A Topological Approach to Fuzzy Sets

This thesis presents a topological approach to studying fuzzy setsby means of modifier operators. Modifier operators are mathematical models, e.g., for hedges, and we present briefly different approaches to studying modifier operators. We are interested in compositional modifier operators, modifiers for short, and these modifiers depend on binary relations. We show that if a modifier depends on a reflexive and transitive binary relation on U, then there exists a unique topology on U such that this modifier is the closure operator in that topology. Also, if U is finite then there exists a lattice isomorphism between the class of all reflexive and transitive relations and the class of all topologies on U. We define topological similarity relation "≈" between L-fuzzy sets in an universe U, and show that the class LU/ ≈ is isomorphic with the class of all topologies on U, if U is finite and L is suitable. We consider finite bitopological spaces as approximation spaces, and we show that lower and upper approximations can be computed by means of α-level sets also in the case of equivalence relations. This means that approximations in the sense of Rough Set Theory can be computed by means of α-level sets. Finally, we present and application to data analysis: we study an approach to detecting dependencies of attributes in data base-like systems, called information systems.

A mathematical study of fuzzy logic; an algebraic approach

Fuzzy set theory and Fuzzy logic is studied from a mathematical point of view. The main goal is to investigatecommon mathematical structures in various fuzzy logical inference systems and to establish a general mathematical basis for fuzzy logic when considered as multi-valued logic. The study is composed of six distinct publications. The first paper deals with Mattila'sLPC+Ch Calculus. THis fuzzy inference system is an attempt to introduce linguistic objects to mathematical logic without defining these objects mathematically.LPC+Ch Calculus is analyzed from algebraic point of view and it is demonstratedthat suitable factorization of the set of well formed formulae (in fact, Lindenbaum algebra) leads to a structure called ET-algebra and introduced in the beginning of the paper. On its basis, all the theorems presented by Mattila and many others can be proved in a simple way which is demonstrated in the Lemmas 1 and 2and Propositions 1-3. The conclusion critically discusses some other issues of LPC+Ch Calculus, specially that no formal semantics for it is given.In the second paper the characterization of solvability of the relational equation RoX=T, where R, X, T are fuzzy relations, X the unknown one, and o the minimum-induced composition by Sanchez, is extended to compositions induced by more general products in the general value lattice. Moreover, the procedure also applies to systemsof equations. In the third publication common features in various fuzzy logicalsystems are investigated. It turns out that adjoint couples and residuated lattices are very often present, though not always explicitly expressed. Some minor new results are also proved.The fourth study concerns Novak's paper, in which Novak introduced first-order fuzzy logic and proved, among other things, the semantico-syntactical completeness of this logic. He also demonstrated that the algebra of his logic is a generalized residuated lattice. In proving that the examination of Novak's logic can be reduced to the examination of locally finite MV-algebras.In the fifth paper a multi-valued sentential logic with values of truth in an injective MV-algebra is introduced and the axiomatizability of this logic is proved. The paper developes some ideas of Goguen and generalizes the results of Pavelka on the unit interval. Our proof for the completeness is purely algebraic. A corollary of the Completeness Theorem is that fuzzy logic on the unit interval is semantically complete if, and only if the algebra of the valuesof truth is a complete MV-algebra. The Compactness Theorem holds in our well-defined fuzzy sentential logic, while the Deduction Theorem and the Finiteness Theorem do not. Because of its generality and good-behaviour, MV-valued logic can be regarded as a mathematical basis of fuzzy reasoning. The last paper is a continuation of the fifth study. The semantics and syntax of fuzzy predicate logic with values of truth in ana injective MV-algerba are introduced, and a list of universally valid sentences is established. The system is proved to be semanticallycomplete. This proof is based on an idea utilizing some elementary properties of injective MV-algebras and MV-homomorphisms, and is purely algebraic.

The role of tributary relative timing and sequencing in controlling large floods

Hydrograph convolution is a product of tributary inputs from across the watershed. The time-space distribution of precipitation, the biophysical processes that control the conversion of precipitation to runoff and channel flow conveyance processes, are heterogeneous and different areas respond to rainfall in different ways. We take a subwatershed approach to this and account for tributary flow magnitude, relative timing, and sequencing. We hypothesize that as the scale of the watershed increases so we may start to see systematic differences in subwatershed hydrological response. We test this hypothesis for a large flood (T >100 years) in a large watershed in northern England. We undertake a sensitivity analysis of the effects of changing subwatershed hydrological response using a hydraulic model. Delaying upstream tributary peak flow timing to make them asynchronous from downstream subwatersheds reduced flood magnitude. However, significant hydrograph adjustment in any one subwatershed was needed for meaningful reductions in stage downstream, although smaller adjustments in multiple tributaries resulted in comparable impacts. For larger hydrograph adjustments, the effect of changing the timing of two tributaries together was lower than the effect of changing each one separately. For smaller adjustments synergy between two subwatersheds meant the effect of changing them together could be greater than the sum of the parts. Thus, this work shows that while the effects of modifying biophysical catchment properties diminishes with scale due to dilution effects, their impact on relative timing of tributaries may, if applied in the right locations, be an important element of flood management.

Using the optical flow to implement a relative virtual mouse controlled by head movements

The following paper introduces the work conducted to create a relative virtual mouse based on the interpretation of head movements and face gesture through a low cost camera and the optical flow of the images. This virtual device is designed specifically as an alternative non-contact pointer for people with mobility impairments in the upper extremities and reduced head control. The proposed virtual device was compared with a conventional mouse, a touchpad and a digital joystick. Validation results show performances close to a digital joystick but far away from a conventional mouse.

Integrating defeasible argumentation with fuzzy ART neural networks for pattern classification

Many classification systems rely on clustering techniques in which a collection of training examples is provided as an input, and a number of clusters c1,...cm modelling some concept C results as an output, such that every cluster ci is labelled as positive or negative. Given a new, unlabelled instance enew, the above classification is used to determine to which particular cluster ci this new instance belongs. In such a setting clusters can overlap, and a new unlabelled instance can be assigned to more than one cluster with conflicting labels. In the literature, such a case is usually solved non-deterministically by making a random choice. This paper presents a novel, hybrid approach to solve this situation by combining a neural network for classification along with a defeasible argumentation framework which models preference criteria for performing clustering.

Complex syntax in autism spectrum disorders: a study of relative clauses.

BACKGROUND: The few studies that have evaluated syntax in autism spectrum disorder (ASD) have yielded conflicting findings: some suggest that once matched on mental age, ASD and typically developing controls do not differ for grammar, while others report that morphosyntactic deficits are independent of cognitive skills in ASD. There is a need for a better understanding of syntax in ASD and its relation to, or dissociation from, nonverbal abilities. AIMS: Syntax in ASD was assessed by evaluating subject and object relative clause comprehension in adolescents and adults diagnosed with ASD with a performance IQ within the normal range, and with or without a history of language delay. METHODS & PROCEDURES: Twenty-eight participants with ASD (mean age 21.8) and 28 age-matched controls (mean age 22.07) were required to point to a character designated by relative clauses that varied in syntactic complexity. OUTCOMES & RESULTS: Scores indicate that participants with ASD regardless of the language development history perform significantly worse than age-matched controls with object relative clauses. In addition, participants with ASD with a history of language delay (diagnosed with high-functioning autism in the DSM-IV-TR) perform worse on subject relatives than ASD participants without language delay (diagnosed with Asperger syndrome in the DSM-IV-TR), suggesting that these two groups do not have equivalent linguistic abilities. Performance IQ has a positive impact on the success of the task for the population with ASD. CONCLUSIONS & IMPLICATIONS: This study reveals subtle grammatical difficulties remaining in adult individuals with ASD within normal IQ range as compared with age-matched peers. Even in the absence of a history of language delay in childhood, the results suggest that a slight deficit may nevertheless be present and go undetected by standardized language assessments. Both groups with and without language delay have a similar global performance on relative clause comprehension; however, the study also indicates that the participants with reported language delay show more difficulty with subject relatives than the participants without language delay, suggesting the presence of differences in linguistic abilities between these subgroups of ASD.

Adding similarity-based reasoning capabilities to a Horn fragment of possibilistic logic with fuzzy constants

PLFC is a first-order possibilistic logic dealing with fuzzy constants and fuzzily restricted quantifiers. The refutation proof method in PLFC is mainly based on a generalized resolution rule which allows an implicit graded unification among fuzzy constants. However, unification for precise object constants is classical. In order to use PLFC for similarity-based reasoning, in this paper we extend a Horn-rule sublogic of PLFC with similarity-based unification of object constants. The Horn-rule sublogic of PLFC we consider deals only with disjunctive fuzzy constants and it is equipped with a simple and efficient version of PLFC proof method. At the semantic level, it is extended by equipping each sort with a fuzzy similarity relation, and at the syntactic level, by fuzzily “enlarging” each non-fuzzy object constant in the antecedent of a Horn-rule by means of a fuzzy similarity relation.

Formalizing argumentative reasoning in a possibilistic logic programming setting with fuzzy unification

Possibilistic Defeasible Logic Programming (P-DeLP) is a logic programming language which combines features from argumentation theory and logic programming, incorporating the treatment of possibilistic uncertainty at the object-language level. In spite of its expressive power, an important limitation in P-DeLP is that imprecise, fuzzy information cannot be expressed in the object language. One interesting alternative for solving this limitation is the use of PGL+, a possibilistic logic over Gödel logic extended with fuzzy constants. Fuzzy constants in PGL+ allow expressing disjunctive information about the unknown value of a variable, in the sense of a magnitude, modelled as a (unary) predicate. The aim of this article is twofold: firstly, we formalize DePGL+, a possibilistic defeasible logic programming language that extends P-DeLP through the use of PGL+ in order to incorporate fuzzy constants and a fuzzy unification mechanism for them. Secondly, we propose a way to handle conflicting arguments in the context of the extended framework.