Problem-solving: Combinatorial-algebraic approach

The present paper deals with general approach for solutions' design. Considered model is based on assumption that there is situations' set's partial-ordering relation which is in accordance with all of elementary actions. Proposed approach isn't only a natural generalization of classical set-theoretical (linguistic, in essence) one. Many fundamental problems of discrete mathematics lead to considered model. The most important examples are state-identification problems for finite automata, design of prime implicants for boolean functions and design of disjunctive normal forms (DNF) consisting of prime implicants only. Our aim is to elaborate theory which includes as special cases set-theoretical approach and one based on estimation function's conception. We solve general problem of solutions' design (solutions are defined by special predicate). Detalizations for design of important special types of solutions (i.e. minimal, irreducible, cooperative and branching) are considered. On this basis exaustive solutions of controllability/observability (C/O) problems for boolean functions and finite automata are obtained.