Matrix Semigroup Freeness Problems in SL (2, \mathbb Z).



Potapov, I and Ko, S
(2017) Matrix Semigroup Freeness Problems in SL (2, \mathbb Z). In: Theory and Practice of Computer Science - 43rd International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2017, 2017-01-16 - 2017-01-20, 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: 19 Apr 2017 10:22
Last Modified: 03 Mar 2021 09:55
DOI: 10.1007/978-3-319-51963-0_21
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3007005