Learning semilinear sets from examples and via queries

Abstract

AbstractSemilinear sets play an important role in parallel computation models such as matrix grammars, commutative grammars, and Petri nets. In this paper, we consider the problems of learning semilinear sets from examples and via queries. We shall show that (1) the family of semilinear sets is not learnable only from positive examples, while the family of linear sets is learnable only from positive examples, although the problem of learning linear sets from positive examples seems to be computationally intractable; (2) if for any unknown semilinear set Su and any conjectured semilinear set S′, queries whether or not Su⊆S′ and queries whether or not S′⊆Su can be made, there exists a learning procedure which identifies any semilinear set and halts, although the procedure is time-consuming; (3) however, under the same condition, for each fixed dimension, there exist meaningful subfamilies of semilinear sets learnable in polynomial time of the minimum size of representations and, in particular, for any variable dimension, if for any unknown linear set Lu and any conjectured semilinear set S′, queries whether or not Lu⊆S′ can be made, the family of linear sets is learnable in polynomial time of the minimum size of representations and the dimension