Matrix Semigroup Freeness Problems in SL(2, Z)



Ko, Sang-Ki and Potapov, Igor
(2017) Matrix Semigroup Freeness Problems in SL(2, Z). In: 43rd International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM2017), 2017-1-16 - 2017-1-20, Lero – Limerick, Ireland.

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Abstract

In this paper we study decidability and complexity of decision problems on matrices from the special linear group SL(2, 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 SL(2, 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 SL(2, ℤ), for example we show that to decide whether every prime matrix has at most k factorizations is PSPACE-hard.

Item Type: Conference or Workshop Item (Unspecified)
Uncontrolled Keywords: Matrix semigroups, Freeness, Decision problems, Decidability, Computational complexity
Depositing User: Symplectic Admin
Date Deposited: 02 Nov 2016 09:04
Last Modified: 19 Jan 2023 07:26
DOI: 10.1007/978-3-319-51963-0_21
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3004256