John N. Mordeson

Fuzzy automata and languages : theory and applications

INTRODUCTION Sets Relations Functions Fuzzy Subsets Semigroups Finite-State Machines Finite State Automata Languages and Grammars Nondeterministic Finite-State Automata Relationships Between Languages and Automata Pushdown Automata MAX-MIN AUTOMATA Max-Min Automata General Formulation of Automata Classes of Automata Behavior of Max-Min Automata Equivalences and Homomorphisms of Max-Min Automata Reduction of Max-Min Automata Definite Max-Min Automata Reduction of Max-Min Machines Equivalences Irreducibility and Minimality Nondeterministic and Deterministic Case FUZZY MACHINES, LANGUAGES, AND GRAMMARS Max-Product Machines Equivalences Irreducibility and Minimality Max-Product Grammars and Languages Weak Regular Max-Product Grammars Weak Regular Max-Product Languages Properties of GBP Exercises FUZZY LANGUAGES AND GRAMMARS Fuzzy Languages Types of Grammars Fuzzy Context-Free Grammars Context-Free Max-Product Grammars Context-Free Fuzzy Languages On the Description of the Fuzzy Meaning of Context-Free Languages Trees and Pseudoterms Fuzzy Dendrolanguage Generating Systems Normal Form of F-CFDS Sets of Derivation Trees of Fuzzy Context-Free Grammars Fuzzy Tree Automaton Fuzzy Tree Transducer Fuzzy Meaning of Context-Free Languages PROBABILISTIC AUTOMATA AND GRAMMARS Probabilistic Automata and their Approximation e-Approximating by Nonprobability Devices e-Approximating by Finite Automata Applications The Pe Relation Fuzzy Stars Acceptors and Probabilistic Acceptors Characterizations and the Re -Relation Probabilistic and Weighted Grammars Probabilistic and Weighted Grammars of Type 3 Interrelations with Programmed and Time-variant Grammars Probabilistic Grammars and Automata Probabilistic Grammars Weakly Regular Grammars and Asynchronous Automata Type-0 Probabilistic Grammars and Probabilistic Turing Machines Context-Free Probabilistic Grammars and Pushdown Automata Realization of Fuzzy Languages by Various Automata Properties of Lk, k - 1,2,3 Further Properties of L3 ALGEBRAIC FUZZY AUTOMATA THEORY Fuzzy Finite State Machines Semigroups of Fuzzy Finite State Machines Homomorphisms Admissible Relation Fuzzy Transformation Semigroups Products of Fuzzy Finite State Machines Submachines of a Fuzzy Finite State Machine Retrievability, Separability, and Connectivity Decomposition of Fuzzy Finite State Machines Subsystems of a Fuzzy Finite State Machine Strong Subsystems Cartesian Composition of Fuzzy Finite State Machines Cartesian Composition Admissible Partitions Coverings of Products of Fuzzy Finite State Machines Associative Properties of Products Covering Properties of Products Fuzzy Semiautomaton over a Finite Group MORE ON FUZZY LANGUAGES Fuzzy Regular Languages On Fuzzy Recognizers Minimal Fuzzy Recognizers Fuzzy Recognizers and Recognizable Sets Operation on (Fuzzy) Subsets Construction of Recognizers and Recognizable Sets Accessible and Coaccessible Recognizers Complete Fuzzy Machines Fuzzy Languages on a Free Monoid Algebraic Character and Properties of Fuzzy Regular Languages Deterministic Acceptors of Regular Fuzzy Languages MINIMIZATION OF FUZZY AUTOMATA Equivalence, Reduction, and Minimization of Finite Fuzzy Automata Equivalence of Fuzzy Automata: An Algebraic Approach Reduction and Minimization of Fuzzy Automata Minimal Fuzzy Finite State Automata Behavior, Reduction, and Minimization of Finite L-Automata Matrices over a Bounded Chain Systems of Linear Equivalences over a Bounded Chain Finite L-Automata-Behavior Matrix e-Equivalence e-Irreducibility Minimization L-FUZZY AUTOMATA, GRAMMARS, AND LANGUAGES Fuzzy Recognition of Fuzzy Languages Fuzzy Languages Fuzzy Recognition by Machines Cutpoint Languages Fuzzy Languages not Fuzzy Recognized by Machines in DT2 Rational Probabilistic Events Recursive Fuzzy Languages Closure Properties Fuzzy Grammars and Recursively Enumerable Fuzzy Languages Recursively Enumerable L-Subsets Various Kind of Automata with Weights APPLICATIONS A Formulation of Fuzzy Automata and its Application as a Model of Learning Systems Formulation of Fuzzy Automata Special Cases of Fuzzy Automata Fuzzy Automata as Models of Learning Systems Applications and Simulation Results Properties of Fuzzy Automata Fractionally Fuzzy Grammars with Application to Pattern Recognition Fractionally Fuzzy Grammars A Pattern Recognition Experiment General Fuzzy Acceptors for Syntactic Pattern Recognition e-Equivalence by Inputs Fuzzy-State Automata: Their Stability and Fault Tolerance Relational Description of Automata Fuzzy-State Automata Stable and Almost Stable Behavior of Fuzzy-State Automata Fault Tolerance of Fuzzy-State Automata Clinical Monitoring with Fuzzy Automata Fuzzy Systems REFERENCES INDEX Each chapter also includes a section of exercises

Fuzzy Graphs and Fuzzy Hypergraphs

Fuzzy Subsets: Fuzzy Relations.- Fuzzy Equivalence Relations.- Pattern Classification.- Similarity Relations.- References.- Fuzzy Graphs: Paths and Connectedness. Bridges and Cut Vertices. Forests and Trees. Trees and Cycles. A Characterization of Fuzzy Trees. (Fuzzy) Cut Sets. (Fuzzy Chords, (Fuzzy) Cotrees, and (Fuzzy) Twigs. (Fuzzy) 1-Chain with Boundary 0, (Fuzzy) Coboundary, and (Fuzzy) Cocycles. (Fuzzy) Cycle Set and (Fuzzy) Cocycle Set.- Fuzzy Line Graphs.- Fuzzy Interval Graphs. Fuzzy Intersection Graphs. Fuzzy Interval Graphs. The Fulkerson and Gross Characterization. The Gilmore and Hoffman Characterization.- Operations on Fuzzy Graphs: Cartesian Product and Composition. Union and Join.- On Fuzzy Tree Definition.- References.- Applications of Fuzzy Graphs: Clusters.- Cluster Analysis. Cohesiveness. Slicing in Fuzzy Graphs.- Application to Cluster Analysis.- Fuzzy Intersection Equations. Existence of Solutions.- Fuzzy Graphs in Database Theory. Representation of Dependency Structure r(X,Y) by Fuzzy Graphs.- A Description of Strengthening and Weakening Members of a Group. Connectedness Criteria. Inclusive Connectedness Categories. Exclusive Connectedness Categories.- An Application of Fuzzy Graphs to the Problem Concerning Group Structure. Connectedness of a Fuzzy Graph. Weakening and Strenghtening Points of a Fuzzy Directed Graph.- References.- Fuzzy Hypergraphs: Fuzzy Hypergraphs.- Fuzzy Transversals of Fuzzy Hypergraphs. Properties of Tr(H). Construction of H3.- Coloring of Fuzzy Hypergraphs. beta-degree Coloring Procedures. Chromatic Values of Fuzzy Colorings.- Intersecting Fuzzy Hypergraphs. Characterization of Strongly Intersecting Hypergraphs. Simply Ordered Intersecting Hypergraphs. H-dominant Transversals.- Hebbian Structures.- Additional Applications.- References.

Fuzzy commutative algebra

L-subsets and L-subgroups L-subgroups of Abelian groups L-submodules L-subrings and L-ideals L-subfields structure of L-subrings and L-ideals algebraic L-varieties and intersection equations L-subspaces Galois theory and group L-subalgebras.

The Association of Bone Mineral Density with Vitamin D Receptor Gene Polymorphisms

Abstract: A recent meta-analysis of 16 publications suggested that bone mineral density (BMD) is not associated with vitamin D receptor (VDR) gene polymorphism (VDRGP) at the 0.05 significance level when a study with genotyping mistakes is excluded. We wished to determine whether ‘positive’ findings supporting the BMD–VDRGP association may be explained by chance, and what factors affect the outcomes of these studies. Seventy-five articles and abstracts on the association of VDRGP with BMD and related skeletal phenotypes published before January 1997 were identified. Twenty-three of 67 (34.3%) studies on spinal BMD and 22 of 51 (43.1%) on femoral neck BMD had found a BMD–VDRGP association at p<0.05, significantly (p= 7 × 10–14 for spinal BMD, p= 9 × 10–16 for hip BMD) higher than the expected 5% false positive rate under the null hypothesis of ‘no association’. ‘Positive’ results were more frequently observed in studies on females before the menopause than those on females after the menopause (p<0.02) or on male and female subjects combined (p<0.05) when skeletal phenotypes at any bone sites were considered. The ‘positive rate’ among studies was also influenced by the age range of subjects studied and by the inclusion of subjects with osteoporosis. It is concluded that: (1) BMD is associated with VDRGP with high levels of confidence and (2) non-genetic factors and genetic heterogeneity interfere with the detection of the effects of VDRGP on bone phenotypes.

Operations on fuzzy graphs

We define the operations of Cartesian product, composition, union, and join on fuzzy subgraphs of graphs G1 and G2. If the graph G is formed from G1 and G2 by one of these operations, we determine necessary and sufficient conditions for an arbitrary fuzzy subgraph of G also to be formed by the same operation from fuzzy subgraphs of G1 and G2.

Rough set theory applied to (fuzzy) ideal theory

We use covers of the universal set to define approximation operators on the power set of the given set. In Section 1, we determine basic properties of the upper approximation operator and show how it can be used to give algebraic structural properties of certain subsets. We define a particular cover on the set of ideals of a commutative ring with identity in such a way that both the concepts of the (fuzzy) prime spectrum of a ring and rough set theory can simultaneously be brought to bear on the study of (fuzzy) ideals of a ring.

Fuzzy Group Theory

Fuzzy Subsets and Fuzzy Subgroups.- Fuzzy Caleys Theorem and Fuzzy Lagranges Theorem.- Nilpotent, Commutator, and Solvable Fuzzy Subgroups.- Characterization of Certain Groups and Fuzzy Subgroups.- Free Fuzzy Subgroups and Fuzzy Subgroup Presentations.- Fuzzy Subgroups of Abelian Groups.- Direct Products of Fuzzy Subgroups and Fuzzy Cyclic Subgroups.- Equivalence of Fuzzy Subgroups of Finite Abelian Groups.- Lattices of Fuzzy Subgroups.- Membership Functions From Similarity Relations.

Fuzzy discrete structures

Foreword.- Preface.- Fuzzy Logic Functions: Sets. Relations. Functions. Fuzzy Sets. Semigroups. Fuzzy Logic. Fuzzy Functions and Decomposition. Solution of Fuzzy Logic Inequalities. References.- Decision Trees: Decision Trees. Fuzzy Decision Tree Algorithms. Analysis of the BBB Algorithm. References.- Networks: Network Models. A Maximum Flow Algorithm. The Max Flow, Min Cut Theorem. Maximum Flow in a Network with Fuzzy Arc Capacities. The Maximum Flow with Integer Values. Integer Flows in Network with Two-Sided Fuzzy Capacities Constraints. Real-Valued Flows in a Network with Fuzzy Arc Capacities. Petri Nets. Fuzzy Petri Nets for Rule-Based Decisionmaking. References. - Fuzzy Graphs and Shortest Paths.- Fuzzy Shortest Paths. Analysis of the Fuzzy Path Models. On Valuation and Optimization Problems. References.- Fuzzy Machines, Languages, and Grammars: Max-Product Machines. Irreducibility and Minimality. On Reductions of Maximin Machines. Context-Free Max-Product Grammars. Context-Free Fuzzy Languages. Deterministic Acceptors of Regular Fuzzy Languages. Fuzzy Languages on a Free Monoid. Algebraic Character and Properties of F-Regular Languages. References.- Algebraic Fuzzy Automata. Semigroups of Fuzzy Finite State Machines. Homomorphisms. Admissible Relations. Fuzzy Transformation Semigroups. Submachines. Retrievability, Separability and Connectivity. Decomposition of Fuzzy Finite State Machines. Admissible Partitions. On Fuzzy Recognizers. Minimal Fuzzy Recognizers. References.- Appendix.- Index.- List of Symbols.

Fuzzy maximal, radical, and primary ideals of a ring

We introduce the concepts of fuzzy maximal ideal, the fuzzy radical of a fuzzy ideal and fuzzy primary ideal of a ring. We show that a fuzzy left (right) ideal A of a ring R is a fuzzy maximal ideal if and only A(0) = 1 and A∗ = {x ϵ R:A(x) = A(0)} is a maximal left (right) ideal of R. We also show that a fuzzy ideal A of a commutative ring R with unity is a fuzzy primary ideal of R if and only A(0) = 1, A is two-valued and A∗ is a primary ideal of R.

On subsystems of a fuzzy finite state machine

Abstract In this paper we study subsystems and strong subsystems of a fuzzy finite state machine. The ideas of cyclic subsystem, and simple strong subsystem are introduced and their basic properties are examined.