Abstract

This chapter is dedicated to a new knowledge representation model, providing convergence of classical operations research and modern knowledge engineering. Kernel of the introduced model is the recursively generated multisets, selected according to the predefined restrictions and optimization criteria. Sets of multisets are described by the so-called multiset grammars (MGs), being projection of a conceptual background of well-known string-generating grammars on the multisets universum. Syntax and semantics of MGs and their practice-oriented development—unitary multiset grammars and metagrammars—are considered.

Keywords: systems analysis, operations research, knowledge engineering, digital economy, multisets, recursive multisets, multiset grammars, unitary multiset grammars and multimetagrammars, sociotechnical systems assessment and optimization

## 1. Introduction

Large-scale sociotechnical systems (STS) usually have hierarchical structure, including personnel and various technical devices, which, in turn, consume various material, financial, information resources, as well as energy. As a result, they produce new resources (objects), which are delivered to other similar systems. Main features of such STS are large dimensionality and high volatility of their structures, equipment, consumed/produced objects, and at all, operation logics and dynamics [1–5].

Knowledge and data representation models, used in STS, provide comparatively easy and comfortable management of very large knowledge and data bases with dynamic structures and content [6–10]. These model bases are objects other than matrices, vectors, and graphs, traditionally used in operations research and systems analysis [11–14], and they are much more convenient for practical problem consideration. But, on the other hand, aforementioned models in general case do not incorporate strict theoretical background and fundamental algorithmics, compared with, for example, mathematical programming, which provides strictly optimal solutions for decision-makers. So, practically all decision-support software in the considered STS is based on various heuristics, which correctness and

adequacy are not proved usually in the mathematical sense. As a consequence, quality of the adopted decisions, based on such heuristics, in many cases may be far of optimal.

This chapter is dedicated to a primary survey of the developed knowledge representation model, providing convergence of classical operations research and modern knowledge engineering. This convergence creates new opportunities for complicated problem formalization and solution by integrating best features of mathematical programming (strict optimal solution search in solution space, defined by goal functions and boundary conditions) and constraint programming [15–17] (natural and easily updated top-down representation of logic of the decision-making in various situations). Kernel of the considered model is multisets (MS)—relatively long ago known and in the last 20 years intensively applied object of classical mathematics [18–29]. This background is generalized to the recursively generated, or, for short, recursive multisets (RMS) by introduction of so-called multiset grammars, or, again for short, multigrammars (MGs), which were described by the author in [30, 31]. Last, in turn, are peculiar "projection" of the conceptual basis of classical formal grammars by Chomsky [32, 33], operating strings of symbols, to the multisets universum in such a way, that MGs provide generation of one multiset from another and selection (filtration) of those, which satisfy necessary integral conditions: boundary restrictions and/or optimality criteria.

MGs may be considered as prolog-like constraint programming language for solution of problems in operations research and systems analysis areas. Taking into account relative novelty of the multigrammatical approach and absence of any substantial associations with mathematical constructions presented lower, we introduce main content of the chapter by short informal description of the main elements of this approach in Section 2. Basic formal definitions are presented in Section 3. Section 4 is dedicated to multiset grammars, while Section 5—to detailed consideration of the so-called unitary multigrammars (UMGs) and unitary multimetagrammars (UMMGs), which are main tool of the aforementioned problem formalization and solution.
