Natalia Serdyukova Vladimir Serdyukov (cid:129) Algebraic Formalization of Smart Systems Theory and Practice 123 Natalia Serdyukova Vladimir Serdyukov Plekhanov Russian University Bauman MoscowState Technical ofEconomics University Moscow Moscow Russia Russia and Institute of Education Management ofthe Russian Academy of Education Moscow Russia ISSN 2190-3018 ISSN 2190-3026 (electronic) Smart Innovation,Systems andTechnologies ISBN978-3-319-77050-5 ISBN978-3-319-77051-2 (eBook) https://doi.org/10.1007/978-3-319-77051-2 LibraryofCongressControlNumber:2018933497 ©SpringerInternationalPublishingAG2018 Preface In1937L.BertalanffyproposedtheconceptofaSystemandthedevelopmentofa mathematicalapparatusfordescribingsystems.In1970sA.I.Mal’tsevdevelopeda theoryofalgebraicsystemsconnectingalgebraandlogicforstudyingalgebraicand logical objects. In 1990s the concept of purities by predicates was introduced by one of the authors and we found out some applications of this concept to practice. Thisconceptionbasedonthetheoryofalgebraicsystemsallowstodeepandclarify connections between quantitative and qualitative analysis of a system. The book which is offering to you, “The Algebraic Theory of Smart Systems. Theoryandpractice”,isanattempttorevealthegenerallawsofthetheoryofSmart systems with the help of a very powerful and expressive language of algebraic formalizationandalsoanefforttousethislanguagetosubstantiatepracticalresults inthefieldofSmart systems,whichpreviously hadonlyanempiricaljustification. Infact,thisisatranslationofthetheoryofSmartsystemsfromverballanguagetoa much more expressive language of algebraic formalization allowing in a different light to see the laws of the theory of Smart systems is proposed to the reader. The key users of this book are persons which using elements of artificial intelligence in their work. Moscow, Russia Natalia Serdyukova Moscow, Russia Vladimir Serdyukov Contents 1 The Problem of General Systems Theory’s Formalization . . . . . . . 1 1.1 The Concept of Formalization as a Tool to Study the Phenomena, Processes and Practical Outcomes on a Theoretical Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Two Directions of Development of Logic. From Deductive Systems to A. I. Mal’tsev’s Systems . . . . . . . . . . . . . . . . . . . . 3 1.3 AlgebraicFormalizationoftheGeneralConceptoftheSystem, Based on the Factors Determining the System . . . . . . . . . . . . . 12 1.4 The Hierarchy of Algebraic Formalizations . . . . . . . . . . . . . . . 14 1.5 Probabilistic Algebraic Formalization. . . . . . . . . . . . . . . . . . . . 15 1.6 ASeriesofDistributionofaCompleteCountableDistributive Lattice of Algebraic Systems. A Distribution Function of a Random Function of a Lattice of Algebraic Formalizations . . . . 17 1.7 Examples of Usage of Hierarchies of Algebraic Formalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 The Performance of a System by Using an Algebraic System of Factors Determining the System. P-Properties of a System. . . . . . . 21 2.1 Factors Determining the System . . . . . . . . . . . . . . . . . . . . . . . 22 2.1.1 Static and Dynamic Predicates . . . . . . . . . . . . . . . . . . 22 2.2 The Scheme of the Dynamic Predicates’ Functioning in Models that Are Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Cycles in the System’s Development and Functioning. . . . . . . . 28 2.3.1 Cycle Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.2 Practical Examples of the Smart System Cyclic Functioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.3 Kondratiev’s Cycles in Economic Theory . . . . . . . . . . 29 2.3.4 J. Schumpeter Theory of Cyclic Development . . . . . . . 30 2.4 Algorithm for Determining and Regulating Smart System’s Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4.1 Vronsky’s Determinant. . . . . . . . . . . . . . . . . . . . . . . . 34 2.4.2 TheoremsontheStructureof theGeneral Solution of a Homogeneous System of Linear ODEs. . . . . . . . . . . 35 2.4.3 Algorithm of Determining the Possibility of Regulating the Properties of the System S . . . . . . . . . . 38 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3 The Simulation of the System with the Help of Finite Group of Factors Determining the System. P-Properties of the System. Cayley Tables and Their Role in Modeling Associative Closed System with Feedback. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1 P-Properties of the Smart System. Sustainability of Smart Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Example. Smart Systems Modeling by a Group of Four Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3 Relationship Between Factors Determining a System and Elements of a System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 Substitution of Functions of a System. System’s Compensational Possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.5 Compensational Functions of a Quotient—Flexible Smart System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.6 Sustainability upon the Smart System Functioning . . . . . . . . . . 51 3.7 Loss Detection Point of Sustainability of a System Algorithm that Uses Models of Groups of Factors Describing the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4 External and Internal Properties of a System. Integrity and P-Integrity of a System by Predicate P. Formalization Smart Systems’ Axiomatic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 System Approach Basic Principles. System’s Links. Connection with Synergetics . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3 The Model of Hierarchy of Structural Links of the System . . . . 63 4.4 Types of System Connections. Different Types of Classifications. Classification of Binary Links of the First Level of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.4.1 Operations Over System Links . . . . . . . . . . . . . . . . . . 66 4.5 Closed Associative Systems with Feedback. Partial Classification on the System Links Levels and the Number of Synergistic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.6 System Binary Links and Mappings. . . . . . . . . . . . . . . . . . . . . 68 4.7 AlgorithmofAnalysisandDecompositionoftheSystembyIts Links Levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.8 Example. System Decomposition. Smart System The World University Rankings. Evaluation of the The World University Rankings System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.9 Algebraic Formalization of the Axiomatic Description of Smart Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5 Formalization ofSystem Links:DifferentApproaches. Duality in Smart Systems Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.1 Preliminary Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2 Several Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.3 System Connections Strength. Example: The Social Relationships Strength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.4 Duality in System Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.5 The ConnectionBetweenDualityand theConcept ofa Factor of a System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.6 AlgebraicFormalizationofModelingtheProcessesPreserving the Operation of Composition of Factors of a Closed System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.6.1 Example. Modeling Decomposition Process of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.7 Duality in the Theory of Strong and Weak System’s Links. . . . 90 5.8 Efficiency (Utility of a Smart System). Formalization of Efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.9 Presentation of the General Task of the Smart System Effectiveness Determining in the Form of an Optimization Problem with Risks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.10 Examples. The Use of Duality for Complex Smart Systems ClassificationbytheNumberofSystemGoals.Stabilitybythe Parameter of Achieving the Goal of the System . . . . . . . . . . . . 94 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6 P-Innovative and P-Pseudo-Innovative Systems on the Predicate P and Their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.1 Formalization of Innovation and Effectiveness Concepts . . . . . . 98 6.2 Algorithm for a Comprehensive Assessment of the Effectiveness of a Smart System . . . . . . . . . . . . . . . . . . . . . . . 100 6.2.1 TheAlgorithmofaComplexEstimationofEfficiency of Functioning of the Innovation System . . . . . . . . . . . 101 6.2.2 Quasi Sustainability of Pseudo-innovative Systems. . . . 102 6.3 Example. Decomposition of the Education System. ApproachestotheStudyoftheEffectivenessoftheEducation System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.4 DecompositionoftheKnowledgeSystem.TheRepresentation of the System of Knowledge in the Form of an Algebraic System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.5 Decomposition of the System. Analysis and Synthesis of the Knowledge Base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 7 Algebraic Approach to the Risk Description. Linear Programming Models with Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.2 Known Approaches to the Mathematical Determination of Risk. The Kolmogorov Risk Function . . . . . . . . . . . . . . . . . . . 118 7.3 The Presentation of the General Model of Multi-criteria Optimization Problem in the Form of Linear Programming Task with Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.4 System Approach to Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7.5 Mathematical Model of Risk . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.6 The Use of the Theory of Infinite Products to Quantify Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.7 The Connection Between the Kolmogorov Risk Function h(x) and the Risk Function r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.7.1 The Simplest Examples of Probability Distributions with a Multiplicative Risk Function. . . . . . . . . . . . . . . 128 7.8 Regulated Risks. Semigroup of Systemic Risks. Description of the System’s Risk Semigroup . . . . . . . . . . . . . . . . . . . . . . . 130 7.8.1 SimulationoftheMomentoftheCrisisoftheSystem with the Help of the Kolmogorov-Chapman Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.8.2 Risks of Formalization Changes for the Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.8.3 Algebraic Approach to the Description of Risks. InternalandExternalSystemsRisks.SystemicRiskor System Risk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.8.4 Algorithm for Regulating the Internal Risks of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.8.5 Some Properties of Risk. Examples. . . . . . . . . . . . . . . 135 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 8 TheTransitionfromanInfiniteModelofFactorsthatDetermine the System to a Finite Model. The Model of Algebraic Formalization of Risks of Changing the Scenarios of the Long- Term Development of a Smart System of Six Factors on the Example of a Smart University. . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 8.1 The Transition from an Infinite Model of Factors that Determine the System to a Finite Model of the System. . . . . . . 137 8.2 The Necessary Information from the Finite Groups Theory Useful in the Study of Some Features of the System’s Functioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 8.3 The Model of an Algebraic Formalization of Risks of Changing the Scenarios of the Long-Term Development of a Smart System of Six Factors on the Example of a Smart University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.3.1 Risk Modelling in a Smart University . . . . . . . . . . . . . 142 8.4 A Selection of Factors to Determine Long-Term Risks of a System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 8.4.1 Algorithm of Search the Points of Regulation of Functioning of the Closed Associative System on an Example of the Model Consisting of Six Factors . . . . . 146 8.5 Conclusions. Future Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 9 Pro-P-Groups and Algebraically Closed Groups: Application to Smart Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 9.1 Particular Case: Factors Affecting a System Determine a Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 9.1.1 TheMeaningoftheP-PureEmbeddings.Examplesof P-Purities in the Class of All Groups. . . . . . . . . . . . . . 150 9.2 The Group of Automorphisms of the Group of Factors that Determine the System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 9.2.1 Background of the Issue. Basic Definitions and Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 9.3 Direct and Inverse Spectra of Groups and Their Limits. . . . . . . 152 9.4 The Role of Profinite Groups in Algebra and Topology . . . . . . 154 9.4.1 Profinite Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 9.4.2 Profinite Completion. . . . . . . . . . . . . . . . . . . . . . . . . . 154 9.4.3 P-Finite Groups. Pro-P-Groups . . . . . . . . . . . . . . . . . . 155 9.5 Predicates Defined by Systems of Equations on the Class of Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 9.6 InterpretationofSystemsofEquationsOverGroupsofFactors that Describe a Smart System . . . . . . . . . . . . . . . . . . . . . . . . . 162 9.7 P-Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 9.8 Pro-P-Algebraic Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 9.8.1 Inverse and Direct Spectra of Algebraic Systems and Their Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 10 P-Sustainability of a System. Algebraic Formalization of Sustainability Concept. Sustainability of Ranking Systems in Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 10.1 Sustainability: Ranking Systems . . . . . . . . . . . . . . . . . . . . . . . 171 10.2 Final Sustainability of a System. . . . . . . . . . . . . . . . . . . . . . . . 172 10.3 Time Structure of Algebraic Formalization. . . . . . . . . . . . . . . . 173 10.4 The Algorithm of Determination the Scenarios of DevelopmentoftheSystemSandPointsandIntervalsofLoss of the Sustainability of the System S . . . . . . . . . . . . . . . . . . . . 175 10.4.1 TheAlgorithmofaDeterminationandRegulationthe Scenarios of a Functioning of a System S with a Group of Factors GS of an Order p2, Where p Is a Prime Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 10.5 The Connection Between Notions of Final Sustainability, Stationary Points and Classical Sustainability . . . . . . . . . . . . . . 182 10.6 Practice Example. Algebraic Formalization as a Tool of Assertion the Sustainability of Ranking Systems of an Evaluation of Activities of Universities . . . . . . . . . . . . . . . . . . 185 10.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Introduction The emergence of ideas and Smart technologies has changed the mentality of human society and, in particular, in the field of human communication, i.e., in the sphere of universal and public relations. This change is connected with the appearance of a more expressive language—the language of digits and digital technologies of building connections. At present, Smart technologies and Smart systems have become a common phenomenon in almost all spheres of human life. In1937,LudwigvonBertalanffyproposedtheconceptofasystemapproachanda General Theory of Systems and also the development of a mathematical apparatus for describing typologically dissimilar systems. His main idea is to recognize iso- morphism, that is identity, sameness of laws governing the functioning of system objects. In the 1970s, A. I. Mal’tsev developed a theory of algebraic systems that connects algebra and logic and which is a universal mathematical apparatus for studying both algebraic and logical objects. In 1990s the concept of purities by predicates was introduced by one of the authors, and later on we found out some applications of the theory of purities by predicates to practice. This conception makespossibletogetanewmethodologyforthestudyofsystemstheorybasedon theideaofformalizinganotionofasystemusingalgebraicsystemsandmethodsof generalalgebra.Itallowstoclarifyconnectionsbetweenquantitativeandqualitative analysis of a system in order to specify the previously known concepts in the deepeningofthestudyofqualitative properties.Thebookwhichisoffered toyou, “The Algebraic Theory of Smart Systems. Theory and practice,” is an attempt to reveal the general laws of the theory of Smart systems with the help of a very powerful and expressive language of algebraic formalization and also an effort to use this language to substantiate practical results in the field of Smart systems, which previously had only an empirical justification. In fact, this book is a trans- lation of the theory of Smart systems from verbal language to a much more expressive language of algebraic formalization. It allows in a different light to see the laws of the theory of Smart systems.