170 research outputs found
Logspace Computations in Coxeter Groups and Graph Groups
Computing normal forms in groups (or monoids) is in general harder than
solving the word problem (equality testing). However, normal form computation
has a much wider range of applications. It is therefore interesting to
investigate the complexity of computing normal forms for important classes of
groups.
For Coxeter groups we show that the following algorithmic tasks can be solved
by a deterministic Turing machine using logarithmic work space, only: 1.
Compute the length of any geodesic normal form. 2. Compute the set of letters
occurring in any geodesic normal form. 3. Compute the Parikh-image of any
geodesic normal form in case that all defining relations have even length
(i.e., in even Coxeter groups.) 4. For right-angled Coxeter groups we can do
actually compute the short length normal form in logspace. (Note that short
length normal forms are geodesic.)
Next, we apply the results to right-angled Artin groups. They are also known
as free partially commutative groups or as graph groups. As a consequence of
our result on right-angled Coxeter groups we show that shortlex normal forms in
graph groups can be computed in logspace, too. Graph groups play an important
role in group theory, and they have a close connection to concurrency theory.
As an application of our results we show that the word problem for free
partially commutative inverse monoids is in logspace. This result generalizes a
result of Ondrusch and the third author on free inverse monoids. Concurrent
systems which are deterministic and co-deterministic can be studied via inverse
monoids.Comment: An extended abstract of this paper appears in Proceedings of LATIN
201
Languages of Dot-depth One over Infinite Words
Over finite words, languages of dot-depth one are expressively complete for
alternation-free first-order logic. This fragment is also known as the Boolean
closure of existential first-order logic. Here, the atomic formulas comprise
order, successor, minimum, and maximum predicates. Knast (1983) has shown that
it is decidable whether a language has dot-depth one. We extend Knast's result
to infinite words. In particular, we describe the class of languages definable
in alternation-free first-order logic over infinite words, and we give an
effective characterization of this fragment. This characterization has two
components. The first component is identical to Knast's algebraic property for
finite words and the second component is a topological property, namely being a
Boolean combination of Cantor sets.
As an intermediate step we consider finite and infinite words simultaneously.
We then obtain the results for infinite words as well as for finite words as
special cases. In particular, we give a new proof of Knast's Theorem on
languages of dot-depth one over finite words.Comment: Presented at LICS 201
Construction of a cms on a given cpo
In dealing with denotational semantics of programming languages partial orders resp. metric spaces have been used with great benefit in order to provide a meaning to recursive and repetitive constructs. This paper presents two methods to define a metric on a subset M of a cpo D such that M is a complete metric spaces and the metric semantics on M coincides with the cpo semantics on D when the same semantic operators are used. The first method is to add a 'length' on a cpo which means a function ρ : D → IN 0 ∪{∞} of increasing power. The second is based on the ideas of [9] and uses pseudo rank orderings, i.e. monotone sequences of monotone functions ϖn : D → D. We show that SFP domains can be characterized as special kinds of rank orderded cpo's. We also discuss the connection between the Lawson topology and the topology induced by the metric
Fragments of first-order logic over infinite words
We give topological and algebraic characterizations as well as language
theoretic descriptions of the following subclasses of first-order logic FO[<]
for omega-languages: Sigma_2, FO^2, the intersection of FO^2 and Sigma_2, and
Delta_2 (and by duality Pi_2 and the intersection of FO^2 and Pi_2). These
descriptions extend the respective results for finite words. In particular, we
relate the above fragments to language classes of certain (unambiguous)
polynomials. An immediate consequence is the decidability of the membership
problem of these classes, but this was shown before by Wilke and Bojanczyk and
is therefore not our main focus. The paper is about the interplay of algebraic,
topological, and language theoretic properties.Comment: Conference version presented at 26th International Symposium on
Theoretical Aspects of Computer Science, STACS 200
Graph products of spheres, associative graded algebras and Hilbert series
Given a finite, simple, vertex-weighted graph, we construct a graded
associative (non-commutative) algebra, whose generators correspond to vertices
and whose ideal of relations has generators that are graded commutators
corresponding to edges. We show that the Hilbert series of this algebra is the
inverse of the clique polynomial of the graph. Using this result it easy to
recognize if the ideal is inert, from which strong results on the algebra
follow. Non-commutative Grobner bases play an important role in our proof.
There is an interesting application to toric topology. This algebra arises
naturally from a partial product of spheres, which is a special case of a
generalized moment-angle complex. We apply our result to the loop-space
homology of this space.Comment: 19 pages, v3: elaborated on connections to related work, added more
citations, to appear in Mathematische Zeitschrif
On Infinite Real Trace Rational Languages of Maximum Topological Complexity
We consider the set of infinite real traces, over a dependence alphabet
(Gamma, D) with no isolated letter, equipped with the topology induced by the
prefix metric. We then prove that all rational languages of infinite real
traces are analytic sets and that there exist some rational languages of
infinite real traces which are analytic but non Borel sets, and even
Sigma^1_1-complete, hence of maximum possible topological complexity
On distributed monitoring of asynchronous systems
Distributed systems are notoriously difficult to understand and analyze in
order to assert their correction w.r.t. given properties. They often exhibit a
huge number of different behaviors, as soon as the active entities (peers,
agents, processes, etc) behave in an asynchronous manner. Already the
modelization of such systems is a non-trivial task, let alone their formal
verification.
The purpose of this paper is to discuss the problem of distributed monitoring
on a simple model of finite-state distributed automata based on shared actions,
called asynchronous automata. Monitoring is a question related to runtime
verification: assume that we have to check a property against an unknown or
very complex system , so that classical static analysis is not possible.
Therefore instead of model-checking a monitor is used, that checks the property
on the underlying system at runtime.
We are interested here in monitoring distributed systems modeled as
asynchronous automata. It is natural to require that monitors should be of the
same kind as the underlying system, so we consider here distributed monitoring.
A distributed monitor does not have a global view of the system, therefore we
propose the notion of locally monitorable trace language. Our main result shows
that if the distributed alphabet of actions is connected and if is a set of
infinite traces such that both and its complement are countable
unions of locally safety languages, then is locally monitorable. We also
show that over infinite traces, recognizable countable unions of locally safety
languages are precisely the complements of deterministic languages.Comment: Paper appears as an invited lecture at WoLLIC 2012, 19th Workshop on
Logic, Language, Information and Computation. September 3rd to 6th, 2012
University of Buenos Aires, Buenos Aires, Argentin
Finite transducers for divisibility monoids
Divisibility monoids are a natural lattice-theoretical generalization of
Mazurkiewicz trace monoids, namely monoids in which the distributivity of the
involved divisibility lattices is kept as an hypothesis, but the relations
between the generators are not supposed to necessarily be commutations. Here,
we show that every divisibility monoid admits an explicit finite transducer
which allows to compute normal forms in quadratic time. In addition, we prove
that every divisibility monoid is biautomatic.Comment: 20 page
The Complexity of Dynamic Data Race Prediction
Writing concurrent programs is notoriously hard due to scheduling
non-determinism. The most common concurrency bugs are data races, which are
accesses to a shared resource that can be executed concurrently. Dynamic
data-race prediction is the most standard technique for detecting data races:
given an observed, data-race-free trace , the task is to determine whether
can be reordered to a trace that exposes a data-race. Although the
problem has received significant practical attention for over three decades,
its complexity has remained elusive. In this work, we address this lacuna,
identifying sources of intractability and conditions under which the problem is
efficiently solvable. Given a trace of size over threads, our main
results are as follows.
First, we establish a general upper-bound, as
well as an upper-bound when certain parameters of are constant. In
addition, we show that the problem is NP-hard and even W[1]-hard parameterized
by , and thus unlikely to be fixed-parameter tractable. Second, we study the
problem over acyclic communication topologies, such as server-clients
hierarchies. We establish an upper-bound,
where is the number of shared variables accessed in . In addition, we
show that even for traces with threads, the problem has no
algorithm under Orthogonal Vectors. Since any trace with 2
threads defines an acyclic topology, our upper-bound for this case is optimal
wrt polynomial improvements for up to moderate values of and . Finally,
we study a distance-bounded version of the problem, where the task is to expose
a data race by a witness trace that is similar to . We develop an algorithm
that works in time when certain parameters of are constant
Fixed points of endomorphisms of trace monoids
It is proved that the fixed point submonoid and the periodic point submonoid
of a trace monoid endomorphism are always finitely generated. Considering the
Foata normal form metric on trace monoids and uniformly continuous
endomorphisms, a finiteness theorem is proved for the infinite fixed points of
the continuous extension to real traces
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