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.

Role of Phosphorylation of the Oxysterol Binding Protein (OSBP) in (PI(4)P) and Sterol-Containing Membrane Recognition and Binding

Studies have demonstrated that the oxysterol binding protein (OSBP) acts as a phosphatidylinositol phosphate (PIP)-sterol exchanger at membrane contact sites (MCS) of the endoplasmic reticulum (ER) and Golgi. OSBP is known to pick up phosphatidylinositol-4-phosphate (PI(4)P) from the ER, transfer it to the trans-Golgi in exchange for a cholesterol molecule that is then transferred from the trans-Golgi to the ER. Upon further examination of this pathway by Ridgway et al. (1), it appeared that phosphorylation of OSBP played a role in the localization of OSBP. The dephosphorylation state of OSBP was linked to Golgi localization and the depletion of cholesterol at the ER. To mimic the phosphorylated state of OSBP, the mutant OSBP-S5E was designed by Ridgway et al. (1). The lipid and sterol recognition by wt-OSBP and its phosphomimic mutant OSBP-S5E were investigated using immobilized lipid bilayers and dual polarization interferometry (DPI). DPI is a technique in which the protein binding affinity to immobilized lipid bilayers is measured and the binding behavior is examined through real time. Lipid bilayers containing 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) and varying concentrations of PI(4)Ps or sterols (cholesterol or 25-hydroxycholesterol) were immobilized on a silicon nitride chip. It was determined that wt-OSBP binds differently to PI(4)P-containing bilayers compared to OSBP-S5E. The binding behavior suggested that wt-OSBP extracts PI(4)P and the change in the binding behavior, in the case of OSBP-S5E, suggested that the phosphorylation of OSBP may prevent the recognition and/or extraction of PI(4)P. In the presence of sterols, the overall binding behavior of OSBP, regardless of phosphorylation state, was fairly similar. The maximum specific bound mass of OSBP to sterols did not differ as the concentration of sterols increased. However, comparing the maximum specific bound mass of OSBP to cholesterol with oxysterol (25-hydroxycholesterol), OSBP displayed nearly a 2-fold increase in bound mass. With the absence of the wt-OSBP-PI(4)P binding behavior, it can be speculated that the sterols were not extracted. In addition, the binding behavior of OSBP was further tested using a fluorescence based binding assay. Using 22-(N-(7-nitrobenz-2-oxa-1,3-diazol-4-yl)amino)-23,24-bisnor-5-cholen-3β-ol (22-NBD cholesterol), wt-OSBP a one site binding dissociation constant Kd, of 15 ± 1.4 nM was determined. OSBP-S5E did not bind to 22-NBD cholesterol and Kd value was not obtained.