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.

No abstract

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.

This paper establishes an upper bound on the size of a concept class with given recursive teaching dimension (RTD, a teaching complexity parameter). The upper bound coincides with Sauer's well-known bound on classes with a fixed VC-dimension. Our result thus supports the recently emerging conjecture that the combinatorics of VC-dimension and those of teaching complexity are intrinsically interlinked. We further introduce and study RTD-maximum classes (whose size meets the upper bound) and RTD-maximal classes (whose RTD increases if a concept is added to them), showing similarities but also differences to the corresponding notions for VC-dimension. Another contribution is a set of new results on maximal classes of a given VC-dimension. Methodologically, our contribution is the successful application of algebraic techniques, which we use to obtain a purely algebraic characterization of teaching sets (sample sets that uniquely identify a concept in a given concept class) and to prove our analog of Sauer's bound for RTD. Such techniques have been used before to prove results relevant to computational learning theory, e.g., by Smolensky [13], but are not standard in the field.

No abstract

No abstract