Mathematical Foundations for Manufacturing

We introduce geometric consideration into the theory of formal languages. We aim to shed light on our understanding of global patterns that occur on infinite strings. We utilise methods of geometric group theory. Our emphasis is on large scale geometries. Two infinite strings have the same large scale geometry if there are colour preserving bi-Lipschitz maps with distortions between the strings. Call these maps quasi-isometries. Introduction of large scale geometries poses several questions. The first question asks to study the partial order induced by quasi-isometries. This partial order compares large scale geometries; as such it presents an algebraic tool for classification of global patterns. We prove there is a greatest large scale geometry and infinitely many minimal large scale geometries. The second question is related to understanding the quasi-isometric maps on various classes of strings. The third question investigates the sets of large scale geometries of strings accepted by computational models, e.g. Buchi automata. We provide an algorithm that describes large scale geometries of strings accepted by Buchi automata. This links large scale geometries with automata theory. The fourth question studies the complexity of the quasi-isometry problem. We show the problem is Σ 3 0-complete thus providing a bridge with computability theory. Finally, the fifth question asks to build algebraic structures that are invariants of large scale geometries. We invoke asymptotic cones, a key concept in geometric group theory, defined via model-theoretic notion of ultra-product. Partly, we study asymptotic cones of algorithmically random strings thus connecting the topic with algorithmic randomness.

Cubic sets are the very useful generalization of fuzzy sets where one is allowed to extend the output through a subinterval of [ 0 , 1 ] and a number from [ 0 , 1 ] . Generalized cubic sets generalized the cubic sets with the help of cubic point. On the other hand Soft sets were proved to be very effective tool for handling imprecision. Semigroups are the associative structures have many applications in the theory of Automata. In this paper we blend the idea of cubic sets, generalized cubic sets and semigroups with the soft sets in order to develop a generalized approach namely generalized cubic soft sets in semigroups. As the ideal theory play a fundamental role in algebraic structures through this we can make a quotient structures. So we apply the idea of neutrosophic cubic soft sets in a very particular class of semigroups namely weakly regular semigroups and characterize it through different types of ideals. By using generalized cubic soft sets we define different types of generalized cubic soft ideals in semigroups through three different ways. We discuss a relationship between the generalized cubic soft ideals and characteristic functions and cubic level sets after providing some basic operations. We discuss two different lattice structures in semigroups and show that in the case when a semigroup is regular both structures coincides with each other. We characterize right weakly regular semigroups using different types of generalized cubic soft ideals. In this characterization we use some classical results as without them we cannot prove the inter relationship between a weakly regular semigroups and generalized cubic soft ideals. This generalization leads us to a new research direction in algebraic structures and in decision making theory.

In this study, the basic characteristics of semigroups, a basic idea in algebraic structures, are examined, along with their several classifications. A single associative binary operation defines semigroups, which have a variety of characteristics that affect how they behave structurally. The fundamental characteristics that set semigroups apart from other algebraic structures like groups and monoids such as associativity, identity elements, and idempotency—are thoroughly examined at the outset of this research. The classification of semigroups according to particular characteristics like commutativity, regularity, and simplicity is further examined in this study. The significance of Green’s relations for categorizing semigroups and comprehending their internal organization is highlighted in particular. The research also looks at the applications of these categories in automata theory, coding theory, and dynamic system modelling, among other more general mathematical contexts. This study attempts to give a clearer knowledge of the fundamental nature of semigroups and their role within algebra through an extensive survey of the existing literature and the introduction of new viewpoints on semigroup properties. The results provide information that could stimulate additional investigation into semigroups and applications in a variety of mathematical fields.

In [Church 56, sect.07] mathematical logic or symbolic logic, or logistics, is defined as “the subject of formal logic when treated by the method of setting up a formalized language”. Among formalized languages there are purely logical (propositional and predicate) languages, languages of arithmetic and languages of set theory. Predicate languages — elementary (=first-order) as well as non-elementary (of higher orders) — are used for formal descriptions of properties of mathematical structures (first of all algebraic structures). We can use them for an axiomatic description of different classes of structures. Languages of arithmetic describe the set of natural numbers (which hardly can be described axiomatically and must anyway be regarded as existing before any axiomatic considerations). Languages of set theory have no clear semantics and are intended to formulate various axiomatic theories.

Application of symbolic logic to the design axioms

Formal logic provides us with a powerful set of techniques for criticizing some arguments and showing others to be valid. These techniques are relevant to all of us with an interest in being skilful and accurate reasoners. In this very accessible book, extensively revised and rewritten for the second edition, Peter Smith presents a guide to the fundamental aims and basic elements of formal logic. He introduces the reader to the languages of propositional and predicate logic, and develops natural deduction systems for evaluating arguments translated into these languages. His discussion is richly illustrated with worked examples and exercises, and alongside the formal work there is illuminating philosophical commentary. This book will make an ideal text for a first logic course and will provide a firm basis for further work in formal and philosophical logic.

This chapter demonstrates the application of a qualitative formal method, logical formalization, to organization and management theory. Organizational arguments are usually phrased out in some natural language in the first place. After separating the premises (facts, definitions) of a natural language argument from its conclusions (predictions), this preprocessed text is translated into a logical language. Then, experimentation can begin if the logical formulae standing for the verbal premises imply the putative conclusions as formal theorems. If not, what kind of modifications can make these outcomes follow? What other theorems are implied from the same argument core? A substantial advantage of using symbolic logic over many branches of applied mathematics is that logical models can quite closely map the intended meaning of assertive sentences, while the deduction of conclusions can proceed with the rigor of mathematical proofs. The examples highlight how different logical languages, different dialects, can be used to the idiosyncrasies of the subject. The proof and the translation process from natural language statements to logical models are supported by user-friendly theorem-prover softwares. The appliers of the method need not be logic experts; what they need are analytical skills, sharp eyes at formula evaluation, and some stamina. The promise of using symbolic logic is combining the flexibility of qualitative reasoning with exactness in drawing conclusions from complex arguments. The chapter is to show how and in which extent logical formalization can fulfill this promise.

A problem which confronts the developers of concurrent logic programming (CLP) systems concerns the design of the programming environment, particularly the provision of debugging tools. Debugging tools are useful for many activities besides identifying bugs: they can help in program testing and demonstration, in software experiments, and in teaching the language semantics. For CLP languages the questions of what debugging tools should be provided, and of how they should be used, are still open. Although the languages are closely related to other logic programming and concurrent programming languages, they are sufficiently different that new debugging techniques are required. This paper describes a primarily channel-oriented debugging methodology and a set of debugging tools that we have developed in the light of our experience in using and teaching the CLP language Parlog. With these tools a programmer can test a program by observing communication on channels, opening up a process to examine the activity on internal channels and, if necessary, to check in detail the execution steps of a process. Almost all of the ideas, which are equally applicable to other CLP languages, have been implemented in two commercial Parlog systems. As well as fulfilling the needs of existing users of CLP languages, we believe that the tools emphasise the attractions of CLP as a concurrent programming paradigm, since they are made possible by the unique attributes of CLP.

This chapter presents an overview of prospective new conceptual categories expected to characterize the new or second-generation systemics. We refuse here to call it systemics 2.0 (Minati, 2016; Minati, Abram, & Pessa, 2016) even though GOFS, or systemics 1.0, had several releases, as mentioned in Sect. 1.1, including the theory of dynamical systems, automata theory, control theory, cybernetics, games theory and system dynamics.

Simulation of three-dimensional cell growth with application to optimal tumor treatment

The purpose of this paper is to design an expert system for control system design. The architecture of ESCSD is designed and implemented using CLIPS, which is an expert system building too...

A topology-dependent expert system for DC design of a self-biased single transistor stage was developed and tested. The test case system was implemented in Turbo Prolog and was run on an XT-compatible personal computer. The rule-based system includes four groups of rules: heuristic, general network, topology-dependent network, and default rules. It is shown that the expert system mimics the human design procedure and can handle mathematically ill-defined design problems. A main inherent feature of the proposed expert system is its ability to produce a design solution for any given combination of non-conflicting input specifications. >

An Expert System for Tunnel Design

This work presents the methodology utilized for the development of the numerical module of SIPREX, an expert system for digital filter design. This system integrates knowledge on the different techniques and application areas of digital signal processing with powerful numerical algorithms, representing a valuable tool for the solution of problems involving analysis and synthesis of digital filters.

The TRANSEPT Family of Expert Systems for the Preliminary Design of Power Networks