Quasi - Fractal Propositional Algebra Digitalization of Propositional Algebra and NPC

Nowadays, the functioning of human society cannot be imagined outside the processes of digital transformation. Digitalization penetrates almost all spheres of functioning of human society and creates a symbiosis of the virtual and real human environment. Concepts such as digital twins, innovative digital business models, and so on, have appeared and are used in practice. Naturally, the question arises about the digital evaluation of logical statements. In this chapter we want to consider the possibility of digitalization for Boolean propositional algebra and Narrow Predicate Calculus (NPC). We shall strictly follow here our paper (Serdyukova and Serdyukov in Procedia Comput Sci 192:1471–1483, 2021) in which digitalization function for propositional algebra and Narrow Predicate Calculus (NPC) is constructed to provide wholeness of the book. The usage of Boolean propositional algebra and Narrow Predicate Calculus (NPC) is a convenient instrument for building various assessments of the truth values of NPC closed formulas. In this chapter we shall consider constructions from Serdyukova and Serdyukov (Procedia Comput Sci 192:1471–1483, 2021) in more details. For that we shall begin from different approaches to digitalization of propositional logic and NPC. Also, we need to remind shortly the base of our considerations. As we already marked in Serdyukova and Serdyukov (Procedia Comput Sci 192:1471–1483, 2021), in work (Serdyukova and Serdyukov in Algebraic formalization of smart systems. Theory and practice, smart innovation, systems and technologies. Springer Nature, Switzerland, 2018), on the basis of works (Serdyukova in Algebra Logic 30:432–456, 1991; Serdyukova in Optimization of tax system of Russia, Parts I and II. Budget and Treasury Academy, Rostov State Economic University, Moscow, 2002; Serdyukova in The new scheme of a formalization of an expert system in teaching, 2014; Serdyukova et al. in Algebraic formalization of sustainability in smart university ranking system, pp. 459–474, 2017; Serdyukova et al. in Formalization of knowledge systems on the basis of system approach, pp. 371–381, 2015; Uskov et al. in Int J Knowl-Based Intell Eng Syst 20(3):175–188, 2016), methods of algebraic formalization of smart systems were proposed. These methods make it possible to connect qualitative and quantitative indicators that assess the state, structure, connections and functioning of smart systems in whole complex mechanism of smart systems management. In Serdyukova and Serdyukov (Algebraic identification of smart systems. Theory and practice, intelligent systems reference library. Springer Nature, Switzerland, 2021) new notions of a quasi - fractal algebraic system and mathematical operator \(QF\) were introduced. On this basis a methodology that allows to investigate a smart system in more details, and managed it using different levels of a quasi - fractal algebraic system with the help of which the smart system is modeled. In this chapter we also spread digitalization function onto quasi - fractal Boolean propositional algebra. This chapter is based on our paper Serdyukova and Serdyukov (Procedia Comput Sci 192:1471–1483, 2021). Chapters 4 , 6 , and 7 are based on the notion of digitalization of NPC.