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This chapter presents a binary logic framework whose function elements are invariant under permutation and complementary operations. The entire framework is described using 4 levels of hierarchy: variables, states, functions, and logic functionals. Under the proposed framework, it is possible to determine higher level function complexity by analysing lower levels of organisation characteristics. These characteristics can be determined quite accurately because the symmetry conditions of variable and state organisations have invariant logic functions and a corresponding logic functional organisation. More symmetrical arrangement at state level creates more symmetrical permutations within the function space. Lower level properties are highly influential on the higher level properties of function components within a logic functional space. The proposed framework provides a logic foundation to describe complex binary systems using lower level properties, making analysis of systems more efficient and less calculation intensive. Different global coding schemes are discussed and typical two-variable cases of logic functionals are illustrated.

In modern logic, various systems have been proposed extending classical Boolean logic & switching theory. Such logic frameworks include multiple-valued logic, probability logic, fuzzy logic, module logic, quantum logic and various other frameworks. Although these extensions have been applied to many applications in mathematics, in science and in engineering, all extensions to Boolean logic invalidates at least one of the six fundamental rules of Boolean logic shown in L1 to L6. We propose a new framework of logic, variant logic, extending Boolean logic whilst satisfying the six fundamental rules (L1–L6). By defining the Variant–Invariant behaviour of logical operations, this framework can be constructed using four types of general operators. Main results of the chapter are summarized in , respectively. To show significant differences between classical logic and new variant logic, invariant properties of this hierarchical organization are discussed. Simplest cases of one-variable conditions are illustrated. Variant logic can provide the necessary framework to support analysis and description of Cellular Automata, Fractal Theory, Chaos Theory and other systems dealing with complexity. Such applications of this framework will be explored in future papers.

Four variant measures are used to represent combinatorial functions including binomial coefficients. These variant measures are based on two types of -bit vectors. Type A corresponds to non-periodic boundary conditions, while Type B corresponds to periodic boundary conditions. For each type, groups containing the four variant measures are formed, which are invariant against permutative and associative operations. By mapping two group elements of Type B on coefficients of binomial decompositions, patterns similar to Pascal’s triangle are observed.

A method to classify one-dimensional binary sequences using three parameters intrinsic to the sequence itself is introduced. The classification scheme creates combinatorial patterns that can be arranged in a two-dimensional triangular structure. Projections of this structure contain interesting properties related to the Pascal triangle numbers. The arrangement of numbers within the triangular structure has been named “triangular numbers”, and the essential parameters, elementary equation, and sequencing schemes are discussed as well as visualizations of sample distributions, special cases, and search results. We believe this to be a novel finding as sequences generated using this method are not contained in the On-Line Encyclopedia of Integer Sequences or OEIS.

Symmetric Boolean functions play a key role in stream ciphers. Symmetric constructions provide core components in cryptographic applications. In this chapter, four meta symmetric clustering schemes (combination, crossing, variant and rotation) are organized in a hierarchy for variables of 0–1 vectors in measuring phase spaces. Local counting properties in a cluster and global counting properties in a given level are formulated. From selected symmetric clusters, a number of various symmetric Boolean functions are formulated. Counting properties on symmetric clusters, vectors in selected clusters and special symmetric Boolean functions are listed. Four sets of symmetric Boolean functions are compared. Properties of symmetric clusters and Boolean functions are discussed. Main results are expressed in theorems and tables. Among four meta schemes, the variant scheme presents novel properties approximately with clusters on a 2D phase space different from other schemes: combinatorial , crossing and rotation on 1D measuring phase spaces, respectively. The variant pseudorandom number generator is a similar approach on RC4 and HC128 stream ciphers using word-oriented 0–1 vectors. Further advanced researches and explorations on relevant optimal configurations are required.

Using four measures in Type B, there are 11 invariant expressions to form elementary equations of variant measurement. In this chapter, two invariant expressions are selected to illustrate sample procedures from elementary equations to relevant variant maps. Using various projections and multiple levels of representations, complicated binomial coefficients and their variations are illustrated under various conditions. Using multinomial coefficients, multiple viewpoints are used for references. Due to this type of variation framework contains rich structures, further explorations are required from multiple levels on both theoretical foundation and practical applications.

Sequences of random variables play a key role in probability theory, stochastic processes, and statistics to analyze dynamic behavior. Speckle patterns have emerged as useful tools to explore space–time variations of random sequences in various measurement applications of comprehensive properties in complex space–time variation events. In this chapter, a variant map system is proposed to analyze statistical properties of random sequences in visual representations. An input 0–1 sequence will be divided into multiple segments and each segment of a fixed length will be transformed into a 2-tuple pair of measures. Five measuring sets are identified and rearranged in a 1D or 2D numerical array as a histogram representing a visual map. These five types of maps consist of two types in 1D format as classical maps and three types in 2D format as variant maps. Properties are analyzed on all five types of maps. A cryptographic sequence of the AES cipher is selected as a sample stream. The five types of visual maps are generated and refined clustering characteristics are organized into four groups on changes of segmented and shifted lengths for visual comparisons on enlarged 2DP maps. Speckle patterns of various distributions are observed. Three variant maps with distinct statistic distributions could be useful to provide new visual tools to explore comprehensive cryptographic sequences on complex nonlinear dynamic behavior in global network environments.

Various random streams have different stationary properties. It is necessary to use statistical probability and time series to evaluate quality of stationary randomness. In this chapter, a testing model is used on three maps for a random sequence. Multiple segments are divided on the shifted sequence as three measuring sets. For a map, the maxima are extracted and three maximal values are identified. 2D maps represent stationary randomness. Conditions of station random/stationary sequences are investigated. Testing sets are collected from three types of six random resources: AES, DES, A5, RC4, Australian National University (ANU), and University of Science and Technology of China (USTC) (two block ciphers, two stream ciphers, and two quantum ciphers). Six random sequences are selected. Measurements of stationary randomness are compared. There are only 0.0034–4.27% differences that are recognized. Using variation ratios, six samples are composed of three variation categories on {AES, DES}, {A5, RC4}, and {ANU, USTC}, respectively. From a measuring viewpoint, all six samples are showing distinguished stationary randomness properties.

Applying network topology schemes, two types of three levels of meta knowledge representations have been established. This chapter proposes a meta model on concept cell that provides a meta organisation of knowledge in natural and artificial intelligent systems structurally.

The Simple Ballot Model (SBM) and the Component Ballot Model (CBM)—are proposed for solving uncertainty in an election when two candidates gain the same number of votes under the approval rule. The SBM establishes a framework to support counting. In separating the two candidates, it is essential to extract additional information from dominantly valid votes. The CBM uses probability matrices, vectors and permutation group as components. A stable-voting mechanism under permutation invariant can be created to distinguish candidates. The result of the chapter establishes a voting authority to resolve uncertainty of two candidates under the approval rule.