arXiv subjects

Reino Niskanen

Publications and source records attributed to Reino Niskanen.

6 recordsLinked to original sources

arXiv2026

On Word Representations and Embeddings in Complex Matrices

Embeddings of word structures into matrix semigroups provide a natural bridge between combinatorics on words and linear algebra. However, low-dimensional matrix semigroups impose strong structural restrictions on possible embeddings. Certain finitely generated groups admit faithful representations in SL(2, C) and other similar matrix groups. On the other hand, it is known that the product of two free semigroups on two generators cannot be embedded into the 2x2 complex matrices. In this paper we study embeddings of word structures into low-dimensional matrix semigroups over the complex numbers and develop new techniques for constructing word representations of the Euclidean Bianchi groups. These representations provide a symbolic framework and a natural first step towards analysing fundamental decision problems in 2x2 matrix semigroups.

cs.FL↗

arXiv2025

On the Identity and Group Problems for Complex Heisenberg Matrices

We study the Identity Problem, the problem of determining if a finitely generated semigroup of matrices contains the identity matrix; see Problem 3 (Chapter 10.3) in ``Unsolved Problems in Mathematical Systems and Control Theory'' by Blondel and Megretski (2004). This fundamental problem is known to be undecidable for $\mathbb{Z}^{4 \times 4}$ and decidable for $\mathbb{Z}^{2 \times 2}$. The Identity Problem has been recently shown to be in polynomial time by Dong for the Heisenberg group over complex numbers in any fixed dimension with the use of Lie algebra and the Baker-Campbell-Hausdorff formula. We develop alternative proof techniques for the problem making a step forward towards more general problems such as the Membership Problem. Using our techniques we also show that the problem of determining if a given set of Heisenberg matrices generates a group can be decided in polynomial time.

cs.DM↗

arXiv2019

Monadic Decomposability of Regular Relations

Monadic decomposibility --- the ability to determine whether a formula in a given logical theory can be decomposed into a boolean combination of monadic formulas --- is a powerful tool for devising a decision procedure for a given logical theory. In this paper, we revisit a classical decision problem in automata theory: given a regular (a.k.a. synchronized rational) relation, determine whether it is recognizable, i.e., it has a monadic decomposition (that is, a representation as a boolean combination of cartesian products of regular languages). Regular relations are expressive formalisms which, using an appropriate string encoding, can capture relations definable in Presburger Arithmetic. In fact, their expressive power coincide with relations definable in a universal automatic structure; equivalently, those definable by finite set interpretations in WS1S (Weak Second Order Theory of One Successor). Determining whether a regular relation admits a recognizable relation was known to be decidable (and in exponential time for binary relations), but its precise complexity still hitherto remains open. Our main contribution is to fully settle the complexity of this decision problem by developing new techniques employing infinite Ramsey theory. The complexity for DFA (resp. NFA) representations of regular relations is shown to be NLOGSPACE-complete (resp. \PSPACE-complete).

cs.FL↗

arXiv2018

On the Identity Problem for the Special Linear Group and the Heisenberg Group

We study the identity problem for matrices, i.e., whether the identity matrix is in a semigroup generated by a given set of generators. In particular we consider the identity problem for the special linear group following recent NP-completeness result for ${\rm SL}(2,\mathbb{Z})$ and the undecidability for ${\rm SL}(4,\mathbb{Z})$ generated by $48$ matrices. First we show that there is no embedding from pairs of words into $3\times3$ integer matrices with determinant one, i.e., into ${\rm SL}(3,\mathbb{Z})$ extending previously known result that there is no embedding into $\mathbb{C}^{2\times 2}$. Apart from theoretical importance of the result it can be seen as a strong evidence that the computational problems in \slthreez are decidable. The result excludes the most natural possibility of encoding the Post correspondence problem into ${\rm SL}(3,\mathbb{Z})$, where the matrix products extended by the right multiplication correspond to the Turing machine simulation. Then we show that the identity problem is decidable in polynomial time for an important subgroup of ${\rm SL}(3,\mathbb{Z})$, the Heisenberg group ${\rm H}(3,\mathbb{Z})$. Furthermore, we extend the decidability result for ${\rm H}(n,\mathbb{Q})$ in any dimension $n$. Finally we are tightening the gap on decidability question for this long standing open problem by improving the undecidability result for the identity problem in ${\rm SL}(4,\mathbb{Z})$ substantially reducing the bound on the size of the generator set from $48$ to $8$ by developing a novel reduction technique.

cs.DM↗

arXiv2016

Undecidability of Two-dimensional Robot Games

Robot game is a two-player vector addition game played on the integer lattice $\mathbb{Z}^n$. Both players have sets of vectors and in each turn the vector chosen by a player is added to the current configuration vector of the game. One of the players, called Eve, tries to play the game from the initial configuration to the origin while the other player, Adam, tries to avoid the origin. The problem is to decide whether or not Eve has a winning strategy. In this paper we prove undecidability of the robot game in dimension two answering the question formulated by Doyen and Rabinovich in 2011 and closing the gap between undecidable and decidable cases.

cs.GT↗

arXiv2015

Weighted automata on infinite words in the context of Attacker-Defender games

We consider infinite-state Attacker-Defender games with reachability objectives. The results of the paper are twofold. Firstly we prove a new language-theoretic result for weighted automata on infinite words and show its encoding into the framework of Attacker-Defender games. Secondly we use this novel concept to prove undecidability for checking existence of a winning strategy in several low-dimensional mathematical games including vector reachability games, word games and braid games.

cs.FL↗