Igor Potapov

Igor Potapov
University of Liverpool | UoL · Department of Computer Science

About

140
Publications
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714
Citations

Publications

Publications (140)
Conference Paper
Full-text available
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 proble...
Conference Paper
Full-text available
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 p...
Conference Paper
Full-text available
In this paper we investigate the decidability and complexity of problems related to braid compos-ition. While all known problems for a class of braids with 3 strands, B^3 , have polynomial time solutions we prove that a very natural question for braid composition, the membership problem, is NP-hard for braids with only 3 strands. The membership pro...
Article
Full-text available
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 thos...
Article
Full-text available
The problem of uniformly placing N points onto a sphere finds applications in many areas. For example, points on the sphere correspond to unit quaternions as well as to the group of rotations SO(3) and the online version of generating uniform rotations (known as “incremental generation”) plays a crucial role in a large number of engineering applica...
Article
A temporal graph is a dynamic graph where every edge is assigned a set of integer time labels that indicate at which discrete time step the edge is available. In this paper, we study how changes of the time labels, corresponding to delays on the availability of the edges, affect the reachability sets from given sources. We introduce control mechani...
Article
Full-text available
Crystal Structure Prediction (CSP) is one of the central and most challenging problems in materials science and computational chemistry. In CSP, the goal is to find a configuration of ions in 3D space that yields the lowest potential energy. Finding an efficient procedure to solve this complex optimisation question is a well known open problem. Due...
Article
We consider a discrete system of n devices lying on a 2-dimensional square grid and forming an initial connected shape SI. Each device is equipped with a linear-strength mechanism which enables it to move a whole line of consecutive devices in a single time-step, called a line move. We study the problem of transforming SI into a given connected tar...
Chapter
We study a model of programmable matter systems consisting of n devices lying on a 2-dimensional square grid which are able to perform the minimal mechanical operation of rotating around each other. The goal is to transform an initial shape A into a target shape B. We investigate the class of shapes which can be constructed in such a scenario under...
Chapter
We consider a discrete system of n simple indistinguishable devices, called agents, forming a connected shape SI on a two-dimensional square grid. Agents are equipped with a linear-strength mechanism, called a line move, by which an agent can push a whole line of consecutive agents in one of the four directions in a single time-step. We study the p...
Preprint
We study a model of programmable matter systems consisting of $n$ devices lying on a 2-dimensional square grid which are able to perform the minimal mechanical operation of rotating around each other. The goal is to transform an initial shape A into a target shape B. We investigate the class of shapes which can be constructed in such a scenario und...
Preprint
Full-text available
We consider a discrete system of $n$ simple indistinguishable devices, called \emph{agents}, forming a \emph{connected} shape $S_I$ on a two-dimensional square grid. Agents are equipped with a linear-strength mechanism, called a \emph{line move}, by which an agent can push a whole line of consecutive agents in one of the four directions in a single...
Chapter
In this paper we combine two classical generalisations of finite automata (weighted automata and automata on infinite words) into a model of integer weighted automata on infinite words and study the universality and the emptiness problems under zero weight acceptance. We show that the universality problem is undecidable for three-state automata by...
Preprint
Full-text available
This paper introduces the natural generalisation of necklaces to the multidimensional setting -- multidimensional necklaces. One-dimensional necklaces are known as cyclic words, two-dimensional necklaces correspond to toroidal codes, and necklaces of dimension three can represent periodic motives in crystals. Our central results are two approximati...
Article
We study reachability problems for various nondeterministic polynomial maps in Zn. We prove that the reachability problem for very simple three-dimensional affine maps (with independent variables) is undecidable and is PSPACE-hard for both two-dimensional affine maps and one-dimensional quadratic maps. Then we show that the complexity of the reacha...
Preprint
Full-text available
The main result of the paper is the first polynomial-time algorithm for ranking bracelets. The time-complexity of the algorithm is O(k^2 n^4), where k is the size of the alphabet and n is the length of the considered bracelets. The key part of the algorithm is to compute the rank of any word with respect to the set of bracelets by finding three oth...
Article
We consider the following variant of the mortality problem: given k×k matrices A1,A2,…,At, do there exist t nonnegative integers m1,m2,…,mt such that the product A1m1A2m2⋯Atmt is equal to the zero matrix? It is known that this problem is decidable when t≤2 for matrices over algebraic numbers but becomes undecidable for sufficiently large t and k, e...
Book
Chapter ‘Recent Advances on Reachability Problems for Valence Systems’ is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.
Chapter
We consider a discrete system of n devices lying on a 2-dimensional square grid and forming an initial connected shape . Each device is equipped with a linear-strength mechanism which enables it to move a whole line of consecutive devices in a single time-step. We study the problem of transforming into a given connected target shape of the same num...
Preprint
Full-text available
In graph theory, the objective of the k-centre problem is to find a set of $k$ vertices for which the largest distance of any vertex to its closest vertex in the $k$-set is minimised. In this paper, we introduce the $k$-centre problem for sets of necklaces, i.e. the equivalence classes of words under the cyclic shift. This can be seen as the k-cent...
Preprint
Full-text available
We consider a discrete system of $n$ devices lying on a 2-dimensional square grid and forming an initial connected shape $S_I$. Each device is equipped with a linear-strength mechanism which enables it to move a whole line of consecutive devices in a single time-step. We study the problem of transforming $S_I$ into a given connected target shape $S...
Article
In this work, 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 in parallel in a si...
Preprint
Full-text available
A temporal graph is a dynamic graph where every edge is assigned a set of integer time labels that indicate at which discrete time step the edge is available. In this paper, we study how changes of the time labels, corresponding to delays on the availability of the edges, affect the reachability sets from given sources. The questions about reachabi...
Article
Full-text available
A temporal graph is a dynamic graph where every edge is assigned a set of integer time labels that indicate at which discrete time step the edge is available. In this paper, we study how changes of the time labels, corresponding to delays on the availability of the edges, affect the reachability sets from given sources. The questions about reachabi...
Chapter
Crystal Structure Prediction (csp) is one of the central and most challenging problems in materials science and computational chemistry. In csp, the goal is to find a configuration of ions in 3D space that yields the lowest potential energy. Finding an efficient procedure to solve this complex optimisation question is a well known open problem in c...
Book
This book constitutes the refereed proceedings of the 14th International Conference on Reachability Problems, RP 2020, held in Paris, France in October 2020. The 8 full papers presented were carefully reviewed and selected from 25 submissions. In addition, 2 invited papers were included in this volume. The papers cover topics such as reachability f...
Chapter
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 p...
Preprint
Crystal Structure Prediction (csp) is one of the central and most challenging problems in materials science and computational chemistry. In csp, the goal is to find a configuration of ions in 3D space that yields the lowest potential energy. Finding an efficient procedure to solve this complex optimisation question is a well known open problem in c...
Preprint
Full-text available
In this work we extend previously known decidability results for $2\times 2$ matrices over $\mathbb{Q}$. Namely, we introduce a notion of flat rational sets: if $M$ is a monoid and $N\leq M$ is its submonoid, then flat rational sets of $M$ relative to $N$ are finite unions of the form $L_0g_1L_1 \cdots g_t L_t$ where all $L_i$s are rational subsets...
Article
A robot game, also known as a Z-VAS game, is a two-player vector addition game played on the integer lattice Zn, where one of the players, Adam, aims to avoid the origin while the other player, Eve, aims to reach the origin. The problem is to decide whether or not Eve has a winning strategy. In this paper we prove undecidability of the two-dimensio...
Chapter
The problem of uniformly placing N points onto a sphere finds applications in many areas. An online version of this problem was recently studied with respect to the gap ratio as a measure of uniformity. The proposed online algorithm of Chen et al. was upper-bounded by 5.99 and then improved to 3.69, which is achieved by considering a circumscribed...
Preprint
Full-text available
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 $n$...
Preprint
Full-text available
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 algebra...
Book
This book constitutes the refereed proceedings of the 13th International Conference on Reachability Problems, RP 2019, held in Brussels, Belgium, in September 2019.The 14 full papers presented were carefully reviewed and selected from 26 submissions. The papers cover topics such as reachability for infinite state systems; rewriting systems; reachab...
Preprint
Full-text available
In this paper we consider the set ${\mathbb Z}^{\pm\omega}_{6}$ of two-way infinite words $\xi$ over the alphabet $\{0,1,2,3,4,5\}$ with the integer left part $\lfloor\xi\rfloor$ and the fractional right part $\{\xi\}$ separated by a radix point. For such words, the operation of multiplication by integers and division by $6$ are defined as the colu...
Preprint
Full-text available
In this paper we consider the set ${\mathbb Z}^{\pm\omega}_{6}$ of two-way infinite words $\xi$ over the alphabet $\{0,1,2,3,4,5\}$ with the integer left part $\lfloor\xi\rfloor$ and the fractional right part $\{\xi\}$ separated by a radix point. For such words, the operation of multiplication by integers and division by $6$ are defined as the colu...
Preprint
Full-text available
The problem of uniformly placing N points onto a sphere finds applications in many areas. An online version of this problem was recently studied with respect to the gap ratio as a measure of uniformity. The proposed online algorithm of Chen et al. is upper-bounded by 5.99, which is achieved by considering a circumscribed dodecahedron followed by a...
Article
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 SL(2,Z). Our approach to solving these problems is based on the characterization of reachability paths between vector...
Article
Full-text available
Piecewise affine maps (PAMs) are frequently used as a reference model to discuss the frontier between known and open questions about the decidability for reachability questions. In particular, the reachability problem for one-dimensional PAM is still an open problem, even if restricted to only two intervals. As the main contribution of this paper w...
Article
Full-text available
The paper is devoted to several infinite-state Attacker–Defender games with reachability objectives. We prove the undecidability of checking for the existence of a winning strategy in several low-dimensional mathematical games including vector reachability games, word games and braid games. To prove these results, we consider a model of weighted au...
Article
Full-text available
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 me...
Article
Full-text available
We study the Identity problem for matrix semigroups. The Identity problem is to decide whether there exists the identity matrix in the given matrix semigroup. It has been recently shown that the Identity problem is NP-complete for a matrix semigroup generated by matrices from the Special Linear Group ${\rm SL}(2,\mathbb{Z})$ and undecidable for mat...
Conference Paper
We study the vector ambiguity problem and the vector freeness problem in SL\((2,\mathbb {Z})\). Given a finitely generated \(n \times n\) matrix semigroup S and an n-dimensional vector \(\mathbf {x}\), the vector ambiguity problem is to decide whether for every target vector \(\mathbf {y} = M\mathbf {x}\), where \(M \in S\), M is unique. We also co...
Conference Paper
In this paper we study decidability and complexity of decision problems on matrices from the special linear group 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...
Book
This book constitutes the refereed proceedings of the 11th International Workshop on Reachability Problems, RP 2017, held in London, UK, in September 2017. The 12 full papers presented together with 1 invited paper were carefully reviewed and selected from 17 submissions. The aim of the conference is to bring together scholars from diverse fields...
Article
Full-text available
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...
Conference Paper
We introduce a new notion of a relational word as a finite totally ordered set of positions endowed with two 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 corres...
Article
Full-text available
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...
Conference Paper
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-dimensional 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 b...
Book
This book constitutes the refereed proceedings of the 10th International Workshop on Reachability Problems, RP 2016, held in Aalborg, Denmark, in September 2016. The 11 full papers presented together with2 invited papers and 3 abstracts of invited talks were carefully reviewed and selected from 18 submissions. The papers cover a range of topics in...
Article
Full-text available
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 b...
Article
Full-text available
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 corr...
Article
Full-text available
This paper is showing the solution for two open problems about decidability of the vector reachability problem in 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 form $\mathrm{SL}(2,\mathbb{Z})...
Conference Paper
Full-text available
We consider several 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 ch...
Conference Paper
Full-text available
Robot Game is a two player vector addition game played in integer lattice Z^n. In a degree k case both players have k 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 Attacker, tries to play the game from the initial configuration to the origin while the othe...
Book
This book constitutes the refereed proceedings of the 9th International Workshop on Reachability Problems, RP 2015, held in Warsaw, Poland, in September 2015. The 14 papers presented together with 6 extended abstracts in this volume were carefully reviewed and selected from 23 submissions. The papers cover a range of topics in the field of reachabi...
Book
This book constitutes the proceedings of the 19th International Conference on Developments in Language Theory, DLT 2015, held in Liverpool, UK. The 31 papers presented together with 5 invited talks were carefully reviewed and selected from 54 submissions. Its scope is very general and includes, among others, the following topics and areas: combinat...
Article
In this paper we show undecidabillity of universality problem for weighted one counter automata over infinite words by direct reduction from any Infinite Post Correspondence Problem (wPCP). Obviously, this increases the structural complexity of weighted automata comparing to our previous work, but provides more general reduction without taking into...
Book
This book constitutes the proceedings of the 8th International Workshop on Reachability Problems, RP 2014, held in Oxford, UK, in September 2014. The 17 papers presented in this volume were carefully reviewed and selected from 25 submissions. The book also contains a paper summarizing the invited talk. The papers offer new approaches for the modell...
Article
Full-text available
In this paper we introduce and apply a novel approach for self-organisation, partitioning and pattern formation on the non-oriented grid environment. The method is based on the generation of nodal patterns in the environment via sequences of discrete waves. The power of the primitives is illustrated by giving solutions to two geometric problems usi...
Conference Paper
Neighbourhood Sequences are deemed to be important in many practical applications within digital imaging through their application in measuring digital distance. Aggregation of neighbourhood sequences based on classical digital distance functions was proposed as an alternative method for organising swarms or robots on the non-oriented grid environm...
Conference Paper
Full-text available
We study the computational complexity of determining whether the zero matrix belongs to a finitely generated semigroup of two dimensional integer matrices (the mortality problem). We show that this problem is NP-hard to decide in the two-dimensional case by using a new encoding and properties of the projective special linear group. The decidability...
Article
Full-text available
Most computational problems for matrix semigroups and groups are inherently difficult to solve and even undecidable starting from dimension three. The questions about the decidability and complexity of problems for two-dimensional matrix semigroups remain open and are directly linked with other challenging problems in the field. In this paper we st...
Conference Paper
In this paper we introduce and apply a novel approach for self-organization, partitioning and pattern formation on the non-oriented grid environment. The method is based on the generation of nodal patterns in the environment via sequences of discrete waves. The power of the primitives is illustrated by giving solutions to two geometric problems usi...
Conference Paper
Full-text available
In this paper we investigate the complexity of planarity of knot diagrams encoded by Gauss words, both in terms of recognition by automata over infinite alphabets and in terms of classical logarithmic space complexity. As the main result, we show that recognition of planarity of unsigned Gauss words can be done in deterministic logarithmic space an...
Article
In this paper, we introduce and study an algebra of languages representable by vertex-labeled graphs. The proposed algebra is equipped with three operations: the union of languages, the merging of languages and the iteration. In contrast to Kleene algebra, which is mainly used for edge-labeled graphs, it can adequately represent many properties of...
Article
Full-text available
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 vi...
Article
Full-text available
This is an introductory paper in which we rise and study two fundamen-tal problems related to the analysis of a computational dynamic object dis-tributed on the environment: • How to define unambiguously what is the state of such object? • How to measure the amount of state transitions in this case? The main idea of the paper is to show that the st...
Article
In this paper, we investigate various decision problems concerning parameterized versions of some classes of machines. Let C(s,m,t) be the class of nondeterministic multitape Turing machine (TM) acceptors with a two-way read-only input, at most s states, at most m read–write worktapes, and at most t symbols in the worktape alphabet, where s,m,t are...