Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

金水暎 著, 「政策學原論」 (법지사, 1992)

ACM-SIAM Symposium on Discrete Algorithms

Review: K. A. Mihajlova, (Problema vhozdenia did pramyh proizvedenij grupp):The Occurrence Problem for Direct Products of Groups

This paper is concerned with various combinatorial parameters of classes that can be learned from a small set of examples. We show that the recursive teaching dimension, recently introduced by Zilles et al. (2008), is strongly connected to known complexity notions in machine learning, e.g., the self-directed learning complexity and the VC-dimension. To the best of our knowledge these are the first results unveiling such relations between teaching and query learning as well as between teaching and the VC-dimension. It will turn out that for many natural classes the RTD is upper-bounded by the VCD, e.g., classes of VC-dimension 1, intersection-closed classes and finite maximum classes. However, we will also show that there are certain (but rare) classes for which the recursive teaching dimension exceeds the VC-dimension. Moreover, for maximum classes, the combinatorial structure induced by the RTD, called teaching plan, is highly similar to the structure of sample compression schemes. Indeed one can transform any repetition-free teaching plan for a maximum class C into an unlabeled sample compression scheme for C and vice versa, while the latter is produced by (i) the corner-peeling algorithm of Rubinstein and Rubinstein (2012) and (ii) the tail matching algorithm of Kuzmin and Warmuth (2007).

/pdf/recursive-teaching-dimension-vc-dimension-and-sample-1vq4ucv53m.pdf

Recursive teaching dimension, VC-dimension and sample compression

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

/pdf/finite-automata-presentable-abelian-groups-1m4c55lwnf.pdf

Finite Automata Presentable Abelian Groups

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.

Decidability of the membership problem for 2 × 2 integer matrices

Let E be a computably enumerable (c.e.) equivalence relation on the set ω of natural numbers. We say that the quotient set (or equivalently, the relation E) realizes a linearly ordered set if there exists a c.e. relation ⊴ respecting E such that the induced structure ( ; ⊴) is isomorphic to . Thus, one can consider the class of all linearly ordered sets that are realized by ; formally, . 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 on the class of all c.e. equivalence relations: 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 . We study the partially ordered set of lo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among the lo-degrees.

Linear orders realized by c.e. equivalence relations

We build an א₁-categorical but not א₀-categorical theory whose only computably presentable model is the saturated one. As a tool, we introduce a notion related to limitwise monotonic functions.

https://projecteuclid.org/download/pdf_1/euclid.ndjfl/1143468311

An Uncountably Categorical Theory Whose Only Computably Presentable Model Is Saturated

The decision problems on matrices were intensively studied for many decades as matrix products play an essential role in the representation of various computational processes. However, many computational problems for matrix semigroups are inherently difficult to solve even for problems in low dimensions and most matrix semigroup problems become undecidable in general starting from dimension three or four.

This paper solves two open problems about the decidability of the vector reachability problem over a finitely generated semigroup of matrices from SL(2, Z) and the point to point reachability (over rational numbers) for fractional linear transformations, where associated matrices are from SL(2, Z). The approach to solving reachability problems is based on the characterization of reachability paths between points which is followed by the translation of numerical problems on matrices into computational and combinatorial problems on words and formal languages. We also give a geometric interpretation of reachability paths and extend the decidability results to matrix products represented by arbitrary labelled directed graphs. Finally, we will use this technique to prove that a special case of the scalar reachability problem is decidable.

Pavel Semukhin

Papers

Finite Automata Presentable Abelian Groups

Decidability of the membership problem for 2 × 2 integer matrices

Linear orders realized by c.e. equivalence relations

An Uncountably Categorical Theory Whose Only Computably Presentable Model Is Saturated

Vector Reachability Problem in SL(2, Z)