Weakly definable relations and special automata

This chapter presents the weakly definable relations and special automata. The monadic second-order theories and study problems of definability are considered. The chapter characterizes the definable relations by means of finite automata operating on infinite trees. The notion of a special automaton on infinite trees is obtained and is used to characterize the weakly definable sets. As a by-product of the characterization of weakly definable relations, the solution of certain decision problems is obtained. It is shown that the weak second-order theory of a unary function and the weak second-order theory of linearly ordered sets are decidable. These results were corollaries of stronger theorems concerning the corresponding full monadic second-order theories. The same decidability results are deduced using the information concerning weakly definable relations and special automata. The chapter characterizes the weakly-defined relations and develops a theory of special automata.