9.0-12.9 years) and adults (18.4-35.6 years). Maximal isometric torque, rate of torque development (RTD), rate of muscle activation (Q30), electromechanical delay (EMD), and time to peak torque and peak RTD were determined by isokinetic dynamometry and surface electromyography (EMG) in elbow and knee flexion and extension. The subjects were 12 endurance-trained and 16 untrained boys, and 15 endurance-trained and 20 untrained men. The adults displayed consistently higher peak torque, RTD, and Q30, in both absolute and normalized values, whereas the boys had longer EMD (64.7+/-17.1 vs. 56.6+/-15.4 ms) and time to peak RTD (98.5+/-32.1 vs. 80.4+/-15.0 ms for boys and men, respectively). Q30, normalized for peak EMG amplitude, was the only observed training effect (1.95+/-1.16 vs. 1.10+/-0.67 ms for trained and untrained men, respectively). This effect could not be shown in the boys. The findings show normalized muscle strength and rate of activation to be lower in children compared with adults, regardless of training status. Because the observed higher Q30 values were not matched by corresponding higher performance measures in the trained men, the functional and discriminatory significance of Q30 remains unclear. Endurance training does not appear to affect muscle strength or rate of force development in either men or boys.

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.