arXiv subjects

Igor Potapov

Publications and source records attributed to Igor Potapov.

35 records · Page 2Linked to original sources

arXiv2019

On the Mortality Problem: from multiplicative matrix equations to linear recurrence sequences and beyond

We consider the following variant of the Mortality Problem: given $k\times k$ matrices $A_1, A_2, \dots,A_{t}$, does there exist nonnegative integers $m_1, m_2, \dots,m_t$ such that the product $A_1^{m_1} A_2^{m_2} \cdots A_{t}^{m_{t}}$ is equal to the zero matrix? It is known that this problem is decidable when $t \leq 2$ for matrices over algebraic numbers but becomes undecidable for sufficiently large $t$ and $k$ even for integral matrices. In this paper, we prove the first decidability results for $t>2$. We show as one of our central results that for $t=3$ this problem in any dimension is Turing equivalent to the well-known Skolem problem for linear recurrence sequences. Our proof relies on the Primary Decomposition Theorem for matrices that was not used to show decidability results in matrix semigroups before. As a corollary we obtain that the above problem is decidable for $t=3$ and $k \leq 3$ for matrices over algebraic numbers and for $t=3$ and $k=4$ for matrices over real algebraic numbers. Another consequence is that the set of triples $(m_1,m_2,m_3)$ for which the equation $A_1^{m_1} A_2^{m_2} A_3^{m_3}$ equals the zero matrix is equal to a finite union of direct products of semilinear sets. For $t=4$ we show that the solution set can be non-semilinear, and thus it seems unlikely that there is a direct connection to the Skolem problem. However we prove that the problem is still decidable for upper-triangular $2 \times 2$ rational matrices by employing powerful tools from transcendence theory such as Baker's theorem and S-unit equations.

cs.DM↗

arXiv2019

Pushing Lines Helps: Efficient Universal Centralised Transformations for Programmable Matter

In this paper, we study a discrete system of entities residing on a two-dimensional square grid. Each entity is modelled as a node occupying a distinct cell of the grid. The set of all $n$ nodes forms initially a connected shape $A$. Entities are equipped with a linear-strength pushing mechanism that can push a whole line of entities, from 1 to $n$, in parallel in a single time-step. A target connected shape $B$ is also provided and the goal is to \emph{transform} $A$ into $B$ via a sequence of line movements. Existing models based on local movement of individual nodes, such as rotating or sliding a single node, can be shown to be special cases of the present model, therefore their (inefficient, $Θ(n^2)$) \emph{universal transformations} carry over. Our main goal is to investigate whether the parallelism inherent in this new type of movement can be exploited for efficient, i.e., sub-quadratic worst-case, transformations. As a first step towards this, we restrict attention solely to centralised transformations and leave the distributed case as a direction for future research. Our results are positive. By focusing on the apparently hard instance of transforming a diagonal $A$ into a straight line $B$, we first obtain transformations of time $O(n\sqrt{n})$ without and with preserving the connectivity of the shape throughout the transformation. Then, we further improve by providing two $O(n\log n)$-time transformations for this problem. By building upon these ideas, we first manage to develop an $O(n\sqrt{n})$-time universal transformation. Our main result is then an $ O(n \log n) $-time universal transformation. We leave as an interesting open problem a suspected $Ω(n\log n)$-time lower bound.

cs.DS↗

arXiv2018

De Bruijn graphs and powers of $3/2$

In this paper we consider the set ${\mathbb Z}^{\pmω}_{6}$ of two-way infinite words $ξ$ over the alphabet $\{0,1,2,3,4,5\}$ with the integer left part $\lfloorξ\rfloor$ and the fractional right part $\{ξ\}$ separated by a radix point. For such words, the operation of multiplication by integers and division by $6$ are defined as the column multiplication and division in base 6 numerical system. The paper develops a finite automata approach for analysis of sequences $\left (\left \lfloor ξ\left (\frac{3}{2} \right)^n \right \rfloor \right)_{n \in {\mathbb Z}}$ for the words $ξ\in {\mathbb Z}^{\pm ω}_{6}$ that have some common properties with $Z$-numbers in Mahler's $3/2$-problem. Such sequence of $Z$-words written under each other with the same digit positions in the same column is an infinite $2$-dimensional word over the alphabet ${\mathbb Z}_6$. The automata representation of the columns in the integer part of $2$-dimensional $Z$-words has the nice structural properties of the de Bruijn graphs. This way provides some sufficient conditions for the emptiness of the set of $Z$-numbers. Our approach has been initially inspirated by the proposition 2.5 in [1] where authors applies cellular automata for analysis of $\left(\left\{ξ\left(\frac{3}{2}\right)^n\right\} \right)_{n\in{\mathbb Z}}$, $ξ\in{\mathbb R}$.

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↗

arXiv2017

Composition problems for braids: Membership, Identity and Freeness

In this paper we investigate the decidability and complexity of problems related to braid composition. While all known problems for a class of braids with three strands, $B_3$, have polynomial time solutions we prove that a very natural question for braid composition, the membership problem, is NP-complete for braids with only three strands. The membership problem is decidable in NP for $B_3$, but it becomes harder for a class of braids with more strands. In particular we show that fundamental problems about braid compositions are undecidable for braids with at least five strands, but decidability of these problems for $B_4$ remains open. Finally we show that the freeness problem for semigroups of braids from $B_3$ is also decidable in NP. The paper introduces a few challenging algorithmic problems about topological braids opening new connections between braid groups, combinatorics on words, complexity theory and provides solutions for some of these problems by application of several techniques from automata theory, matrix semigroups and algorithms.

cs.CC↗

arXiv2016

Matrix Semigroup Freeness Problems in $\mathrm{SL}(2,\mathbb{Z})$

In this paper we study decidability and complexity of decision problems on matrices from the special linear group $\mathrm{SL}(2,\mathbb{Z})$. In particular, we study the freeness problem: given a finite set of matrices $G$ generating a multiplicative semigroup $S$, decide whether each element of $S$ has at most one factorization over $G$. In other words, is $G$ a code? We show that the problem of deciding whether a matrix semigroup in $\mathrm{SL}(2,\mathbb{Z})$ is non-free is NP-hard. Then, we study questions about the number of factorizations of matrices in the matrix semigroup such as the finite freeness problem, the recurrent matrix problem, the unique factorizability problem, etc. Finally, we show that some factorization problems could be even harder in $\mathrm{SL}(2,\mathbb{Z})$, for example we show that to decide whether every prime matrix has at most $k$ factorizations is PSPACE-hard.

cs.CC↗

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↗

arXiv2016

Vector Reachability Problem in $\mathrm{SL}(2,\mathbb{Z})$

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 $\mathrm{SL}(2,\mathbb{Z})$ and the point to point reachability (over rational numbers) for fractional linear transformations, where associated matrices are from $\mathrm{SL}(2,\mathbb{Z})$. 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.

cs.FL↗

arXiv2016

Decidability of the Membership Problem for $2\times 2$ 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.

cs.DM↗

arXiv2015

Reachability problems for PAMs

Piecewise affine maps (PAMs) are frequently used as a reference model to show the openness of the reachability questions in other systems. The reachability problem for one-dimentional PAM is still open even if we define it with only two intervals. As the main contribution of this paper we introduce new techniques for solving reachability problems based on p-adic norms and weights as well as showing decidability for two classes of maps. Then we show the connections between topological properties for PAM's orbits, reachability problems and representation of numbers in a rational base system. Finally we show a particular instance where the uniform distribution of the original orbit may not remain uniform or even dense after making regular shifts and taking a fractional part in that sequence.

math.NA↗

arXiv2015

On insertion-deletion systems over relational words

We introduce a new notion of a relational word as a finite totally ordered set of positions endowed with three binary relations that describe which positions are labeled by equal data, by unequal data and those having an undefined relation between their labels. We define the operations of insertion and deletion on relational words generalizing corresponding operations on strings. We prove that the transitive and reflexive closure of these operations has a decidable membership problem for the case of short insertion-deletion rules (of size two/three and three/two). At the same time, we show that in the general case such systems can produce a coding of any recursively enumerable language leading to undecidabilty of reachability questions.

cs.FL↗

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↗

arXiv2014

Broadcasting Automata and Patterns on Z^2

The Broadcasting Automata model draws inspiration from a variety of sources such as Ad-Hoc radio networks, cellular automata, neighbourhood se- quences and nature, employing many of the same pattern forming methods that can be seen in the superposition of waves and resonance. Algorithms for broad- casting automata model are in the same vain as those encountered in distributed algorithms using a simple notion of waves, messages passed from automata to au- tomata throughout the topology, to construct computations. The waves generated by activating processes in a digital environment can be used for designing a vari- ety of wave algorithms. In this chapter we aim to study the geometrical shapes of informational waves on integer grid generated in broadcasting automata model as well as their potential use for metric approximation in a discrete space. An explo- ration of the ability to vary the broadcasting radius of each node leads to results of categorisations of digital discs, their form, composition, encodings and gener- ation. Results pertaining to the nodal patterns generated by arbitrary transmission radii on the plane are explored with a connection to broadcasting sequences and ap- proximation of discrete metrics of which results are given for the approximation of astroids, a previously unachievable concave metric, through a novel application of the aggregation of waves via a number of explored functions.

cs.FL↗

arXiv2010

The Identity Correspondence Problem and its Applications

In this paper we study several closely related fundamental problems for words and matrices. First, we introduce the Identity Correspondence Problem (ICP): whether a finite set of pairs of words (over a group alphabet) can generate an identity pair by a sequence of concatenations. We prove that ICP is undecidable by a reduction of Post's Correspondence Problem via several new encoding techniques. In the second part of the paper we use ICP to answer a long standing open problem concerning matrix semigroups: "Is it decidable for a finitely generated semigroup S of square integral matrices whether or not the identity matrix belongs to S?". We show that the problem is undecidable starting from dimension four even when the number of matrices in the generator is 48. From this fact, we can immediately derive that the fundamental problem of whether a finite set of matrices generates a group is also undecidable. We also answer several question for matrices over different number fields. Apart from the application to matrix problems, we believe that the Identity Correspondence Problem will also be useful in identifying new areas of undecidable problems in abstract algebra, computational questions in logic and combinatorics on words.

math.GR↗

arXiv2010

A measure of state transition of collective of stateless automata in discrete environment

In this work a collective of interacting stateless automata in a discrete geometric environment is considered as an integral automata-like computational dynamic object. For such distributed on the environment object different approaches to definition of the measure of state transition are possible. We propose a geometric approach for defining what a state is.

cs.FL↗

arXiv2009

On Descriptional Complexity of the Planarity Problem for Gauss Words

In this paper we investigate the descriptional complexity of knot theoretic problems and show upper bounds for planarity problem of signed and unsigned knot diagrams represented by Gauss words. Since a topological equivalence of knots can involve knot diagrams with arbitrarily many crossings then Gauss words will be considered as strings over an infinite (unbounded) alphabet. For establishing the upper bounds on recognition of knot properties, we study these problems in a context of automata models over an infinite alphabet.

cs.FL↗

arXiv2004

Temporal logic with predicate abstraction

A predicate linear temporal logic LTL_{λ,=} without quantifiers but with predicate abstraction mechanism and equality is considered. The models of LTL_{λ,=} can be naturally seen as the systems of pebbles (flexible constants) moving over the elements of some (possibly infinite) domain. This allows to use LTL_{λ,=} for the specification of dynamic systems using some resources, such as processes using memory locations, mobile agents occupying some sites, etc. On the other hand we show that LTL_{λ,=} is not recursively axiomatizable and, therefore, fully automated verification of LTL_{λ,=} specifications is not, in general, possible.

cs.LO↗