4,579 research outputs found
Minsky machines and algorithmic problems
This is a survey of using Minsky machines to study algorithmic problems in
semigroups, groups and other algebraic systems.Comment: 19 page
Multigraded Hilbert Series of noncommutative modules
In this paper, we propose methods for computing the Hilbert series of
multigraded right modules over the free associative algebra. In particular, we
compute such series for noncommutative multigraded algebras. Using results from
the theory of regular languages, we provide conditions when the methods are
effective and hence the sum of the Hilbert series is a rational function.
Moreover, a characterization of finite-dimensional algebras is obtained in
terms of the nilpotency of a key matrix involved in the computations. Using
this result, efficient variants of the methods are also developed for the
computation of Hilbert series of truncated infinite-dimensional algebras whose
(non-truncated) Hilbert series may not be rational functions. We consider some
applications of the computation of multigraded Hilbert series to algebras that
are invariant under the action of the general linear group. In fact, in this
case such series are symmetric functions which can be decomposed in terms of
Schur functions. Finally, we present an efficient and complete implementation
of (standard) graded and multigraded Hilbert series that has been developed in
the kernel of the computer algebra system Singular. A large set of tests
provides a comprehensive experimentation for the proposed algorithms and their
implementations.Comment: 28 pages, to appear in Journal of Algebr
Revisiting the Equivalence Problem for Finite Multitape Automata
The decidability of determining equivalence of deterministic multitape
automata (or transducers) was a longstanding open problem until it was resolved
by Harju and Karhum\"{a}ki in the early 1990s. Their proof of decidability
yields a co_NP upper bound, but apparently not much more is known about the
complexity of the problem. In this paper we give an alternative proof of
decidability, which follows the basic strategy of Harju and Karhumaki but
replaces their use of group theory with results on matrix algebras. From our
proof we obtain a simple randomised algorithm for deciding language equivalence
of deterministic multitape automata and, more generally, multiplicity
equivalence of nondeterministic multitape automata. The algorithm involves only
matrix exponentiation and runs in polynomial time for each fixed number of
tapes. If the two input automata are inequivalent then the algorithm outputs a
word on which they differ
Fraction-free algorithm for the computation of diagonal forms matrices over Ore domains using Gr{\"o}bner bases
This paper is a sequel to "Computing diagonal form and Jacobson normal form
of a matrix using Groebner bases", J. of Symb. Computation, 46 (5), 2011. We
present a new fraction-free algorithm for the computation of a diagonal form of
a matrix over a certain non-commutative Euclidean domain over a computable
field with the help of Gr\"obner bases. This algorithm is formulated in a
general constructive framework of non-commutative Ore localizations of
-algebras (OLGAs). We split the computation of a normal form of a matrix
into the diagonalization and the normalization processes. Both of them can be
made fraction-free. For a matrix over an OLGA we provide a diagonalization
algorithm to compute and with fraction-free entries such that
holds and is diagonal. The fraction-free approach gives us more information
on the system of linear functional equations and its solutions, than the
classical setup of an operator algebra with rational functions coefficients. In
particular, one can handle distributional solutions together with, say,
meromorphic ones. We investigate Ore localizations of common operator algebras
over and use them in the unimodularity analysis of transformation
matrices . In turn, this allows to lift the isomorphism of modules over an
OLGA Euclidean domain to a polynomial subring of it. We discuss the relation of
this lifting with the solutions of the original system of equations. Moreover,
we prove some new results concerning normal forms of matrices over non-simple
domains. Our implementation in the computer algebra system {\sc
Singular:Plural} follows the fraction-free strategy and shows impressive
performance, compared with methods which directly use fractions. Since we
experience moderate swell of coefficients and obtain simple transformation
matrices, the method we propose is well suited for solving nontrivial practical
problems.Comment: 25 pages, to appear in Journal of Symbolic Computatio
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