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1

Vector Ambiguity and Freeness Problems in SL (2, Z)

Ko S.-K., Potapov I.

Fundamenta Informaticae

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2018

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Vol. 162, nr 2/3

161--182

EN

We study the vector ambiguity problem and the vector freeness problem in SL (2, Z). Given a finitely generated n x n matrix semigroup S and an n-dimensional vector x, the vector ambiguity problem is to decide whether for every target vector y = Mx, where M ∈ S, M is unique. We also consider the vector freeness problem which is to show that every matrix M which is transforming x to Mx has a unique factorization with respect to the generator of S. We show that both problems are NP-complete in SL (2, Z), which is the set of 2 x 2 integer matrices with determinant 1. Moreover, we generalize the vector ambiguity problem and extend to the finite and k-vector ambiguity problems where we consider the degree of vector ambiguity of matrix semigroups.

2

On the Computational Complexity of Matrix Semigroup Problems

Bell P. C., Potapov I.

Fundamenta Informaticae

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2012

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Vol. 116, nr 1-4

1-13

EN

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 study the computational complexity of the problem of determining whether the identity matrix belongs to a matrix semigroup (the Identity Problem) generated by a finite set of 2 × 2 integral unimodular matrices. The Identity Problem for matrix semigroups is a well-known challenging problem, which has remained open in any dimension until recently. It is currently known that the problem is decidable in dimension two and undecidable starting from dimension four. In particular, we show that the Identity Problem for 2 × 2 integral unimodular matrices is NP-hard by a reduction of the Subset Sum Problem and several new encoding techniques. An upper bound for the nontrivial decidability result by C. Choffrut and J. Karhum¨aki is unknown. However, we derive a lower bound on the minimum length solution to the Identity Problem for a constructible set of instances, which is exponential in the number of matrices of the generator set and the maximal element of the matrices. This shows that the most obvious candidate for an NP algorithm, which is to guess the shortest sequence of matrices which multiply to give the identity matrix, does not work correctly since the certificate would have a length which is exponential in the size of the instance. Both results lead to a number of corollaries confirming the same bounds for vector reachability, scalar reachability and zero in the right upper corner problems.

3

On the Computational Power of Querying the History

Lisitsa A., Potapov I.

Fundamenta Informaticae

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2009

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Vol. 91, nr 2

395-409

EN

Querying its own history is an important mechanism in the computations, especially those interacting with people or other computations such as transaction processing, electronic data interchange. John McCarthy in his Elephant programming language proposal suggested exploiting the referring to the past as the main programming primitive. In this paper we study the computational power of such primitive. In order to do that we propose a refined formal model, History Dependent Machine (HDM), which uses querying the history as its sole computational primitive. Our main result may be spelled in general terms as: a model with a single agent wandering around a pool of resources and having ability to check its own history for simple temporal properties has a universal computational power. Moreover, HDM can simulate any multicounter machine in real time. Then we show that the computations of HDM may be specified in the extension of propositional linear temporal logic by flexible constants, the abstraction operator and equality. We use then universality of HDM model to show that the above extension with a single flexible constant is not recursively axiomatizable.

4

Time/Space Efficient Compressed Pattern Matching

Gąsieniec L., Potapov I.

Fundamenta Informaticae

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2003

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Vol. 56, nr 1,2

137-154

EN

An exact pattern matching problem is to find all occurrences of a pattern p in a text t. We say that the pattern matching algorithm is optimal if its running time is linear in the sizes of t and p, i.e., O(t+p). Perhaps one of the most interesting settings of the pattern matching problem is when one has to design an efficient algorithm with a help of a small extra space. In this paper we explore this setting to the extreme. We work under an assumption that the text t is available only in a compressed form, represented by a straight-line program. The compression methods based on efficient construction of straight-line programs are as competitive as the compression standards, including the Lempel-Ziv compression scheme and recently intensively studied text compression via block sorting, due to Burrows and Wheeler. Our main result is an algorithm that solves the compressed string matching problem in an optimal linear time, with a help of a constant extra space. We also discuss an efficient implementation of a version our algorithm showing that the new concept may have also some interesting real applications. Our result is in contrast with many other compressed pattern matching algorithms where the goal is to find all pattern occurrences in time related to the size of the compressed text. However one must remember that all previous algorithms used at least a linear (in a compressed text, a dictionary, or a pattern) extra memory while our algorithm can be implemented in a constant size extra space. Also from the practical point of view, when the compression ratio is constant (very rarely smaller than 25%), there is no dramatic difference between the running time based on the size of the compressed text and the size of the original text, while an extra space resources might be strictly limited.