Algebraic Quasi—Fractal Logic of Smart Systems: Theory and Practice

Foreword by Seregina S. F.
Foreword by Vladimir R. Evstigneev
Preface
Contents
1 Quasi - Fractal Propositional Algebra Digitalization of Propositional Algebra and NPC
	1.1 Introduction. Different Approaches to Digitalization of Propositional Logic and NPC
		1.1.1 Basic Required Information: The Concept of Propositional Algebra. The Concept of Boolean Algebra. Lattice Concept. The Concept of Ershov Algebra. Narrow Predicate Calculus
		1.1.2 Main Definitions. Lattice Concept. The Concept of Boolean Algebra. The Concept of Ershov Algebra. Narrow Predicate Calculus
		1.1.3 Practical Outcome. Problem of the Information Reliability
		1.1.4 Conclusions
	References
2 Quasi - Fractal NPC: Digitalization of Connections of Smart Systems (Expressed in the Language of NPC), Quasi - Fractal Smart Systems with Uncountable Levels Number
	2.1 Quasi - Fractals with Uncountable Levels Number
		2.1.1 Operator QF
		2.1.2 The Graph of a Quasi - Fractal Algebraic System with Uncountable Number of Levels
		2.1.3 DQF Operator Dual to QF Operator with Countable Number of Levels
	2.2 Quasi - Fractal First Order Logic Application of the Fixed - Point Theorem
		2.2.1 Measures, Homomorphisms, Metrics
		2.2.2 Quasi - Fractal Metric Spaces. Contraction Mappings Examples
		2.2.3 Interpretation of Distance of Boolean Ring and Fixed - Point Theorem
		2.2.4 Practical Outcomes. Quasi - Fractal Homomorphisms that Define Quasi - Fractal Measures
	2.3 Quasi - Fractal First Order Logic and Connections Between Different First Order Logics, Defined by Digitalization Functions
		2.3.1 Kripke Timeline and Fixed - Point
		2.3.2 Relevance Logic (Not a Paradoxical Theory of Succession)
		2.3.3 Fixed - Point Logic
		2.3.4 Sustainable Logical Subsystems of First Order Logic and Digital Estimation of Their True Values
		2.3.5 Practical Outcomes: Risks Sustainable Subsystems
		2.3.6 Interpretation of Distances Between the Edges of an Edge Graph of Quasi - Fractal Boolean Ring Corresponding to Quasi - Fractal Boolean Algebra with Time’s Operator
		2.3.7 Quasi - Fractal Digitalization Function. Quasi - Fractal Boolean Algebras
		2.3.8 Quasi - Fractal Narrow Predicate Calculus  QFk<ωγα(NPC): Quasi - Fractal Digitalization Function. Distributive Quasi - Fractal Digitalization Function
		2.3.9 The Kernel of Sustainability of Quasi - Fractal Narrow Predicate Calculus
		2.3.10 Examples: Contraction Mappings
		2.3.11 Refinement of Estimates of Statements of a Quasi - Fractal NPC Using the Digitalization Function
	References
3 Quasi - Fractal Temporal Logic
	3.1 Problem Statement
	3.2 Introduction. Temporal Logic
		3.2.1 The Emergence of Temporal Logic
		3.2.2 Modal µ-Calculus
	3.3 Basic Tense Logic
		3.3.1 System State Concept
		3.3.2 Prior’s Basic Quasi - Fractal Tense Logic
		3.3.3 Computation Tree Logic (CTL)
		3.3.4 Linear Time Logic
		3.3.5 Quasi - Fractal Kripke Timeline with Countable Number of Levels
	3.4 Logic in Terms of Different Algebraic Systems
		3.4.1 Necessary Definitions. Quasi - Fractal Predicates. Quasi - Fractal Normal Functions. Quasi - Fractal Additive Functions. Quasi - Fractal Operations
		3.4.2 Basic Tense Logic as Boolean Algebra with Operators. Representation
		3.4.3 Basic Tense Quasi - Fractal Logic as Quasi - Fractal Boolean Algebra with Operators. Representation
		3.4.4 Practical Outcomes
		3.4.5 Practical Outcomes. Quasi - Fractal Scenario of System Development
		3.4.6 Practical Outcomes. Equivalent Actions to Achieve the Result of Quasi - Fractal System
		3.4.7 Practical Outcomes. Critical Path Method in Project Management in Economics for Refine Planning Models
		3.4.8 Practical Outcomes. Risk Analysis of Smart Planning Systems
		3.4.9 Practical Outcomes. Risk Analysis of Smart Control Systems
		3.4.10 Logic in Terms of Different Algebraic Systems. Logic in Terms of Category Theory and Its Applications
		3.4.11 P - purities (Predicate Purities, Ershov’s Purities) in the Class of Algebras of Finite Signature with Finite Operations)
	References
4 Fixed - Point Logic. Quasi - Fractal Aspects
	4.1 Problem Statement
	4.2 Introduction. Fixed - Point Logic
		4.2.1 Quasi - Fractal Inflation Subscale of Quasi - Fractal Kripke Timeline
		4.2.2 Shelah and Gurevich Iterations as Contractive Mappings
		4.2.3 Quasi - Fractal Functions. Example of a Quasi - Fractal Sustainable Subsystems with a Property P (as a Fixed - Point Subsystem)
		4.2.4 Further Generalizations of Boolean Algebras. Generalizations of Boolean Algebras as Instruments of Quasi - Fractal Lattices
		4.2.5 Algebraic Lattices. Quasi - Fractal Algebraic Lattices
		4.2.6 The Density Property as an Analogue of the Purities by Predicate in the Class of Boolean Algebras
	4.3 Dual Lattices. Dual Quasi - Fractal Lattices
		4.3.1 The Notion of a Dual Lattice and Its Usage in Logic
		4.3.2 Practical Outcome. The Structure of the Evolution of the System S Simulation
		4.3.3 Algorithm of Tracing the Structure of the Evolution of the System S Simulation
	4.4 Universal Coalgebra and Quasi - Fractal Dual Operator DQF
		4.4.1 Quasi - Fractal Dual Operator DQF with Infinite (Not Necessarily Countable) Number of Levels
		4.4.2 LFP (Least Fixed - Point), IFP (Inflationary Fixed - Point), NFP (Nondeterministic Fixed - Point), PFP (Partial Fixed - Point)
	References
5 Quasi - Fractal Temporal Topological Logic with Time Parameter Over Topological Space
	5.1 Introduction. Problem Statement
	5.2 Topological Logic. Main Definitions
		5.2.1 Definition of Topology Logic
		5.2.2 Application of Various Agebraic Systems for the Description of Temporal Topological Logic
	5.3 Approaches to Logic from the Point of View of Its Description Using Various Mathematical Structures
		5.3.1 Logic in Terms of Category Theory
		5.3.2 Applications of P - purities to Logic in Terms of Category Theory. P - purities in the Class of Semigroups
	5.4 Topological Space as an Algebraic System
	5.5 Tense Topological Logics Over Euclidean Spaces
		5.5.1 Building Kripke Quasi - Fractal Timeline with Uncountable Number of Levels
		5.5.2 Kripke Structure/Frame from the Point of View of Boolean Algebras
	5.6 Practice Applications of 2 P  - Purities (Predicate Purities, Ershov’s purities) in the Class of Quasi - Fractal Algebras of Finite Signature with Finite–ary Operations and in the Class of Algebraic Systems with Finite–Ary Operations and Finite–Ary Predicates, Including Quasi - Fractal Kripke Time Line
		5.6.1 System Properties Control Algorithm
	5.7 Conclusion
	References
6 Presystem Concept: Quasi - Fractal Probabilistic Logic: The Conditional Digitalization Function: Modeling of Smart System States Using Homological Algebra Method
	6.1 Introduction
	6.2 Main Results
		6.2.1 Presystem Concept: Chaotic Presystem Models: Quasi - Fractal Chaotic Presystem Models
		6.2.2 Auxiliary Results: Giant Component in Chaotic Presystem: Two Approaches
		6.2.3 Erdős–Rényi Algorithms Towards Logic: Application of the Erdős–Rényi Algorithm Using the Digitization Function
		6.2.4 The “Almost All” First Order Languages L NPC of the Signature with Equality  Ω= langle{ f1 , , fn |n <0 } , { P1 , , Pm |m <0  }, =  rangle Property
	6.3 Auxiliary Results. Generalizations. Conditional Groups’ Probability p( A|B ), Where [A,A] leB, A = langleA,*, - 1 , e rangle. Conditional Groups’ Function of Digitalization
	6.4 Automorphisms of the Model GS as Motions of the System s. Temporal Dynamic Model of a System S Modelled by a Group of Factors GS
		6.4.1 Modelling System’s Motion with the Help of an Automorphisms Group Aut GS of a Group GS
		6.4.2 Topological Mixing: Systems, Similar to Chaotic Presystems
		6.4.3 Properties of Analogically Topological Mixing Systems
	6.5 The Role of the Commutator in Algebraic/Group-Theoretic Models of Systems Theory
		6.5.1 The Commuter of the Group
		6.5.2 Rows of Commutants
		6.5.3 Fisical Sense of Group Hom(G,A): Comparative Analysis of Various System’s Models Specified Using a Group of Factors Determined the System
		6.5.4 Modelling Quasi - Fractal State of a System: Invariant of a Smart System’s State
		6.5.5 Practice Applications
	References
7 Quasi - Fractal Probabilistic Logic. Application to Brownian Motion
	7.1 Introduction
	7.2 Problem Statement
		7.2.1 Detailed Problem Statement Description and Partial Decision
		7.2.2 Practical Problem Part. Problem 2
	7.3 Main Results
		7.3.1 Main Notions and Definitions. Chaotic First Order Logic. Definitions of Metric and Measure: Difference
		7.3.2 Practice Application
		7.3.3 Modeling Distance in the Models of Factors GS of a Chaotic Closed Associative System S
		7.3.4 The Distance Between Elements in the Model. Overview of Results
		7.3.5 Word - Metric
	7.4 Main Question About the Existence of a Giant Component in a Chaotic Closed System
		7.4.1 Erdős–Rényi Theorem for the Cayley Graph of a Group and Amenability
		7.4.2 Quasi - Fractal Graph ΓLA1   of the Links System of the Quasi - Fractal System A1 = langleA1 ;Ω1  rangle. Distance on Quasi - Fractal Graph
		7.4.3 The Connection Between Theorem on the Distribution of Prime Numbers with the Number of Synergistic Effects of a Closed Associative System
		7.4.4 Conclusions. Future Steps
	References
8 The Erdős–Rényi Algorithm from the Point of View of Category Theory. 2P  - Giant Component of an Algebraic System
	8.1 Problem Statement
	8.2 Introduction. Background
	8.3 Category. Main Definitions
	8.4 Algebraic System as a Category Written in the Form of a Graph
	8.5 Erdős–Rényi Model. P - Giant Component in Algebraic Systems
		8.5.1 Categorical Models in Probability Theory
		8.5.2 P  - Giant Component in Algebras. Erdesch–Renyi Algorithm on the Language of Category Theory
		8.5.3 Erdős–Rényi Model of a Random Quasi - Fractal Finite Graph. 2P  - Giant Component in Quasi - Fractal Finite Algebras
		8.5.4 The Inverse Erdős–Rényi Algorithm
	References