Decidability of the Membership Problem for 2×22\times 2 integer matrices

The main result of this paper is the decidability of the membership problem for $2\times 2$ nonsingular integer matrices. Namely, we will construct the first algorithm that for any nonsingular $2\times 2$ integer matrices $M_1,\dots,M_n$ and $M$ decides whether $M$ belongs to the semigroup generated by $\{M_1,\dots,M_n\}$. Our algorithm relies on a translation of the numerical problem on matrices into combinatorial problems on words. It also makes use of some algebraical properties of well-known subgroups of $\mathrm{GL}(2,\mathbb{Z})$ and various new techniques and constructions that help to limit an infinite number of possibilities by reducing them to the membership problem for regular languages

Semigroup identities of tropical matrices through matrix ranks

We prove the conjecture that, for any $n$, the monoid of all $n \times n$ tropical matrices satisfies nontrivial semigroup identities. To this end, we prove that the factor rank of a large enough power of a tropical matrix does not exceed the tropical rank of the original matrix.Comment: 13 page

Vector Reachability Problem in SL(2,Z)

This paper solves three open problems about the decidability of the vector and scalar reachability problems and the point to point reachability by fractional linear transformations over finitely generated semigroups of matrices from . Our approach to solving these problems is based on the characterization of reachability paths between vectors or points, which is then used to translate the numerical problems on matrices into computational problems on words and regular languages. We will also give geometric interpretations of these results

Some aspects of linear space automata

Linear space automaton is introduced as a generalization of probabilistic automaton and its various properties are investigated.Linear space automaton has the abilities equivalent to probabilistic automaton but we can treat the former more easily than the latter because we can make use of properties of the linear space, successfully.First the solutions are given for the problems of connectivity, state equivalence, reduction and identification of linear space automata. Second, the matrix representation of linear space automaton is investigated and the relations between linear space automaton and probabilistic automaton are shown. Third, we discuss the closure properties of the family of all real functions on a free semigroup Σ* which are defined by linear space automata and then give a solution to the synthesis problem of linear space automata.Finally, some considerations are given to the problems of sets of tapes accepted by l.a.'s and also of operations under which the family of all the output functions of l.a.'s is not closed

To the Fiftieth Anniversary of the Department of Algebra and Mathematical Logic of Kyiv Taras Shevchenko University

History of algebraic research in Kyiv University starts in 1902 when Professor D. Grave who was a pupil of the St.Petersburg mathematical school started his work at Kyiv University. After moving to Kyiv Professor Grave began intensive research in the modern at that time branches of mathematics such as group theory, Galois theory, theory of algebraic numbers

Algebraic Aspects of Abelian Sandpile Models

The abelian sandpile models feature a finite abelian group G generated by the operators corresponding to particle addition at various sites. We study the canonical decomposition of G as a product of cyclic groups G = Z_{d_1} X Z_{d_2} X Z_{d_3}...X Z_{d_g}, where g is the least number of generators of G, and d_i is a multiple of d_{i+1}. The structure of G is determined in terms of toppling matrix. We construct scalar functions, linear in height variables of the pile, that are invariant toppling at any site. These invariants provide convenient coordinates to label the recurrent configurations of the sandpile. For an L X L square lattice, we show that g = L. In this case, we observe that the system has nontrivial symmetries coming from the action of the cyclotomic Galois group of the (2L+2)th roots of unity which operates on the set of eigenvalues of the toppling matrix. These eigenvalues are algebraic integers, whose product is the order |G|. With the help of this Galois group, we obtain an explicit factorizaration of |G|. We also use it to define other simpler, though under-complete, sets of toppling invariants.Comment: 39 pages, TIFR/TH/94-3

On Reachability Problems for Low-Dimensional Matrix Semigroup

We consider the Membership and the Half-Space Reachability problems for matrices in dimensions two and three. Our first main result is that the Membership Problem is decidable for finitely generated sub-semigroups of the Heisenberg group over rational numbers. Furthermore, we prove two decidability results for the Half-Space Reachability Problem. Namely, we show that this problem is decidable for sub-semigroups of GL(2,Z) and of the Heisenberg group over rational numbers