A new order theory of set systems and better quasi-orderings

By reformulating a learning process of a set system L as a game between Teacher (presenter of data) and Learner (updater of abstract independent set), we define the order type dim L of L to be the order type of the game tree. The theory of this new order type and continuous, monotone function between set systems corresponds to the theory of well quasi-orderings (WQOs). As Nash-Williams developed the theory of WQOs to the theory of better quasiorderings (BQOs), we introduce a set system that has order type and corresponds to a BQO. We prove that the class of set systems corresponding to BQOs is closed by any monotone function. In (Shinohara and Arimura. "Inductive inference of unbounded unions of pattern languages from positive data." Theoretical Computer Science, pp. 191-209, 2000), for any set system L, they considered the class of arbitrary (finite) unions of members of L. From view point of WQOs and BQOs, we characterize the set systems L such that the class of arbitrary (finite) unions of members of L has order type. The characterization shows that the order structure of the set system L with respect to the set inclusion is not important for the resulting set system having order type. We point out continuous, monotone function of set systems is similar to positive reduction to Jockusch-Owings' weakly semirecursive sets.