"The origin of this book goes back to a talk - I remember it as if it were yesterday - that I had in 1927 with a mathematics professor of mine, Dr. George D. Birkhoff, when I was a sophomore at Harvard. I remarked to him that I wished there were an algebra of language - an algebra by means of which one could calculate the answer to an argument instead of going to sea in an ocean of words. He said to me that there was, and told me of a book called "The Laws of Thought" by George Boole, published in 1854. I quickly sought the book in the library, and although it confused me, since I did not then know about later improvements in Boolean algebra, nevertheless, the fact that here was an algebra dealing with things like classes and statements, and operations like AND, OR, NOT, opened wide for me a most exhilarating vista - the possibility that all the language of thought could really become calculable like mathematics. This possibility has become more real than ever before because of the calculating power of automatic computers. So, I hope that the contents of this book may likewise stir the imagination and spirits of many readers". (This is an extraction from the preface to the book "Symbolic Logic and Intelligent Machines" by Edmund C. Berkeley, written in Newtonville, Massachusetts, and published in 1959)
Decades later, another researcher, Gennady Melnikov, in his book "The ABC of Mathematical Logic", published in 1967, in the section: "Why Mathematical Logic Gravitates to Proper Logic", in a journalistic form expressed his observation that despite the fact that Boolean algebra, originally created for calculating logical reasonings, due to development of many engineering branches of knowledge, has expanded its application as a means of describing and modeling discrete systems whose property states can, as in logic, take only two states. The author draws attention to the fact that in 1938, Claude Shannon used Boolean algebra for analysing and calculating relay circuits. And suggests that: "Since by the time of publication of the book, objects, whose elements are capable of entering into discrete relations (for example, electrical or electronic circuits), became widespread — the mathematical logic should be either considered as the mathematics of the finite discrete qualitative relations, modeling structures, or as the science of discrete modeling of systems whose properties have finite number of states". (Unfortunately, this book, written originally in Russian, has never been translated into other languages.)
Kaleidoscope Project is a pilot research and development initiative intended to introduce the notion of Qualitative Dynamical Systems and to explore the feasibility of mathematical modeling of behavior of the qualitative dynamical systems by representing their qualitative dependence relations in the form of algebraic expressions. The Qualitative Dynamical Systems, the Qualitative Dynamical Systems Mathematical Modeling Technique, and the Modeling Solutions proposed by the Modeling Technique are referred to below, in the course of annotations of the content of the cluster of the webpages of the Project.
A note. The last section, named "Downloadables", located below all other sections, contains a reference to the page: "Download Kaleidoscope Project Applications", which suggests downloading two Java Desktop Applications developed in the Kaleidoscope project for macOS and Windows operating systems.
Page: Qualitative Dynamical Systems, Their
Behavior and Mathematical Modeling
The objective of this page is to introduce the concept of the class of Qualitative Dynamical Systems and to describe the characteristics of their nature and behavior, which constitute the foundation for creating mathematical models of these systems. This page identifies the atomic elements of behavior of the systems and describes basic ways of composing these elements. Open the Page
Page: Qualitative Dynamical Systems Mathematical Modeling Technique
This page is to introduce Qualitative Dynamical Systems Mathematical Modeling Technique, developed as part of the Kaleidoscope Project, and to describe modeling solutions proposed by the Modeling Technique. The purpose of the technique and the solutions is to enable the creation of qualitative systems simulation models, capable of reproducing the behavior of existing or being designed systems. Open the Page
Page: The Algebra of Symbols
This page is the description of the algebraic system, called "The Algebra of Symbols" (AoS), which is used for the computation of the functions of models of qualitative systems. The AoS is defined as part of the Qualitative Dynamical Systems Mathematical Modeling Technique, described on the previous page. It is the base for qualitative systems modeling formalisms, descriptions of which are presented right after this page. Open the Page
Qualitative Dynamical Systems Modeling
Technique's Modeling Solutions
There are three major modeling solutions based on the Algebra of Symbols.
Solution 1: Extended Disjunctive Normal Form
Expression Calculator
Page, titled: "Extended Disjunctive Normal Form Expression Calculator", illustrates a way of describing the dependence of states of a single qualitative variable on the state of several groups of qualitative variables. This dependence is computed by a qualitative function, implemented in the form of an algebraic expression written as the composition of operations of the Algebra of Symbols. The structure of the expression is called Extended Disjunctive Normal Form (XDNF). Open the Page
Solution 2:
Qualitative Dynamical Systems Vector
State-Space Matrix Equation Calculator
The page is titled: "Qualitative Systems Vector State-Space and Matrix Model Equation Calculator". It describes the qualitative dynamical systems mathematical modeling technique based on the representation of qualitative systems in the form of matrix state equations. Using several simple models and the step-by-step computation style, this page illustrates the mechanism of simulation of behavior of the modeled systems as the sequence of transformations of states of the models and visualization of the results of the computation of each step. Open the Page
Solution 3: Multicolored Logical Net Modeling
Formalism
This page describes a way of modeling qualitative systems in the form of graphically depicted properties of the systems, facts of occurrence of situations in them, dependencies of facts on the current state of properties, and dependencies of the next state of properties on the state of facts. This makes it possible to represent these models on such drawing surfaces as a computer screen or a printed sheet of paper. Open the Page
Desktop Applications Developed by and Used
In the Kaleidoscope Project
In addition to the pages that introduced: "Qualitative Dynamical Systems", "Qualitative Dynamical Systems Modeling Technique", and the pages that described modeling solutions, suggested by the Technique, the following two pages provide several screenshots illustrating different views of the software tools developed in the Kaleidoscope project for supporting creation and simulation of the McLN models.
The page titled: "Qualitative Dynamical Systems Modeling & Simulation Environment" is named after the application described on it. It presents several snapshots of the application screens and provides explanations of the content of these screenshots. Open the Page
The page titled: "Space Exploration Mission // An Agent-Robot Behavior Simulator", is also named after the application described on it. This application imitates the Research Facility that is monitoring the state and actions of the remote robot-rover, whose controlling unit is implemented as Multicolored Logical Net (McLN). The page is created for testing if the expressive power of the McLN is sufficient for it to be used as a behavior controller. Open the Page
Downloadables
Both presented software applications support Qualitative Dynamical Systems Mathematical Modeling Technique, and are open-source Java desktop applications, provided in the form of certified installers. They are available for macOS and Windows computers, and can be downloaded from the page "Download Kaleidoscope Project Applications". Open the Page