2 research outputs found
Matrix Semigroup Freeness Problems in
In this paper we study decidability and complexity of decision problems on matrices from the special linear group . In particular, we study the freeness problem: given a finite set of matrices generating a multiplicative semigroup , decide whether each element of has at most one factorization over . In other words, is a code? We show that the problem of deciding whether a matrix semigroup in 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 , for example we show that to decide whether every prime matrix has at most factorizations is PSPACE-hard
The Identity Problem for Matrix Semigroups in SL2(Z) is NP-complete
In this paper, we show that the problem of determining if the identity matrix belongs to a finitely generated semigroup of matrices from the modular group and thus the Special Linear group is solvable in . From this fact, we can immediately derive that the fundamental problem of whether a given finite set of matrices from or generates a group or free semigroup is also decidable in . The previous algorithm for these problems, shown in 2005 by Choffrut and Karhum\"aki, was in \EXPSPACE mainly due to the translation of matrices into exponentially long words over a binary alphabet and further constructions with a large nondeterministic finite state automaton that is built on these words. Our algorithm is based on various new techniques that allow us to operate with compressed word representations of matrices without explicit expansions. When combined with the known -hard lower bound, this proves that the membership problem for the identity problem, the group problem and the freeness problem in are -complete