The syntactic inference problem for dol-sequences
The syntactic inference problem consists of deciding, for a given set of words, whether there exists a grammar such that its language includes these given words; and also of actually finding any such grammars. In this paper, the problem is considered for DOL-systems. The stress is on the second, constructive, part of the problem. The initial information may have various forms. Most of the results deal with cases in which
— the words are given as a sequence (i.e., with their rank order numbers), which may be either consecutive or scattered.
— the size of the alphabet is given.
From the decidability point of view most of the results are not new. The proposed decision method, however, represents a considerable speed-up by passing the initial data through a number of algebraic "sieves" which turn out to be quite dense.
The method depends on there being enough information to establish a linear dependence relation between the Parikh-vectors of the given words.
Several variants of the problem are discussed. One subcase of a hitherto open problem is solved; other problems remain open.