Bipolar Cognitive Mapping and Decision Analysis

The focus of this chapter is on cognitive mapping and cognitive-map-based (CM-based) decision analysis. This chapter builds a bridge from mental quantum gravity to social quantum gravity. It is shown that bipolar relativity, as an equilibrium-based unification of nature, agent and causality, is naturally the unification of quantum bioeconomics, brain dynamics, and socioeconomics as well. Simulated examples are used to illustrate the unification with cognitive mapping and CM-based multiagent decision, coordination, and global regulation in international relations.

YinYang Bipolar Quantum Bioeconomics for Equilibrium-Based Biosystem Simulation and Regulation

As a continuation of Chapter 8, this chapter presents a theory of bipolar quantum bioeconomics (BQBE) with a focus on computer simulation and visualization of equilibrium, non-equilibrium, and oscillatory properties of YinYang-N-Element cellular network models for growing and degenerating biological processes. From a modern bioinformatics perspective, it provides a scientific basis for simulation and regulation in genomics, bioeconomics, metabolism, computational biology, aging, artificial intelligence, and biomedical engineering. It is also expected to serve as a mathematical basis for biosystem inspired socioeconomics, market analysis, business decision support, multiagent coordination and global regulation. From a holistic natural medicine perspective, diagnostic decision support in TCM is illustrated with the YinYang-5-Element bipolar cellular network; the potential of YinYang-N-Element BQCA in qigong, Chinese meridian system, and innate immunology is briefly discussed.

Bipolar Fuzzy Sets and Equilibrium Relations

Based on bipolar sets and quantum lattices, the concepts of bipolar fuzzy sets and equilibrium relations are presented in this chapter for bipolar fuzzy clustering, coordination, and global regulation. Related theorems are proved. Simulated application examples in multiagent macroeconomics are illustrated. Bipolar fuzzy sets and equilibrium relations provide a theoretical basis for cognitive-map-based bipolar decision, coordination, and global regulation.

Introduction

(Note: In this book we refer to relativity theories defined in spacetime geometry as spacetime relativity. Thus, all previous relativity theories by Galileo, Newton, Lorenz, and Einstein belong to spacetime relativity. This terminological treatment is for distinguishing YinYang bipolar geometry from spacetime.)

Bipolar Quantum Linear Algebra (BQLA) and Bipolar Quantum Cellular Automata (BQCA)

This chapter brings bipolar relativity from the logical and relational levels to the algebraic level. Following a brief review on traditional cellular automata and linear algebra, bipolar quantum linear algebra (BQLA) and bipolar quantum cellular automata (BQCA) are presented. Three families of YinYang-N-Element bipolar cellular networks (BCNs) are developed, compared, and analyzed; YinYang bipolar dynamic equations are derived for YinYang-N-Element BQCA. Global (system level) and local (element level) energy equilibrium and non-equilibrium conditions are established and axiomatically proved for all three families of cellular structures that lead to the concept of collective bipolar equilibrium-based adaptivity. The unifying nature of bipolar relativity in the context of BQCA is illustrated. The background independence nature of YinYang bipolar geometry is demonstrated with BQLA and BQCA. Under the unifying theory, it is shown that the bipolar dimensional view, cellular view, and bipolar interactive view are logically consistent. The algebraic trajectories of bipolar agents in YinYang bipolar geometry are illustrated with simulations. Bipolar cellular processes in cosmology, brain, and life sciences are hypothesized and discussed.

Bipolar Quantum Lattice and Dynamic Triangular Norms

Bipolar quantum lattice (BQL) and dynamic triangular norms (t-norms) are presented in this chapter. BQLs are defined as special types of bipolar partially ordered sets or posets. It is shown that bipolar quantum entanglement is definable on BQLs. With the addition of fuzziness, BDL is extended to a bipolar dynamic fuzzy logic (BDFL). The essential part of BDFL consists of bipolar dynamic triangular norms (t-norms) and their co-norms which extend their truth-based counterparts from a static unipolar fuzzy lattice to a bipolar dynamic quantum lattice. BDFL has the advantage in dealing with uncertainties in bipolar dynamic environments. With bipolar quantum lattices (crisp or fuzzy), the concepts of bipolar symmetry and quasi-symmetry are defined which form a basis toward a logically complete quantum theory. The concepts of strict bipolarity, linearity, and integrity of BQLs are introduced. A recovery theorem is presented for the depolarization of any strict BQL to Boolean logic. The recovery theorem reinforces the computability of BDL or BDFL.

MentalSquares

While earlier chapters have focused on the logical, physical, and biological aspects of the Q5 paradigm, this chapter shifts focus to the mental aspect. MentalSquares (MSQs) - an equilibrium-based dimensional approach is presented for pattern classification and diagnostic analysis of bipolar disorders. While a support vector machine is defined in Hilbert space, MSQs can be considered a generic dimensional approach to support vector machinery for modeling mental balance and imbalance of two opposite but bipolar interactive poles. A MSQ is dimensional because its two opposite poles form a 2-dimensional background independent YinYang bipolar geometry from which a third dimension – equilibrium or non-equilibrium – is transcendental with mental fusion or mental separation measures. It is generic because any multidimensional mental equilibrium or non-equilibrium can be deconstructed into one or more bipolar equilibria which can then be represented as a mental square. Different MSQs are illustrated for bipolar disorder (BPD) classification and diagnostic analysis based on the concept of mental fusion and separation. It is shown that MSQs extend the traditional categorical standard classification of BPDs to a non-linear dynamic logical model while preserving all the properties of the standard; it supports both classification and visualization with qualitative and quantitative features; it serves as a scalable generic dimensional model in computational neuroscience for broader scientific discoveries, and it has the cognitive simplicity for clinical and computer operability. From a broader perspective, the agent-oriented nature of MSQs provides a basis for multiagent data mining (Zhang & Zhang, 2004) and cognitive informatics of brain and behaviors (Wang, 2004).

YinYang Bipolar Quantum Entanglement

YinYang bipolar relativity leads to an equilibrium-based logically complete quantum theory which is presented and discussed in this chapter. It is shown that bipolar quantum entanglement and bipolar quantum computing bring bipolar relativity deeper into microscopic worlds. The concepts of bipolar qubit and YinYang bipolar complementarity are proposed and compared with Niels Bohr’s particle-wave complementarity. Bipolar qubit box is compared with Schrödinger’s cat box. Since bipolar quantum entanglement is fundamentally different from classical quantum theory (which is referred to as unipolar quantum theory in this book), the new approach provides bipolar quantum computing with the unique features: (1) it forms a key for equilibrium-based quantum controllability and quantum-digital compatibility; (2) it makes bipolar quantum teleportation theoretically possible for the first time without conventional communication between Alice and Bob; (3) it enables bitwise encryption without a large prime number that points to a different research direction of cryptography aimed at making prime-number-based cryptography and quantum factoring algorithm both obsolete; (4) it shows potential to bring quantum computing and communication closer to deterministic reality; (5) it leads to a unifying Q5 paradigm aimed at revealing the ubiquitous effects of bipolar quantum entanglement with the sub theories of logical, physical, mental, social, and biological quantum gravities and quantum computing.

Agents, Causality, and YinYang Bipolar Relativity

This chapter presents the theory of bipolar relativity–a central theme of this book. The concepts of YinYang bipolar agents, bipolar adaptivity, bipolar causality, bipolar strings, bipolar geometry, and bipolar relativity are logically defined. The unifying property of bipolar relativity is examined. Space and time emergence from YinYang bipolar geometry is proposed. Bipolar relativity provides a number of predictions. Some of them are domain dependent and some are domain independent. In particular, it is conjectured that spacetime relativity, singularity, gravitation, electromagnetism, quantum mechanics, bioinformatics, neurodynamics, and socioeconomics are different phenomena of YinYang bipolar relativity; microscopic and macroscopic agent interactions in physics, socioeconomics, and life science are directly or indirectly caused by bipolar causality and regulated by bipolar relativity; all physical, social, mental, and biological action-reaction forces are fundamentally different forms of bipolar quantum entanglement in large or small scales; gravity is not necessarily limited by the speed of light; graviton does not necessarily exist.

Bipolar Sets and YinYang Bipolar Dynamic Logic (BDL)

In this chapter an equilibrium-based set-theoretic approach to mathematical abstraction and axiomatization is presented for resolving the LAFIP paradox (Ch. 1) and for enabling logically definable causality (Ch. 2). Bipolar set theory is formally presented, which leads to YinYang bipolar dynamic logic (BDL). BDL in zeroth-order, 1st-order, and modal forms are presented with four pairs of dynamic De Morgan’s laws and a bipolar universal modus ponens (BUMP). BUMP as a key element of BDL enables logically definable causality and quantum computing. Soundness and completeness of a bipolar axiomatization are asserted; computability is proved; computational complexity is analyzed. BDL can be considered a non-linear bipolar dynamic generalization of Boolean logic plus quantum entanglement. Despite its non-linear bipolar dynamic quantum property, it does not compromise the basic law of excluded middle. The recovery of BDL to Boolean logic is axiomatically proved through depolarization and the computability of BDL is proved. A redress on the ancient paradox of the liar is presented with a few observations on Gödel’s incompleteness theorem. Based on BDL, bipolar relations, bipolar transitivity, and equilibrium relations are introduced. It is shown that a bipolar equilibrium relation can be a non-linear bipolar fusion of many equivalence relations. Thus, BDL provides a logical basis for YinYang bipolar relativity–an equilibrium-based axiomatization of social and physical sciences.