152 research outputs found
The submonoid and rational subset membership problems for graph groups
We show that the membership problem in a finitely generated submonoid of a
graph group (also called a right-angled Artin group or a free partially
commutative group) is decidable if and only if the independence graph
(commutation graph) is a transitive forest. As a consequence we obtain the
first example of a finitely presented group with a decidable generalized word
problem that does not have a decidable membership problem for finitely
generated submonoids. We also show that the rational subset membership problem
is decidable for a graph group if and only if the independence graph is a
transitive forest, answering a question of Kambites, Silva, and the second
author. Finally we prove that for certain amalgamated free products and
HNN-extensions the rational subset and submonoid membership problems are
recursively equivalent. In particular, this applies to finitely generated
groups with two or more ends that are either torsion-free or residually finite
Submonoids and rational subsets of groups with infinitely many ends
In this paper we show that the membership problems for finitely generated
submonoids and for rational subsets are recursively equivalent for groups with
two or more ends
Branching-time model checking of one-counter processes
One-counter processes (OCPs) are pushdown processes which operate only on a
unary stack alphabet. We study the computational complexity of model checking
computation tree logic (CTL) over OCPs. A PSPACE upper bound is inherited from
the modal mu-calculus for this problem. First, we analyze the periodic
behaviour of CTL over OCPs and derive a model checking algorithm whose running
time is exponential only in the number of control locations and a syntactic
notion of the formula that we call leftward until depth. Thus, model checking
fixed OCPs against CTL formulas with a fixed leftward until depth is in P. This
generalizes a result of the first author, Mayr, and To for the expression
complexity of CTL's fragment EF. Second, we prove that already over some fixed
OCP, CTL model checking is PSPACE-hard. Third, we show that there already
exists a fixed CTL formula for which model checking of OCPs is PSPACE-hard. To
obtain the latter result, we employ two results from complexity theory: (i)
Converting a natural number in Chinese remainder presentation into binary
presentation is in logspace-uniform NC^1 and (ii) PSPACE is AC^0-serializable.
We demonstrate that our approach can be used to obtain further results. We show
that model-checking CTL's fragment EF over OCPs is hard for P^NP, thus
establishing a matching lower bound and answering an open question of the first
author, Mayr, and To. We moreover show that the following problem is hard for
PSPACE: Given a one-counter Markov decision process, a set of target states
with counter value zero each, and an initial state, to decide whether the
probability that the initial state will eventually reach one of the target
states is arbitrarily close to 1. This improves a previously known lower bound
for every level of the Boolean hierarchy by Brazdil et al
The First-Order Theory of Ground Tree Rewrite Graphs
We prove that the complexity of the uniform first-order theory
of ground tree rewrite graphs is in ATIME(2^{2^{poly(n)}},O(n). Providing a matching lower bound, we show that there is some
fixed ground tree rewrite graph whose first-order theory is hard
for ATIME(2^{2^{poly(n)}},poly(n)) with respect to logspace reductions. Finally, we prove that there exists a fixed ground tree rewrite graph together with a single unary predicate in form of a regular tree language such that the resulting structure has a non-elementary first-order theory
Subgroup Membership in GL(2,Z)
It is shown that the subgroup membership problem for a virtually free group can be decided in polynomial time where all group elements are represented by so-called power words, i.e., words of the form p_1^{z_1} p_2^{z_2} ? p_k^{z_k}. Here the p_i are explicit words over the generating set of the group and all z_i are binary encoded integers. As a corollary, it follows that the subgroup membership problem for the matrix group GL(2,?) can be decided in polynomial time when all matrix entries are given in binary notation
Branching-time Model Checking of One-counter Processes
One-counter processes (OCPs) are pushdown processes which operate only on a unary stack alphabet. We study the computational complexity of model checking computation tree logic () over OCPs. A upper bound is inherited from the modal -calculus for this problem. First, we analyze the periodic behaviour of over OCPs and derive a model checking algorithm whose running time is exponential only in the number of control locations and a syntactic notion of the formula that we call leftward until depth. Thus, model checking fixed OCPs against formulas with a fixed leftward until depth is in . This generalizes a result of the first author, Mayr, and To for the expression complexity of \u27s fragment . Second, we prove that already over some fixed OCP, model checking is -hard. Third, we show that there already exists a fixed formula for which model checking of OCPs is -hard. For the latter, we employ two results from complexity theory: (i) Converting a natural number in Chinese remainder presentation into binary presentation is in logspace-uniform and (ii) is -serializable. We demonstrate that our approach can be used to answer further open questions
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