Algebraic structure of hierarchical semantic networks

Abstract

The mathematical description of hierarchical semantic networks class is specified. Such networks are natural as coherent semantic representations of logical-mathematical models of subject domain knowledge spaces. Such models are realized in format similar to G. Frege's triangle and D. Pospelov's square. The description is agreed with the unified format for abstract knowledge representation formalisms and includes sets of knowledge representations and knowledge fragments, operation of knowledge fragments composition and the relation of fragments inclusion. The system of requirements to the sets of the objects making networks and network fragments structures is specified. The algorithm of creation of any fragments of hierarchical networks from basic fragments of such networks by operation of composition is defined. Existence of algorithm of finding the shortest sequence of the composition operation that create any fragment of a network is proved. Any sequence of compositions defines algebraic structure of the received semantic network. That structure consists of network fragments ordered by relation of transformation possibility of one fragment of a network into another fragment. This relation is not the order relation on a set of fragments of semantic networks, but it is an order on a set of networks. Variants of monotony relations for composition and inclusions of semantic networks fragments are considered.