Finitely distinguishable erasing pattern languages

Pattern languages have been an object of study in various subfields of computer science for decades. This paper introduces and studies a decision problem on patterns called the finite distinguishability problem: given a pattern π, are there finite sets T+ and T− of strings such that the only pattern language containing all strings in T+ and none of the strings in T− is the language generated by π? This problem is related to the complexity of teacher-directed learning, as studied in computational learning theory, as well as to the long-standing open question whether the equivalence of two patterns is decidable. We show that finite distinguishability is decidable if the underlying alphabet is of size other than 2 or 3, and provide a number of related results, such as (i) partial solutions for alphabet sizes 2 and 3, and (ii) decidability proofs for variants of the problem for special subclasses of patterns, namely, regular, 1-variable, and non-cross patterns. For the same subclasses, we further determine the values of two complexity parameters in teacher-directed learning, namely the teaching dimension and the recursive teaching dimension.