Extraction of a common up to the names of arguments sub-formula of two elementary conjunctions and some AI problems

The paper is devoted to the use of the notion of a common up to the names of arguments sub-formula of two elementary conjunctions of predicate atomic formulas. It allows to construct a level description of a set of goal formulas and to decrease the number of proof steps for one of NP-complete problems. Logic-predicate networks which may change its configuration (the number of layers and the number of cells in the layer) during the process of training are described. The problem of multi-agent description when every agent can describe only a part of an object (these parts are intersected) but every agent gives its own names to the elements of the whole object may be solved with the use of the notion of such a common sub-formula.