Elementary Set Theory Research Articles

A random set description of a possibility measure and its natural extension

The relationship is studied between possibility and necessity measures defined on arbitrary spaces, the theory of imprecise probabilities, and elementary random set theory. It is shown how special random sets can be used to generate normal possibility and necessity measures, as well as their natural extensions. This leads to interesting alternative formulas for the calculation of these natural extensions.

Application of Statistical Processes on a Set Theory Model for Anomaly Handling in System Requirements Analysis

AbstractIn 1995 Ronald Carson developed a mathematical model for system requirements analysis (SRA) utilizing set theory to explicitly treat unwanted or unexpected conditions that may be experienced by the system being analyzed. He showed that the SRA model can be used to derive completeness criteria. He claimed that the highest utility of the model would be found in the area of software requirements because of the focus on system states.A simple example using a Power Distribution Unit (PDU) was made in order to clarify the salient points. This simple example had only three states and six transitions, which were identified as nine antecedents (conditions which precede). Eight requirements were generated for the PDU.In this paper, the author assumes that the requirements for this simple example are stochastic, rather than deterministic, in nature. The arguments in support of this assumption are compelling. One is that many processes which are defined by a set of requirements are themselves stochastic in nature. Another is that a near deterministic model can be developed from a stochastic model in many applications. One can also argue that a stochastic process is more robust and better suited to a set of requirements subject to change due to ambiguities in syntax or lexicon or in how a system is thought to behave.The entire set of requirements constitutes the probabilistic universe of the system. Each requirement has a probability associated with it that the requirement is satisfied. Furthermore, the requirements may have dependencies.The methodology is developed from simple Bayesian relationships that have been used in a number of computer automated tools. What may seem different is that the probability is a characteristic of the antecedent rather than the requirement itself.This model, though not desirable for all classes of requirements, may fit a number of situations where the outcome of the requirement is not certain.In conclusion, the SRA model utilizing elementary set theory is expanded from a deterministic system to a stochastic system. The method is rigorous, and can account for 1) ambiguities in language, 2) unforeseen conditions, 3) undefined initial conditions, and 4) probability density function of primitives. It can also make the system defined by a set of requirements more robust.

Software Design of System Requirements Utilizing Set Theory Model for Anomaly Handling

AbstractIn 1995 Ronald Carson developed a mathematical model for system requirements analysis (SRA) utilizing set theory to explicitly treat unwanted or unexpected conditions that may be experienced by the system being analyzed. He showed that the SRA model can be used to derive completeness criteria. He claimed that the highest utility of the model would be found in the area of software requirements because of the focus on system states.A simple example using a Power Distribution Unit (PDU) was made in order to clarify the salient points. This simple example had only three states and six transitions, which were identified as nine antecedents (conditions which precede). Eight requirements were generated for the PDU.In this paper, the requirements for this simple example, along with its principles, are used to construct a simple software implementation in graphical fashion in order to test the effectiveness of the methodology for software.In conclusion, the SRA model utilizing elementary set theory needs some refinement for software applications. Though the method is rigorous, problems surface because of 1) ambiguities in language, 2) unforeseen conditions, 3) undefined initial conditions, and 4) probability density function of primitives.In addition, the start condition of the timers are undefined, as are the initial states of primitives, e.g. valid input, degraded input and inhibit output and degraded output.

7.3.3 A SET THEORY MODEL FOR ANOMALY HANDLING IN SYSTEM REQUIREMENTS ANALYSIS

AbstractA mathematical model for system requirements analysis (SRA) is developed utilizing elementary set theory. The model provides a means to explicitly treat unwanted or unexpected conditions that may be experienced by the system being analyzed. The SRA model can also be used to derive completeness criteria for the development of system states, as well as the completeness of the SRA itself. The requirement antecedents (the “under what conditions” for each function and performance requirement) are categorized in accordance with their associated consequences. Although generally applicable, the model finds its highest utility in the area of software requirements because of their unique design processes which focus on system states. As such, a relationship with software development “formal methods” is described.

The generative power of feature geometry

This paper presents a formalization of the common structure underlying different proposals concerning the geometry of phonological features. Using the concepts and tools of elementary set theory, lattice theory, and the theory of independent Boolean algebras, ametatheory of autosegmental feature structure is developed. This metatheory enables us to compare contemporary autosegmental phonology to the now abandoned “standard” phonology and to the historically important Pānini-type phonology as well, and to investigate the restrictions on the use of features in phonological theory.

On plural reference and elementary set theory

The view that plural reference is reference to a set is examined in light of George Boolos's treatment of second-order quantification as plural quantification in English. I argue that monadic second-order logic does not, in Boolos's treatment, reflect the behavior of plural quantifiers under negation and claim that any sentence that properly translates a second-order formula, in accordance with his treatment, has a first-order formulation. Support for this turns on the use of certain partitive constructions to assign values to variables in a way that makes Boolos's reading of second-order variables available for a first-order language and, with it, the possibility of interpreting quantification in an unrestricted domain.

Object Command Language: a formalism to build molecular models and to analyze structural parameters in macromolecules, with applications to nucleic acids

We have written a programming language OCL (Object Command Language) to solve, in a general way, two recurring problems that arise during the construction of molecular models and during the geometrical characterization of macromolecules: how to move precisely and reproducibly any part of a molecular model in any user-defined local reference axes; and how to calculate standard or userdefined structural parameters that characterize the complex geometries of any macromolecule. OCL endows the user with three main capabilities: the definition of subsets of the macromolecule, called objects in OCL, with a formalism from elementary set theory or lexical analysis; the definition of sequences of elementary geometrical operations, called procedures in OCL, enabling one to build arbitrary three-dimensional (3D) orthonormal reference frames, to be associated with previously defined objects; and the transmission of these definitions to programs that allow one to display, to modify and to analyze interactively the molecular structure, or to programs that perform energy minimizations or molecular dynamics. Several applications to nucleic acids are presented.

Automated deduction in von Neumann-Bernays-G�del set theory

I present a new clausal version of NGB set theory, and compare my version with that first given by Boyer et al. [4]. A complete set of reductions for Boolean rings is given, derived from those of Hsiang [7]. I list over 400 theorems proved semiautomatically in elementary set theory, and supply the proofs of several of these, including Cantor's theorem. I present a semiautomated proof that the composition of homomorphisms is a homomorphism, thus solving a challenge problem given in [4]. Using the clauses and heuristics presented, there is no apparent obstacle to the semiautomated development of set theory through considerably more difficult theorems.

Standard and nonstandard analysis: fundamental theory, techniques and applications

Elementary set theory and logic number systems real numbers and hyperreal numbers applications of infinitesimal techniques in real analysis the transfer principle and the use of general nonstandard techniques.

Some logical issues in Madhyamaka thought

In this paper we should like to argue that the prasajya negation of the Madhyamaka school of Buddhist philosophy is not the same as that of the non-Madhyamaka schools (that is, that the distinction between prasajya and paryudâsa negation is not drawn in the same way). We should also like to argue that the terms and concepts of elementary set theory, employed in conjunction with the elementary predicate calculus, are useful in the explication of the laws of the excluded middle and of contradiction and also in the clarification of the

Variations on a theme of Euclid

Moritz Pasch's axiomatic definition of a Euclidean line interval [1, 2] is generalized to allow arbitrarily many multiple points and to apply especially to the point sets associated with unclosed geodesics having two endpoints. Many other ‘uniquely extensible’ examples of the theory exist (see section 2). The main primitive ideas are a set of objects S(points), a set of subsets of S(G strings, generalizing intervals) and end points of G strings. It is a theory in the large, described with elementary set theory. To construct a theory of linear order we (1) define (in either of two ways) a relation on the set of Gstrings having endpoint A,(2) show that the relation is an equivalence relation, assuming several generalizations of Pasch's order properties of intervals, (3) define a ray emanating from Aas an equivalence class of Gstrings with endpoint Aand show that a ray is linearly ordered. A G string is ‘oriented’ with theorem 1.4. To specialize this order theory to the Euclidean or projective lines read the...

Programming Vision Systems and Integrating them with Robots

We present LIVIA, a language for writing application programs for industrial vision systems. LIVIA is a Pascal-like language with specific instructions for vision. The objects are identified by the values of their invariant features. Operations on the objects can be done using the elementary set theory. The recognition of objects in a data base and the selection of any of their features is easily programmed

Specification of the UNIX Filing System

A specification of the UNIX filing system is given using a notation based on elementary mathematical set theory. The notation used involves very few special constructs of its own. The specification is detailed enough to capture the filing system's behavior at the system call level, yet abstracts from issues of data representation, whether in programs or on the storage medium, and from the description of any algorithms which might be used to implement the system. The presentation of the specification is in several stages, each new stage building on its predecessors; major concepts are introduced separately so that they may be easily understood. The notation used allows these separate stages to be joined together to give a complete description of each filing system operation-including its error conditions. Features of the specification notation are explained as they are used, and the Appendix gives the definitions of the symbols drawn from set theory.

Anatomy of a text analysis package

A model of a text processing package is described which is based on elementary set theory. This is used to describe procedures for computing indexes, concordances, and collocations. Aspects of the implementation of the model are given.

Eurisko: A program that learns new heuristics and domain concepts: The nature of Heuristics III: Program design and results

The AM program, an early attempt to mechanize learning by discovery, has recently been expanded and extended to several other task domains. AM's ultimate failure apparently was due to its inability to discover new, powerful, domain-specific heuristics for the various new fields it uncovered. At that time, it seemed straight-forward to simply add ‘Heuristics’ as one more field in which to let AM explore, observe, define, and develop. That task—learning new heuristics by discovery—turned out to be much more difficult than was realized initially, and we have just now achieved some successes at it. Along the way, it became clearer why AM had succeeded in the first place, and why it was so difficult to use the same paradigm to discover new heuristics. In essence, AM was an automatic programming system, whose primitive actions were modifications to pieces of LISP code, code which represented the characteristic functions of various math concepts. It was only because of the deep relationship between LISP and Mathematics that these operations (loop unwinding, recursion elimination, composition, argument elimination, function substitution, etc.) which were basic LISP mutators also turned out to yield a high ‘hit rat’ of viable, useful new math concepts when applied to previously-known, useful math concepts. But no such deep relationship existed between LISP and Heuristics, and when the basic automatic programming operators were applied to viable, useful heuristics, they almost always produced useless (often worse than useless) new rules. Our work on the nature of heuristics has enabled the construction of a new language in which the statement of heuristics is more natural and compact. Briefly, the vocabulary includes many types of conditions, actions, and descriptive properties that a heuristic might possess; instead of writing a large lump of LISP code to represent the heuristic, one spreads the same information out across dozens of ‘slots’. By employing this new language, the old property that AM satisfied fortuitously is once again satisfied: the primitive syntactic operators usually now produce meaningful semantic variants of what they operate on. The ties to the foundations of Heuretics have been engineered into the syntax and vocabulary of the new language, partly by design and partly by evolution, much as John McCarthy engineered ties to the foundations of Mathematics into LISP. The EURISKO program embodies this language, and it is described in this paper, along with its results in eight task domains: design of naval fleets, elementary set theory and number theory, LISP programming, biological evolution, games in general, the design of three-dimensional VLSI devices, the discovery of heuristics which help the system discover heuristics, and the discovery of appropriate new types of ‘slots’ in each domain. Along the way, some very powerful new concepts, designs, and heuristics were indeed discovered mechanically. Characteristics that make a domain ripe for AM-like exploration for new concepts and conjectures are explicated, plus features that make a domain especially suitable for EURISKO-level exploration for new heuristics.

A BASIC MODEL OF INTERACTING SUBJECTS

Abstract In studying the problem of interaction between subjects, an approach which allows us to define in an unambiguous way the concepts of symmetrical, complementary and parallel interaction is proposed. This approach makes use of a point of view within which it is possible to develop a rational model based only on some fundamental elements of set theory and mathematical logic The model allows us to develop a probabilistic theory of change, the representation basis of which is furnished by the 16 basic set functions. The functions represent operations on two partially overlapping sets, which will be called the “worlds”, of interacting subjects.. Both an interaction test that allows a “measure” of the relation parameters and the preliminary results of a dyadic interaction experiment are presented Some remarks can be made from the results of the present experiment and from other similar ones. Among these there are evidences that the symmetrical, complementary and parallel classes of interactions belong t...

Towards the automation of set theory and its logic

We give a description of a prototype extensible deductive system for elementary set theory which is based on truth-value preserving transformations, and then describe an experimental application of it to the first few chapters of W. V. O. Quine's book: Set Theory and its Logic.

Elements of Set Theory.

Elements of Set Theory.

Chemistry and Logical Structures. 2. The Role of Set Theory

Summary In this second paper the role of set theory in chemistry at the present time is considered. After an introduction to basic set theory, several of the more important applications to chemistry are explained and discussed. It is pointed out that much of our existing knowledge of theoretical chemistry can easily be accommodated within the framework of elementary set theory.

Twin concordance: A set theoretic and probability theory approach

The concepts and assumptions used in twin concordance studies are made explicit with a mathematical treatment. The logic connecting those assumptions to the usual interpretations of twin studies is substantiated in a rigorous manner through the use of elementary set and probability theory.