170 research outputs found
Matrix Semigroup Freeness Problems in SL(2, Z)
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
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
Vector Ambiguity and Freeness Problems in SL (2, ℤ).
We study the vector ambiguity problem and the vector freeness problem in SL(2,Z). Given a finitely generated n×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 2×2 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
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
Vector Reachability Problem in
The decision problems on matrices were intensively studied for many decades
as matrix products play an essential role in the representation of various
computational processes. However, many computational problems for matrix
semigroups are inherently difficult to solve even for problems in low
dimensions and most matrix semigroup problems become undecidable in general
starting from dimension three or four.
This paper solves two open problems about the decidability of the vector
reachability problem over a finitely generated semigroup of matrices from
and the point to point reachability (over rational
numbers) for fractional linear transformations, where associated matrices are
from . The approach to solving reachability problems
is based on the characterization of reachability paths between points which is
followed by the translation of numerical problems on matrices into
computational and combinatorial problems on words and formal languages. We also
give a geometric interpretation of reachability paths and extend the
decidability results to matrix products represented by arbitrary labelled
directed graphs. Finally, we will use this technique to prove that a special
case of the scalar reachability problem is decidable
Decidability of the Membership Problem for integer matrices
The main result of this paper is the decidability of the membership problem
for nonsingular integer matrices. Namely, we will construct the
first algorithm that for any nonsingular integer matrices
and decides whether belongs to the semigroup generated
by .
Our algorithm relies on a translation of the numerical problem on matrices
into combinatorial problems on words. It also makes use of some algebraical
properties of well-known subgroups of and various
new techniques and constructions that help to limit an infinite number of
possibilities by reducing them to the membership problem for regular languages
Vector Reachability Problem in SL(2,Z)
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 . Our approach to solving these problems is based on the characterization of reachability paths between vectors or points, which is then used to translate the numerical problems on matrices into computational problems on words and regular languages. We will also give geometric interpretations of these results
Acceptance Ambiguity for Quantum Automata
We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield Measure Once Quantum Finite Automata (MO-QFA). We study the distribution of acceptance probabilities of such MO-QFA, which is partly motivated by similar freeness problems for matrix semigroups and other computational models. We show that determining if the acceptance probabilities of all possible input words are unique is undecidable for 32 state MO-QFA, even when all unitary matrices and the projection matrix are rational and the initial configuration is defined over real algebraic numbers. We utilize properties of the skew field of quaternions, free rotation groups, representations of tuples of rationals as a linear sum of radicals and a reduction of the mixed modification Post\u27s correspondence problem
The Identity Correspondence Problem and its Applications
In this paper 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 via several new encoding
techniques.
In the second part of the paper we use ICP to answer a long standing open
problem concerning matrix semigroups: "Is it decidable for a finitely generated
semigroup S of square integral matrices whether or not the identity matrix
belongs to S?". We show that the problem is undecidable starting from dimension
four even when the number of matrices in the generator is 48. From this fact,
we can immediately derive that the fundamental problem of whether a finite set
of matrices generates a group is also undecidable. We also answer several
question for matrices over different number fields. Apart from the application
to matrix problems, we believe that the Identity Correspondence Problem will
also be useful in identifying new areas of undecidable problems in abstract
algebra, computational questions in logic and combinatorics on words.Comment: We have made some proofs clearer and fixed an important typo from the
published journal version of this article, see footnote 3 on page 1
Composition problems for braids: Membership, Identity and Freeness
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, , 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 membership problem is decidable in NP for , but it becomes harder for a class of braids with more strands. In particular we show that fundamental problems about braid compositions are undecidable for braids with at least five strands, but decidability of these problems for remains open. Finally we show that the freeness problem for semigroups of braids from is also decidable in NP. The paper introduces a few challenging algorithmic problems about topological braids opening new connections between braid groups, combinatorics on words, complexity theory and provides solutions for some of these problems by application of several techniques from automata theory, matrix semigroups and algorithms
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