Commutative regular shuffle closed languages, noetherian property, and learning theory

To develop computational learning theory of commutative regular shuffle closed languages, we study finite elasticity for classes of (semi)group-like structures. One is the class of ANd + F such that A is a matrix of size e × d with nonnegative integer entries and F consists of at most k number of e-dimensional nonnegative integer vectors, and another is the class Xd k of AZd + F such that A is a square matrix of size d with integer entries and F consists of at most k number of ddimensional integer vectors (F is repeated according to the lattice AZd). Each class turns out to be the elementwise unions of k-copies of a more manageable class. So we formulate "learning time" of a class and then study in general setting how much "learning time" is increased by the elementwise union, by using Ramsey number. We also point out that such a standpoint can be generalized by using Noetherian spaces.