The main result of this paper is the decidability of the membership problem for 2 × 2 nonsingular integer matrices. Namely, we will construct the first algorithm that for any nonsingular 2 × 2 integer matrices M1, . . . , Mn and M decides whether M belongs to the semigroup generated by {M1, . . . , Mn}. Our algorithm relies on a translation of numerical problems on matrices into combinatorial problems on words. It also makes use of some algebraic properties of well-known subgroups of GL(2, Z) and various new techniques and constructions that help to convert matrix equations into the emptiness problem for intersection of regular languages.

Let E be a computably enumerable (c.e.) equivalence relation on the set ω of natural numbers. We say that the quotient set ω/E (or equivalently, the relation E) realizes a linearly ordered set L if there exists a c.e. relation respecting E such that the induced structure (ω/E; ) is isomorphic to L. Thus, one can consider the class of all linearly ordered sets that are realized by ω/E; formally, K(E) = {L | the order-type L is realized by E}. In this paper we study the relationship between computability-theoretic properties of E and algebraic properties of linearly ordered sets realized by E. One can also define the following pre-order lo on the class of all c.e. equivalence relations: E 1 lo E 2 if every linear order realized by E 1 is also realized by E 2 . Following the tradition of computability theory, the lo-degrees are the classes of equivalence relations induced by the pre-order lo . We study the partially ordered set of lo-degrees. For instance, we construct various chains and antichains and show the existence of a maximal element among the lo-degrees.

We give new examples of FA presentable torsion-free abelian groups. Namely, for every n 2, we construct a rank n indecomposable torsion-free abelian group which has an FA presentation. We also construct an FA presentation of the group (Z, +) 2 in which every nontrivial cyclic subgroup is not FA recognizable.

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Automatic classes are classes of languages for which a finite automaton can decide the membership problem for the languages in the class, in a uniform way, given an index for the language. For alphabet size of at least 4, every automatic class of erasing pattern languages is contained, for some constant n, in the class of all languages generated by patterns which contain (1) every variable only once and (2) at most n symbols after the first occurrence of a variable. It is shown that such a class is automatically learnable using a learner with the length of the long-term memory being bounded by the length of the first example seen. The study is extended to show the learnability of related classes such as the class of unions of two pattern languages of the above type.