TOPOLOGICALLY CONVEX SETS AND FIXED POINT THEORY
Topologically convex sets and fixed point theory - mapping
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Intelligent System Development Using a Rough Sets Methodology
The purpose of this research was to examine the potential of the rough sets technique for developing intelligent models of complex systems from limited information. Rough sets a simple but promising technology to extract easily understood rules from data. The rough set methodology has been shown to perform well when used with a large set of exemplars, but its performance with sparse data sets is less certain. The difficulty is that rules will be developed based on just a few examples, each of which might have a large amount of noise associated with them. The question then becomes, what is the probability of a useful rule being developed from such limited information? One nice feature of rough sets is that in unusual situations, the technique can give an answer of 'I don't know'. That is, if a case arises that is different from the cases the rough set rules were developed on, the methodology can recognize this and alert human operators of it. It can also be trained to do this when the desired action is unknown because conflicting examples apply to the same set of inputs. This summer's project was to look at combining rough set theory with statistical theory to develop confidence limits in rules developed by rough sets. Often it is important not to make a certain type of mistake (e.g., false positives or false negatives), so the rules must be biased toward preventing a catastrophic error, rather than giving the most likely course of action. A method to determine the best course of action in the light of such constraints was examined. The resulting technique was tested with files containing electrical power line 'signatures' from the space shuttle and with decompression sickness data.
Anderson, Gray T.↗
Optimal Scheduling of a Multi-User, Multi-Tasking Automatic Robotic Telescope
An algorithm is introduced to generate the optimal schedule of observations for a multi-user, multi-tasking automatic robotic telescope. The schedule is optimal m that it will take the highest possible quality data, perform the largest number of tasks, and fairly serve multiple users in terms of data quality, allocated telescope time, and observation priorities. In addition, the algorithm addresses problems due to overloading and underloading the telescope. The algorithm, which employs dynamic-programming-based decision theory, is analytically proven to produce the optimal schedule using set theory. Simulations conducted using actual astronomical data are presented which show significant improvement over existing scheduling techniques. Once completed the new scheduling algorithm will be tested on the Tennessee State University/Harvard-Smithsonian Center for Astrophysics 32-inch automatic Photoelectric Telescope at Fairborn Observatory, Arizona.
Tantaris, R. N.↗
Causal sets and an emerging continuum
Abstract Causal set theory offers a simple and elegant picture of discrete physics. But the vast majority of causal sets look nothing at all like continuum spacetimes, and must be excluded in some way to obtain a realistic theory. I describe recent results showing that almost all non-manifoldlike causal sets are, in fact, very strongly suppressed in the gravitational path integral. This does not quite demonstrate the emergence of a continuum—we do not yet understand the remaining unsuppressed causal sets well enough—but it is a significant step in that direction.
Astronomy & Astrophysics↗
North American Fuzzy Logic Processing Society (NAFIPS 1992), volume 1
This document contains papers presented at the NAFIPS '92 North American Fuzzy Information Processing Society Conference. More than 75 papers were presented at this Conference, which was sponsored by NAFIPS in cooperation with NASA, the Instituto Tecnologico de Morelia, the Indian Society for Fuzzy Mathematics and Information Processing (ISFUMIP), the Instituto Tecnologico de Estudios Superiores de Monterrey (ITESM), the International Fuzzy Systems Association (IFSA), the Japan Society for Fuzzy Theory and Systems, and the Microelectronics and Computer Technology Corporation (MCC). The fuzzy set theory has led to a large number of diverse applications. Recently, interesting applications have been developed which involve the integration of fuzzy systems with adaptive processes such as neural networks and genetic algorithms. NAFIPS '92 was directed toward the advancement, commercialization, and engineering development of these technologies.
Villarreal, James A.↗
North American Fuzzy Logic Processing Society (NAFIPS 1992), volume 2
This document contains papers presented at the NAFIPS '92 North American Fuzzy Information Processing Society Conference. More than 75 papers were presented at this Conference, which was sponsored by NAFIPS in cooperation with NASA, the Instituto Tecnologico de Morelia, the Indian Society for Fuzzy Mathematics and Information Processing (ISFUMIP), the Instituto Tecnologico de Estudios Superiores de Monterrey (ITESM), the International Fuzzy Systems Association (IFSA), the Japan Society for Fuzzy Theory and Systems, and the Microelectronics and Computer Technology Corporation (MCC). The fuzzy set theory has led to a large number of diverse applications. Recently, interesting applications have been developed which involve the integration of fuzzy systems with adaptive processes such a neural networks and genetic algorithms. NAFIPS '92 was directed toward the advancement, commercialization, and engineering development of these technologies.
Villarreal, James A.↗
Two Methods for Efficient Solution of the Hitting-Set Problem
A paper addresses much of the same subject matter as that of Fast Algorithms for Model-Based Diagnosis (NPO-30582), which appears elsewhere in this issue of NASA Tech Briefs. However, in the paper, the emphasis is more on the hitting-set problem (also known as the transversal problem), which is well known among experts in combinatorics. The authors primary interest in the hitting-set problem lies in its connection to the diagnosis problem: it is a theorem of model-based diagnosis that in the set-theory representation of the components of a system, the minimal diagnoses of a system are the minimal hitting sets of the system. In the paper, the hitting-set problem (and, hence, the diagnosis problem) is translated from a combinatorial to a computational problem by mapping it onto the Boolean satisfiability and integer- programming problems. The paper goes on to describe developments nearly identical to those summarized in the cited companion NASA Tech Briefs article, including the utilization of Boolean-satisfiability and integer- programming techniques to reduce the computation time and/or memory needed to solve the hitting-set problem.
Vatan, Farrokh↗
Deriving Laws from Ordering Relations
It took much effort in the early days of non-Euclidean geometry to break away from the mindset that all spaces are flat and that two distinct parallel lines do not cross. Up to that point, all that was known was Euclidean geometry, and it was difficult to imagine anything else. We have suffered a similar handicap brought on by the enormous relevance of Boolean algebra to the problems of our age-logic and set theory. Previously, I demonstrated that the algebra of questions is not Boolean, but rather is described by the free distributive algebra. To get to this stage took much effort, as many obstacles-most self-placed-had to be overcome. As Boolean algebras were all I had ever known, it was almost impossible for me to imagine working with an algebra where elements do not have complements. With this realization, it became very clear that the sum and product rules of probability theory at the most basic level had absolutely nothing to do with the Boolean algebra of logical statements. Instead, a measure of degree of inclusion can be invented for many different partially ordered sets, and the sum and product rules fall out of the associativity and distributivity of the algebra. To reinforce this very important idea, this paper will go over how these constructions are made, while focusing on the underlying assumptions. I will derive the sum and product rules for a distributive lattice in general and demonstrate how this leads to probability theory on the Boolean lattice and is related to the calculus of quantum mechanical amplitudes on the partially ordered set of experimental setups. I will also discuss the rules that can be derived from modular lattices and their relevance to the cross-ratio of projective geometry.
Knuth, Kevin H.↗
An Image Processing Algorithm Based On FMAT
Information deleted in ways minimizing adverse effects on reconstructed images. New grey-scale generalization of medial axis transformation (MAT), called FMAT (short for Fuzzy MAT) proposed. Formulated by making natural extension to fuzzy-set theory of all definitions and conditions (e.g., characteristic function of disk, subset condition of disk, and redundancy checking) used in defining MAT of crisp set. Does not need image to have any kind of priori segmentation, and allows medial axis (and skeleton) to be fuzzy subset of input image. Resulting FMAT (consisting of maximal fuzzy disks) capable of reconstructing exactly original image.
Wang, Lui↗
Category Theory
Functions started with an implicit definition - the dependency of some quantity on others. Functions were coined by Leibniz in 1673, were given the f(x) notation by Euler in 1734, were formalized in the language of set theory from around 1847 to 1939. Functions have since been further abstract, often capture and represent structure.
Alan Hylton↗
Multi-layered reasoning by means of conceptual fuzzy sets
The real world consists of a very large number of instances of events and continuous numeric values. On the other hand, people represent and process their knowledge in terms of abstracted concepts derived from generalization of these instances and numeric values. Logic based paradigms for knowledge representation use symbolic processing both for concept representation and inference. Their underlying assumption is that a concept can be defined precisely. However, as this assumption hardly holds for natural concepts, it follows that symbolic processing cannot deal with such concepts. Thus symbolic processing has essential problems from a practical point of view of applications in the real world. In contrast, fuzzy set theory can be viewed as a stronger and more practical notation than formal, logic based theories because it supports both symbolic processing and numeric processing, connecting the logic based world and the real world. In this paper, we propose multi-layered reasoning by using conceptual fuzzy sets (CFS). The general characteristics of CFS are discussed along with upper layer supervision and context dependent processing.
Takagi, Tomohiro↗
Forced nutations of the earth: Influence of inner core dynamics. I - Theory. II - Numerical results and comparisons. III - Very long interferometry data analysis
A treatment is presented of the nutation problem for an oceanless, elastic, spheroidally stratified earth, with the dynamical role of the inner core explicitly included in the formulation. Solving the enlarged system of equations shows that a new almost diurnal eigenfrequency emerges. A rough estimate places it not far from the prograde annual tidal excitation frequency, so that possible resonance effects on nutation amplitudes need careful consideration. Tables are provided that exhibit the sensitivities of various relevant quantities, the eigenfrequencies and the coefficients which appear in the resonance expansion, as well as the nutation amplitudes at important tidal frequencies, to possible errors in the earth parameters which enter the theory set forth. Finally, the analysis of 798 VLBI experiments performed between July 1980 and February 1989 and the determination from this analysis of corrections to selected coefficients in the International Astronomical Union 1980 theory of the nutations of the earth are discussed.
Mathews, P. M.↗
Application of fuzzy theories to formulation of multi-objective design problems
Much of the decision making in real world takes place in an environment in which the goals, the constraints, and the consequences of possible actions are not known precisely. In order to deal with imprecision quantitatively, the tools of fuzzy set theory can by used. This paper demonstrates the effectiveness of fuzzy theories in the formulation and solution of two types of helicopter design problems involving multiple objectives. The first problem deals with the determination of optimal flight parameters to accomplish a specified mission in the presence of three competing objectives. The second problem addresses the optimal design of the main rotor of a helicopter involving eight objective functions. A method of solving these multi-objective problems using nonlinear programming techniques is presented. Results obtained using fuzzy formulation are compared with those obtained using crisp optimization techniques. The outlined procedures are expected to be useful in situations where doubt arises about the exactness of permissible values, degree of credibility, and correctness of statements and judgements.
Dhingra, A. K.↗
Fuzzy compromise: An effective way to solve hierarchical design problems
In this paper, we present a method for modeling design problems using a compromise decision support problem (DSP) incorporating the principles embodied in fuzzy set theory. Specifically, the fuzzy compromise decision support problem is used to study hierarchical design problems. This approach has the advantage that although the system modeled has an element of uncertainty associated with it, the solution obtained is crisp and precise. The efficacy of incorporating fuzzy sets into the solution process is discussed in the context of results obtained for a portal frame.
Allen, J. K.↗
Profile Evolution Simulation in Etching Systems Using Level Set Methods
Semiconductor device profiles are determined by the characteristics of both etching and deposition processes. In particular, a highly anisotropic etch is required to achieve vertical sidewalls. However, etching is comprised of both anisotropic and isotropic components, due to ion and neutral fluxes, respectively. In Ar/Cl2 plasmas, for example, neutral chlorine reacts with the Si surfaces to form silicon chlorides. These compounds are then removed by the impinging ion fluxes. Hence the directionality of the ions (and thus the ion angular distribution function, or IAD), as well as the relative fluxes of neutrals and ions determines the amount of undercutting. One method of modeling device profile evolution is to simulate the moving solid-gas interface between the semiconductor and the plasma as a string of nodes. The velocity of each node is calculated and then the nodes are advanced accordingly. Although this technique appears to be relatively straightforward, extensive looping schemes are required at the profile corners. An alternate method is to use level set theory, which involves embedding the location of the interface in a field variable. The normal speed is calculated at each mesh point, and the field variable is updated. The profile comers are more accurately modeled as the need for looping algorithms is eliminated. The model we have developed is a 2-D Level Set Profile Evolution Simulation (LSPES). The LSPES calculates etch rates of a substrate in low pressure plasmas due to the incident ion and neutral fluxes. For a Si substrate in an Ar/C12 gas mixture, for example, the predictions of the LSPES are identical to those from a string evolution model for high neutral fluxes and two different ion angular distributions.(2) In the figure shown, the relative neutral to ion flux in the bulk plasma is 100 to 1. For a moderately isotropic ion angular distribution function as shown in the cases in the left hand column, both the LSPES (top row) and rude's string method (bottom row) predict tapered profiles. The LSPES uses an AD with a FWHM = 13.5 degrees, and rude's model uses a ratio of sheath voltage to ion temperature of 50. The more anisotropic IADs produce profiles with more vertical sidewalls and a wider bottom surface, as shown in the right hand column. Here, the LSPES has an AD with a FWHM = 2 degrees, and rude's model uses a ratio of 500 of sheath voltage to ion temperature. The agreement between the LSPES and rude's model is excellent in both cases. We will present etching profiles generated by the LSPES, including calculations of the re-emitted fluxes of both neutrals and ions off of the profile walls. In addition, we will show the effect of geometric structures (overhangs, etc.) on the etching profiles. Other physical aspects, such as surface diffusion, will also be included in the model.
Hwang, Helen H.↗
Evaluation of Fuzzy Rulemaking for Expert Systems for Failure Detection
Computer aids in expert systems were proposed to diagnose failures in complex systems. It is shown that the fuzzy set theory of Zadeh offers a new perspective for modeling for humans thinking and language use. It is assumed that real expert human operators of aircraft, power plants and other systems do not think of their control tasks or failure diagnosis tasks in terms of control laws in differential equation form, but rather keep in mind a set of rules of thumb in fuzzy form. Fuzzy set experiments are described.
Laritz, F.↗
Prospects of a mathematical theory of human behavior in complex man-machine systems tasks
A hierarchy of human activities is derived by analyzing automobile driving in general terms. A structural description leads to a block diagram and a time-sharing computer analogy. The range of applicability of existing mathematical models is considered with respect to the hierarchy of human activities in actual complex tasks. Other mathematical tools so far not often applied to man machine systems are also discussed. The mathematical descriptions at least briefly considered here include utility, estimation, control, queueing, and fuzzy set theory as well as artificial intelligence techniques. Some thoughts are given as to how these methods might be integrated and how further work might be pursued.
Johannsen, G.↗
Mathematical algorithms for approximate reasoning
Most state of the art expert system environments contain a single and often ad hoc strategy for approximate reasoning. Some environments provide facilities to program the approximate reasoning algorithms. However, the next generation of expert systems should have an environment which contain a choice of several mathematical algorithms for approximate reasoning. To meet the need for validatable and verifiable coding, the expert system environment must no longer depend upon ad hoc reasoning techniques but instead must include mathematically rigorous techniques for approximate reasoning. Popular approximate reasoning techniques are reviewed, including: certainty factors, belief measures, Bayesian probabilities, fuzzy logic, and Shafer-Dempster techniques for reasoning. A group of mathematically rigorous algorithms for approximate reasoning are focused on that could form the basis of a next generation expert system environment. These algorithms are based upon the axioms of set theory and probability theory. To separate these algorithms for approximate reasoning various conditions of mutual exclusivity and independence are imposed upon the assertions. Approximate reasoning algorithms presented include: reasoning with statistically independent assertions, reasoning with mutually exclusive assertions, reasoning with assertions that exhibit minimum overlay within the state space, reasoning with assertions that exhibit maximum overlay within the state space (i.e. fuzzy logic), pessimistic reasoning (i.e. worst case analysis), optimistic reasoning (i.e. best case analysis), and reasoning with assertions with absolutely no knowledge of the possible dependency among the assertions. A robust environment for expert system construction should include the two modes of inference: modus ponens and modus tollens. Modus ponens inference is based upon reasoning towards the conclusion in a statement of logical implication, whereas modus tollens inference is based upon reasoning away from the conclusion. These algorithms allow one to reason accurately with uncertain data. The above environment can replicate state-f-the-art expert system environments which provides a continuity between the current expert systems which cannot be validated or verified and future expert systems which should be both validated and verified
Murphy, John H.↗