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.

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.

Light and electron microscope studies of host-parasite relations in a mycoparasite

Light microscope studies of the mycoparasite Piptocephalis virginiana revealed that the cylindrical spores of the parasite became spherical upon germination and produced 1-4 germ tubes. Generally t"l.vO germ tubes were produced by each spore. When this parasite was inoculated on its potential hosts, Choanephora cucurbitarum and Phascolomyces articulosus, the germ tube nearest to the host hypha continued to grow and made contact with the host hypha. The tip of the parasite's germ tube became swollen to form a distinct appressorium. Up to this stage the behavior of the parasite was similar regardless of the nature of the host. In the compatible host-parasite combination, the parasite penetrated the host, established a nutritional relationship and continued to grow to cover the host completely with its buff colored spores in 3-4 days. In the incompatible host-parasite combination, the parasite penetrated the host but its further advance was arrested. As a result of failure to establish a nutritional relationship with the resistant host, the parasite made further attempts to penetrate the host at different sites producing multiple infections. In the absence of nutrition the parasite weakened and the host outgrew the parasite completely. In the presence of a non-host species, Linderina pennispora the parasite continued to grow across the non-host 1).yp_hae vlithout establishing an initial contact. Germination studies showed that the parasite germinated equally well in the presence of host and non-host species. Further electron microscope studies revealed that the host-parasite interaction between P. virginiana and its host, C. cucurbi tarum, was compatible when the host hyphae were young slender, with a thin cell wall of one layer. The parasite appeared to penetrate mechanically by pushing the host-cell wall inward. The host plasma membrane invaginated along the involuted cell wall. The older hyphae of C. cucurbitarum possessed two distinct layers of cell wall and-showed an incompatible interaction when challenged vlith the parasite. At the point of contact, the outer layer of the host-cell wall dissolved, probably by enzymatic digestion, and the inner layer became thickened and developed a papilla as a result of its response to the parasite. The haustoria of the parasite in the old hyphae were always surrounded by a thick, well developed sheath, whereas the haustoria of the same age in the young host mycelium were devoid of a sheath during early stages of infection. Instead, they were in direct contact with the host protoplast. The incompatible interaction between a resistant host, P. articulosus and the parasite showed similar results as with the old hyphae of C. cucurbitarum. The cell wall of P. articulosus appeared thick-with two or more layers even in the 18-22 h-old hyphae. No contact or interaction was established between the parasite and the non-host L. pennispora. The role of cell wall in the resistance mechanism is discussed.

RelMDD-A Library for Manipulating Relations Based on MDDs

Relation algebras is one of the state-of-the-art means used by mathematicians and computer scientists for solving very complex problems. As a result, a computer algebra system for relation algebras called RelView has been developed at Kiel University. RelView works within the standard model of relation algebras. On the other hand, relation algebras do have other models which may have different properties. For example, in the standard model we always have L;L=L (the composition of two (heterogeneous) universal relations yields a universal relation). This is not true in some non-standard models. Therefore, any example in RelView will always satisfy this property even though it is not true in general. On the other hand, it has been shown that every relation algebra with relational sums and subobjects can be seen as matrix algebra similar to the correspondence of binary relations between sets and Boolean matrices. The aim of my research is to develop a new system that works with both standard and non-standard models for arbitrary relations using multiple-valued decision diagrams (MDDs). This system will implement relations as matrix algebras. The proposed structure is a library written in C which can be imported by other languages such as Java or Haskell.