The Mivar-based approach is a mathematical tool for designing artificial intelligence (AI) systems. Mivar ("Multidimensional Informational Variable Adaptive Reality") was developed by combining production and Petri nets. The Mivar-based approach was developed for semantic analysis and adequate representation of humanitarian epistemological and axiological principles in the process of developing artificial intelligence. The Mivar-based approach incorporates computer science, informatics and discrete mathematics, databases, expert systems, graph theory, matrices and inference systems. The Mivar-based approach involves two technologies:
Mivar networks allow us to develop cause-effect dependencies (“If-then”) and create an automated, trained, logical reasoning system.
Representatives of Russian association for artificial intelligence (RAAI) – for example, V. I. Gorodecki, doctor of technical science, professor at SPIIRAS and V. N. Vagin, doctor of technical science, professor at MPEI declared that the term is incorrect and suggested that the author should use standard terminology.
History.
While working in the Russian Ministry of Defense, O. O. Varlamov started developing the theory of “rapid logical inference” in 1985. He was analyzing Petri nets and productions to construct algorithms. Generally, mivar-based theory represents an attempt to combine entity-relationship models and their problem instance – semantic networks and Petri networks.
The abbreviation MIVAR was introduced as a technical term by O. O. Varlamov, Doctor of Technical Science, professor at Bauman MSTU in 1993 to designate a “semantic unit” in the process of mathematical modeling. The term has been established and used in all of his further works.
The first experimental systems operating according to mivar-based principles were developed in 2000. Applied mivar systems were introduced in 2015.
Mivar.
Mivar is the smallest structural element of discrete information space.
Object-property-relation.
Object-Property-Relation (VSO) is a graph, the nodes of which are concepts and arcs are connections between concepts.
Mivar space represents a set of axes, a set of elements, a set of points of space and a set of values of points.
formula_1
where:
Then: formula_4
where:
formula_5 sets form multidimensional space: formula_11
formula_12
where:
There is a set of values of multidimensional space points of formula_17:
 formula_18
where:
For every point of space formula_17 there is a single value from formula_23 set or there is no such value. Thus,formula_23 is a set of data model state changes represented in multidimensional space. To implement a transition between multidimensional space and set of points values the relation formula_25 has been introduced: formula_26
where: 
To describe a data model in mivar information space it is necessary to identify three axes:
These sets are independent. The mivar space can be represented by the following tuple:
 formula_32
Thus, mivar is described by «formula_33» formula, in which «formula_31» denotes an object or a thing, «formula_30» denotes properties, «formula_29» variety of relations between other objects of a particular subject domain. The category “Relations” can describe dependencies of any complexity level: formulae, logical transitions, text expressions, functions, services, computational procedures and even neural networks. A wide range of capabilities complicates description of modeling interconnections, but can take into consideration all the factors. Mivar computations use mathematical logic. In a simplified form they can be represented as implication in the form of an "if…, then …” formula. The result of mivar modeling can be represented in the form of a bipartite graph binding two sets of objects: source objects and resultant objects.
Mivar network.
Mivar network is a method for representing objects of the subject domain and their processing rules in the form of a bipartite directed graph consisting of objects and rules.
A Mivar network is a bipartite graph that can be described in the form of a two-dimensional matrix, in that records information about the subject domain of the current task.
Generally, mivar networks provide formalization and representation of human knowledge in the form of a connected multidimensional space. That is, a mivar network is a method of representing a piece of mivar space information in the form of a bipartite, directed graph. The mivar space information is formed by objects and connections, which in total represent the data model of the subject domain. Connections include rules for objects processing. Thus, a mivar network of a subject domain is a part of the mivar space knowledge for that domain.
The graph can consist of objects-variables and rules-procedures. First, two lists are made that form two nonintersecting partitions: the list of objects and the list of rules. Objects are denoted by circles. Each rule in a mivar network is an extension of productions, hyper-rules with multi-activators or computational procedures. It is proved that from the perspective of further processing, these formalisms are identical and in fact are nodes of the bipartite graph, denoted by rectangles.
Multi-dimensional binary matrices.
Mivar networks can be implemented on single computing systems or service-oriented architectures. Certain constraints restrict their application, in particular, the dimension of matrix of linear matrix method for determining logical inference path on the adaptive rule networks. The matrix dimension constraint is due to the fact that implementation requires sending a general matrix to multiple processors. Since every matrix value is initially represented in symbol form, the amount of sent data is crucial when obtaining, for example, 10000 rules/variables. Classical mivar-based method requires storing three values in each matrix cell:
The analysis of possibility of firing a rule is separated from determining output variables according to stages after firing the rule. Consequently, it is possible to use different matrices for “search for fired rules” and “setting values for output variables”. This allowsthe use of multidimensional binary matrices. Binary matrix fragments occupy much less space and improve possibilities of applying mivar networks.
Logical and computational data processing.
To implement logical-and-computational data processing the following should be done. First, a formalized subject domain description is developed. The main objects-variables and rules-procedures are specified on the basis of mivar-based approach and then corresponding lists of “objects” and “rules” are formed. This formalized representation is analogous to the bipartite logical network graph.
The main stages of mivar-based information processing are:
The first stage is the stage of synthesis of conceptual subject domain model and its formalization in the form of production rules with a transition to mivar rules. “Input objects – rules/procedures – output objects”. Currently, this stage is the most complex and requires involvement of a human expert to develop a mivar model of the subject domain.
Automated solution algorithm construction or logical inference is implemented at the second stage. Input data for algorithm construction are: mivar matrix of subject domain description and a set input of object-variables and required object-variables.
The solution is implemented at the third stage.
Data processing method.
Firstly, the matrix is constructed. Matrix analysis determines whether a successful inference path exists. Then possible logical inference paths are defined and at the last stage the shortest path is selected according to the set optimality criteria.
Let formula_14 rules and formula_38 variables be included in the rules as input variables activating them or as output variables. Then, matrix formula_39, each row of which corresponds to one of the rules and contains the information about variables used in the rule, can represent all the interconnections between rules and variables.
One row and one column are added in the matrix formula_31 to store service information. So, the matrix formula_31 of dimension formula_46, is obtained, which shows the whole structure of the source rule network. The structure of this logical network can change, that is, this is a network of rules with evolutionary dynamics.
Example.
To search for a logical inference path the following actions are implemented: