10 research outputs found
A Counterexample to Thiagarajan\u27s Conjecture on Regular Event Structures
We provide a counterexample to a conjecture by Thiagarajan (1996 and 2002) that regular prime event structures correspond exactly to those obtained as unfoldings of finite 1-safe Petri nets. The same counterexample is used to disprove a closely related conjecture by Badouel, Darondeau, and Raoult (1999) that domains of regular event structures with bounded natural-cliques are recognizable by finite trace automata. Event structures, trace automata, and Petri nets are fundamental models in concurrency theory. There exist nice interpretations of these structures as combinatorial and geometric objects and both conjectures can be reformulated in this framework. Namely, the domains of prime event structures correspond exactly to pointed median graphs; from a geometric point of view, these domains are in bijection with pointed CAT(0) cube complexes.
A necessary condition for both conjectures to be true is that domains of respective regular event structures admit a regular nice labeling. To disprove these conjectures, we describe a regular event domain (with bounded natural-cliques) that does not admit a regular nice labeling. Our counterexample is derived from an example by Wise (1996 and 2007) of a nonpositively curved square complex whose universal cover is a CAT(0) square complex containing a particular plane with an aperiodic tiling
Automaton Semigroups and Groups: On the Undecidability of Problems Related to Freeness and Finiteness
In this paper, we study algorithmic problems for automaton semigroups and
automaton groups related to freeness and finiteness. In the course of this
study, we also exhibit some connections between the algebraic structure of
automaton (semi)groups and their dynamics on the boundary. First, we show that
it is undecidable to check whether the group generated by a given invertible
automaton has a positive relation, i.e. a relation p = 1 such that p only
contains positive generators. Besides its obvious relation to the freeness of
the group, the absence of positive relations has previously been studied and is
connected to the triviality of some stabilizers of the boundary. We show that
the emptiness of the set of positive relations is equivalent to the dynamical
property that all (directed positive) orbital graphs centered at non-singular
points are acyclic.
Gillibert showed that the finiteness problem for automaton semigroups is
undecidable. In the second part of the paper, we show that this undecidability
result also holds if the input is restricted to be bi-reversible and invertible
(but, in general, not complete). As an immediate consequence, we obtain that
the finiteness problem for automaton subsemigroups of semigroups generated by
invertible, yet partial automata, so called automaton-inverse semigroups, is
also undecidable.
Erratum: Contrary to a statement in a previous version of the paper, our
approach does not show that that the freeness problem for automaton semigroups
is undecidable. We discuss this in an erratum at the end of the paper
A counterexample to Thiagarajan's conjecture on regular event structures
We provide a counterexample to a conjecture by Thiagarajan (1996 and 2002)
that regular event structures correspond exactly to event structures obtained
as unfoldings of finite 1-safe Petri nets. The same counterexample is used to
disprove a closely related conjecture by Badouel, Darondeau, and Raoult (1999)
that domains of regular event structures with bounded -cliques are
recognizable by finite trace automata. Event structures, trace automata, and
Petri nets are fundamental models in concurrency theory. There exist nice
interpretations of these structures as combinatorial and geometric objects.
Namely, from a graph theoretical point of view, the domains of prime event
structures correspond exactly to median graphs; from a geometric point of view,
these domains are in bijection with CAT(0) cube complexes.
A necessary condition for both conjectures to be true is that domains of
regular event structures (with bounded -cliques) admit a regular nice
labeling. To disprove these conjectures, we describe a regular event domain
(with bounded -cliques) that does not admit a regular nice labeling.
Our counterexample is derived from an example by Wise (1996 and 2007) of a
nonpositively curved square complex whose universal cover is a CAT(0) square
complex containing a particular plane with an aperiodic tiling. We prove that
other counterexamples to Thiagarajan's conjecture arise from aperiodic 4-way
deterministic tile sets of Kari and Papasoglu (1999) and Lukkarila (2009).
On the positive side, using breakthrough results by Agol (2013) and Haglund
and Wise (2008, 2012) from geometric group theory, we prove that Thiagarajan's
conjecture is true for regular event structures whose domains occur as
principal filters of hyperbolic CAT(0) cube complexes which are universal
covers of finite nonpositively curved cube complexes
ON THE FOUNDATIONS OF COMPUTABILITY THEORY
The principal motivation for this work is the observation that there are significant deficiencies in the foundations of conventional computability theory. This thesis examines the problems with conventional computability theory, including its failure to address discrepancies between theory and practice in computer science, semantic confusion in terminology, and limitations in the scope of conventional computing models. In light of these difficulties, fundamental notions are re-examined and revised definitions of key concepts such as “computer,” “computable,” and “computing power” are provided. A detailed analysis is conducted to determine desirable semantics and scope of applicability of foundational notions. The credibility of the revised definitions is ascertained by demonstrating by their ability to address identified problems with conventional definitions. Their practical utility is established through application to examples. Other related issues, including hidden complexity in computations, subtleties related to encodings, and the cardinalities of sets involved in computing, are examined. A resource-based meta-model for characterizing computing model properties is introduced. The proposed definitions are presented as a starting point for an alternate foundation for computability theory. However, formulation of the particular concepts under discussion is not the sole purpose of the thesis. The underlying objective of this research is to open discourse on alternate foundations of computability theory and to inspire re-examination of fundamental notions
Self-stabilisation of cellular automata on tilings
Given a finite set of local constraints, we seek a cellular automaton (i.e.,
a local and uniform algorithm) that self-stabilises on the configurations that
satisfy these constraints. More precisely, starting from a finite perturbation
of a valid configuration, the cellular automaton must eventually fall back to
the space of valid configurations where it remains still. We allow the cellular
automaton to use extra symbols, but in that case, the extra symbols can also
appear in the initial finite perturbation. For several classes of local
constraints (e.g., -colourings with , and North-East deterministic
constraints), we provide efficient self-stabilising cellular automata with or
without additional symbols that wash out finite perturbations in linear or
quadratic time, but also show that there are examples of local constraints for
which the self-stabilisation problem is inherently hard. We note that the
optimal self-stabilisation speed is the same for all local constraints that are
isomorphic to one another. We also consider probabilistic cellular automata
rules and show that in some cases, the use of randomness simplifies the
problem. In the deterministic case, we show that if finite perturbations are
corrected in linear time, then the cellular automaton self-stabilises even
starting from a random perturbation of a valid configuration, that is, when
errors in the initial configuration occur independently with a sufficiently low
density.Comment: 43 pages, 28 figure
On Undecidable Dynamical Properties of Reversible One-Dimensional Cellular Automata
Cellular automata are models for massively parallel computation. A cellular automaton consists of cells which are arranged in some kind of regular lattice and a local update rule which updates the state of each cell according to the states of the cell's neighbors on each step of the computation.
This work focuses on reversible one-dimensional cellular automata in which the cells are arranged in a two-way in_nite line and the computation is reversible, that is, the previous states of the cells can be derived from the current ones. In this work it is shown that several properties of reversible one-dimensional cellular automata are algorithmically undecidable, that is, there exists no algorithm that would tell whether a given cellular automaton has the property or not.
It is shown that the tiling problem of Wang tiles remains undecidable even in some very restricted special cases. It follows that it is undecidable whether some given states will always appear in computations by the given cellular automaton. It also follows that a weaker form of expansivity, which is a concept of dynamical systems, is an undecidable property for reversible one-dimensional cellular automata.
It is shown that several properties of dynamical systems are undecidable for reversible one-dimensional cellular automata. It shown that sensitivity to initial conditions and topological mixing are undecidable properties. Furthermore, non-sensitive and mixing cellular automata are recursively inseparable. It follows that also chaotic behavior is an undecidable property for reversible one-dimensional cellular automata.Siirretty Doriast
Hierarchy and Expansiveness in Two-Dimensional Subshifts of Finite Type
Using a deterministic version of the self-similar (or hierarchical, or
fixed-point ) method for constructing 2-dimensional subshifts of finite type
(SFTs), we construct aperiodic 2D SFTs with a unique direction of
non-expansiveness and prove that the emptiness problem of SFTs is undecidable
even in this restricted case. As an additional application of our method, we
characterize the sets of directions that can be the set of non-expansive
directions of 2D SFTs.Comment: 72 pages, main body of the author's PhD Thesis, most of the results
obtained in collaboration with Pierre Guillo
Hierarchy and Expansiveness in Two-Dimensional Subshifts of Finite Type
Subshifts are sets of configurations over an infinite grid defined by a set of forbidden patterns. In this thesis, we study two-dimensional subshifts offinite type (2D SFTs), where the underlying grid is Z2 and the set of for-bidden patterns is finite. We are mainly interested in the interplay between the computational power of 2D SFTs and their geometry, examined through the concept of expansive subdynamics. 2D SFTs with expansive directions form an interesting and natural class of subshifts that lie between dimensions 1 and 2. An SFT that has only one non-expansive direction is called extremely expansive. We prove that in many aspects, extremely expansive 2D SFTs display the totality of behaviours of general 2D SFTs.
For example, we construct an aperiodic extremely expansive 2D SFT and we prove that the emptiness problem is undecidable even when restricted to the class of extremely expansive 2D SFTs. We also prove that every Medvedev class contains an extremely expansive 2D SFT and we provide a characterization of the sets of directions that can be the set of non-expansive directions of a 2D SFT. Finally, we prove that for every computable sequence of 2D SFTs with an expansive direction, there exists a universal object that simulates all of the elements of the sequence. We use the so called hierarchical, self-simulating or fixed-point method for constructing 2D SFTs which has been previously used by Ga´cs, Durand, Romashchenko and Shen.Siirretty Doriast