618 research outputs found
Testing Generalised Freeness of Words
Pseudo-repetitions are a natural generalisation of the classical notion of repetitions in sequences: they are the repeated concatenation of a word and its encoding under a certain morphism or antimorphism (anti-/morphism, for short). We approach the problem of deciding efficiently, for a word w and a literal anti-/morphism f, whether w contains an instance of a given pattern involving a variable x and its image under f, i.e., f(x). Our results generalise both the problem of finding fixed repetitive structures (e.g., squares, cubes) inside a word and the problem of finding palindromic structures inside a word. For instance, we can detect efficiently a factor of the form xx^Rxxx^R, or any other pattern of such type. We also address the problem of testing efficiently, in the same setting, whether the word w contains an arbitrary pseudo-repetition of a given exponent
Spectral densities of Wishart-Levy free stable random matrices: Analytical results and Monte Carlo validation
Random matrix theory is used to assess the significance of weak correlations
and is well established for Gaussian statistics. However, many complex systems,
with stock markets as a prominent example, exhibit statistics with power-law
tails, that can be modelled with Levy stable distributions. We review
comprehensively the derivation of an analytical expression for the spectra of
covariance matrices approximated by free Levy stable random variables and
validate it by Monte Carlo simulation.Comment: 10 pages, 1 figure, submitted to Eur. Phys. J.
On Testability of First-Order Properties in Bounded-Degree Graphs and Connections to Proximity-Oblivious Testing
We study property testing of properties that are definable in first-order
logic (FO) in the bounded-degree graph and relational structure models. We show
that any FO property that is defined by a formula with quantifier prefix
is testable (i.e., testable with constant query
complexity), while there exists an FO property that is expressible by a formula
with quantifier prefix that is not testable. In the dense
graph model, a similar picture is long known (Alon, Fischer, Krivelevich,
Szegedy, Combinatorica 2000), despite the very different nature of the two
models. In particular, we obtain our lower bound by an FO formula that defines
a class of bounded-degree expanders, based on zig-zag products of graphs. We
expect this to be of independent interest.
We then use our class of FO definable bounded-degree expanders to answer a
long-standing open problem for proximity-oblivious testers (POTs). POTs are a
class of particularly simple testing algorithms, where a basic test is
performed a number of times that may depend on the proximity parameter, but the
basic test itself is independent of the proximity parameter. In their seminal
work, Goldreich and Ron [STOC 2009; SICOMP 2011] show that the graph properties
that are constant-query proximity-oblivious testable in the bounded-degree
model are precisely the properties that can be expressed as a generalised
subgraph freeness (GSF) property that satisfies the non-propagation condition.
It is left open whether the non-propagation condition is necessary. We give a
negative answer by showing that our property is a GSF property which is
propagating. Hence in particular, our property does not admit a POT. For this
result we establish a new connection between FO properties and GSF-local
properties via neighbourhood profiles.Comment: Preliminary version of this article appeared in SODA'21
(arXiv:2008.05800) and CCC'21 (arXiv:2105.08490
Unifying type systems for mobile processes
We present a unifying framework for type systems for process calculi. The
core of the system provides an accurate correspondence between essentially
functional processes and linear logic proofs; fragments of this system
correspond to previously known connections between proofs and processes. We
show how the addition of extra logical axioms can widen the class of typeable
processes in exchange for the loss of some computational properties like
lock-freeness or termination, allowing us to see various well studied systems
(like i/o types, linearity, control) as instances of a general pattern. This
suggests unified methods for extending existing type systems with new features
while staying in a well structured environment and constitutes a step towards
the study of denotational semantics of processes using proof-theoretical
methods
Reachability problems for systems with linear dynamics
This thesis deals with reachability and freeness problems for systems with linear dynamics, including hybrid systems and matrix semigroups. Hybrid systems are a type of dynamical system that exhibit both continuous and discrete dynamic behaviour. Thus they are particularly useful in modelling practical real world systems which can both flow (continuous behaviour) and jump (discrete behaviour). Decision questions for matrix semigroups have attracted a great deal of attention in both the Mathematics and Theoretical Computer Science communities. They can also be used to model applications with only discrete components.
For a computational model, the reachability problem asks whether we can reach a target point starting from an initial point, which is a natural question both in theoretical study and for real-world applications. By studying this problem and its variations, we shall prove in a formal mathematical sense that many problems are intractable or even unsolvable. Thus we know when such a problem appears in other areas like Biology, Physics or Chemistry, either the problem itself needs to be simplified, or it should by studied by approximation.
In this thesis we concentrate on a specific hybrid system model, called an HPCD, and its variations. The objective of studying this model is twofold: to obtain the most expressive system for which reachability is algorithmically solvable and to explore the simplest system for which it is impossible to solve. For the solvable sub-cases, we shall also study whether reachability is in some sense easy or hard by determining which complexity classes the problem belongs to, such as P, NP(-hard) and PSPACE(-hard). Some undecidable results for matrix semigroups are also shown, which both strengthen our knowledge of the structure of matrix semigroups, and lead to some undecidability results for other models
The AND-Prolog compiler system — Automatic parallelization tools for LP
This report presents an overview of the current work performed by us in the context of the efficient parallel implementation of traditional logic programming systems. The
work is based on the &-Prolog System, a system for the automatic parallelization and execution of logic programming languages within the Independent And-parallelism
model, and the global analysis and parallelization tools which have been developed for this system. In order to make the report self-contained, we first describe the "classical" tools of the &-Prolog system. We then explain in detail the work performed in improving and generalizing the global analysis and parallelization tools. Also, we describe the objectives which will drive our future work in this area
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