Davender S. Malik

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 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.

Minimization of fuzzy finite automata

Abstract In this paper we show that a fuzzy finite automation M 1 has an equivalent minimal fuzzy finite automation M . We also show that M can be chsen so that if M 2 is equivalent to M 1 , then M is a homomorphic image of M 2 .

Products of fuzzy finite state machines

Abstract We introduce the concepts of fuzzy transformation semigroups, coverings, cascade and wreath products for fuzzy finite state machines. In order to overcome some of the difficulties which arise from the fuzzification of these concepts, we also introduce the notions of polysemigroups and weak coverings.

On fuzzy regular languages

Abstract We introduce the concept of a max-min fuzzy language, FLν(M), and a min-max fuzzy language, FL ⋏ (M) , recognized by a type of fuzzy automaton M. We show that if L1 and L2 are finite-valued Fν-regular languages, then so are L1 ∪ L2 and L1 ∩ L2. We give a form of a fuzzified pumping lemma which we use to give a necessary and sufficient condition for FL ν (M) to be nonconstant.

Statistical versus Fuzzy Measures of Variable Interaction in Patients with Stroke

Evidence-based medicine, founded in probability-based statistics, applies what is the case for the collective to the individual patient. An intuitive approach, however, would define structure in the (physiologic) system of interest, the human being, directly relevant to other systems (patients) composed of similar variables. A difference in measure of variable interaction in the patient from that in the collective would show how extrapolation of information from the latter to the single patient is counterintuitive. Methods: We compare statistical to ‘fuzzy’ measures of variable interaction. Three diagnostic variables are considered in 30 stroke patients who underwent the same diagnostic tests. ‘Fit’ (fuzzy information) values [0, 1] for degree of variable severity were expertly assigned by 2 blinded raters for real and fabricated patients. Fabricated patients were composed of real-patient ‘fit’ values after shuffling. Real and fabricated patients were each numerically represented as a set . Three groups of fabricated patients and the real patient group were studied. Statistical [Pearson’s product-moment (regression analysis) and Spearman’s rank correlation] and three different fuzzy measures of variable interaction were applied to patient data. Results: Interaction for blood-vessel measured strong in real patients, and weak after one shuffle, using all fuzzy measures. By comparison, the same interaction was found in real patients by only 1 rater (Rater 2) using 1 statistical technique (Spearman’s rank correlation) which, as did Pearson product-moment correlation, found a ‘significant’ interaction between blood-heart in fabricated patients. Conclusion: Our study suggests that the measure of variable interaction in nature – as combined in the individual (real) patient – is captured robustly by fuzzy measures and not so by standard statistical measures.

Fuzzy ideals of artinian rings

Abstract The object of this paper is to study fuzzy ideals over Artinian rings. Among others we prove that a ring with unity is left (right) Artinian if and only if every fuzzy left (right) ideal μ of R is finite valued. This gives a characterization of artinian rings in terms of fuzzy ideals. Mukherjee and Sen (1987) proved that if every fuzzy ideal of a ring with unity is finite valued then the ring is Noetherian. We give an example to show that the converse of this theorem is not true. We also show that certain results of Pan (1987) are not true.

Extensions of fuzzy subrings and fuzzy ideals

Abstract In this paper we give necessary and sufficient conditions for a fuzzy subring or a fuzzy ideal A of a commutative ring R to be extended to one A e of a commutative ring S containing R as a subring such that A and A e have the same image. One of the applications of these results gives a criterion for a fuzzy subring of an integral domain R to be extendable to a fuzzy subfield of the quotient field of R .