39,215 research outputs found
Decision problems for word-hyperbolic semigroups
This paper studies decision problems for semigroups that are word-hyperbolic in the sense of Duncan & Gilman. A fundamental investigation reveals that the natural definition of a `word-hyperbolic structure' has to be strengthened slightly in order to define a unique semigroup up to isomorphism. The isomorphism problem is proven to be undecidable for word-hyperbolic semigroups (in contrast to the situation for word-hyperbolic groups). It is proved that it is undecidable whether a word-hyperbolic semigroup is automatic, asynchronously automatic, biautomatic, or asynchronously biautomatic. (These properties do not hold in general for word-hyperbolic semigroups.) It is proved that the uniform word problem for word-hyperbolic semigroup is solvable in polynomial time (improving on the previous exponential-time algorithm). Algorithms are presented for deciding whether a word-hyperbolic semigroup is a monoid, a group, a completely simple semigroup, a Clifford semigroup, or a free semigroup.PostprintPeer reviewe
Decision problems for word-hyperbolic semigroups
This paper studies decision problems for semigroups that are word-hyperbolic in the sense of Duncan & Gilman. A fundamental investigation reveals that the natural definition of a `word-hyperbolic structure' has to be strengthened slightly in order to define a unique semigroup up to isomorphism. The isomorphism problem is proven to be undecidable for word-hyperbolic semigroups (in contrast to the situation for word-hyperbolic groups). It is proved that it is undecidable whether a word-hyperbolic semigroup is automatic, asynchronously automatic, biautomatic, or asynchronously biautomatic. (These properties do not hold in general for word-hyperbolic semigroups.) It is proved that the uniform word problem for word-hyperbolic semigroup is solvable in polynomial time (improving on the previous exponential-time algorithm). Algorithms are presented for deciding whether a word-hyperbolic semigroup is a monoid, a group, a completely simple semigroup, a Clifford semigroup, or a free semigroup.PostprintPeer reviewe
Evaluating Matrix Circuits
The circuit evaluation problem (also known as the compressed word problem)
for finitely generated linear groups is studied. The best upper bound for this
problem is , which is shown by a reduction to polynomial
identity testing. Conversely, the compressed word problem for the linear group
is equivalent to polynomial identity testing. In
the paper, it is shown that the compressed word problem for every finitely
generated nilpotent group is in . Within
the larger class of polycyclic groups we find examples where the compressed
word problem is at least as hard as polynomial identity testing for skew
arithmetic circuits
AND and/or OR: Uniform Polynomial-Size Circuits
We investigate the complexity of uniform OR circuits and AND circuits of
polynomial-size and depth. As their name suggests, OR circuits have OR gates as
their computation gates, as well as the usual input, output and constant (0/1)
gates. As is the norm for Boolean circuits, our circuits have multiple sink
gates, which implies that an OR circuit computes an OR function on some subset
of its input variables. Determining that subset amounts to solving a number of
reachability questions on a polynomial-size directed graph (which input gates
are connected to the output gate?), taken from a very sparse set of graphs.
However, it is not obvious whether or not this (restricted) reachability
problem can be solved, by say, uniform AC^0 circuits (constant depth,
polynomial-size, AND, OR, NOT gates). This is one reason why characterizing the
power of these simple-looking circuits in terms of uniform classes turns out to
be intriguing. Another is that the model itself seems particularly natural and
worthy of study.
Our goal is the systematic characterization of uniform polynomial-size OR
circuits, and AND circuits, in terms of known uniform machine-based complexity
classes. In particular, we consider the languages reducible to such uniform
families of OR circuits, and AND circuits, under a variety of reduction types.
We give upper and lower bounds on the computational power of these language
classes. We find that these complexity classes are closely related to tallyNL,
the set of unary languages within NL, and to sets reducible to tallyNL.
Specifically, for a variety of types of reductions (many-one, conjunctive truth
table, disjunctive truth table, truth table, Turing) we give characterizations
of languages reducible to OR circuit classes in terms of languages reducible to
tallyNL classes. Then, some of these OR classes are shown to coincide, and some
are proven to be distinct. We give analogous results for AND circuits. Finally,
for many of our OR circuit classes, and analogous AND circuit classes, we prove
whether or not the two classes coincide, although we leave one such inclusion
open.Comment: In Proceedings MCU 2013, arXiv:1309.104
Knapsack problems in products of groups
The classic knapsack and related problems have natural generalizations to
arbitrary (non-commutative) groups, collectively called knapsack-type problems
in groups. We study the effect of free and direct products on their time
complexity. We show that free products in certain sense preserve time
complexity of knapsack-type problems, while direct products may amplify it. Our
methods allow to obtain complexity results for rational subset membership
problem in amalgamated free products over finite subgroups.Comment: 15 pages, 5 figures. Updated to include more general results, mostly
in Section
On the difficulty of presenting finitely presentable groups
We exhibit classes of groups in which the word problem is uniformly solvable
but in which there is no algorithm that can compute finite presentations for
finitely presentable subgroups. Direct products of hyperbolic groups, groups of
integer matrices, and right-angled Coxeter groups form such classes. We discuss
related classes of groups in which there does exist an algorithm to compute
finite presentations for finitely presentable subgroups. We also construct a
finitely presented group that has a polynomial Dehn function but in which there
is no algorithm to compute the first Betti number of the finitely presentable
subgroups.Comment: Final version. To appear in GGD volume dedicated to Fritz Grunewal
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