184 research outputs found
Subshifts as Models for MSO Logic
We study the Monadic Second Order (MSO) Hierarchy over colourings of the
discrete plane, and draw links between classes of formula and classes of
subshifts. We give a characterization of existential MSO in terms of
projections of tilings, and of universal sentences in terms of combinations of
"pattern counting" subshifts. Conversely, we characterise logic fragments
corresponding to various classes of subshifts (subshifts of finite type, sofic
subshifts, all subshifts). Finally, we show by a separation result how the
situation here is different from the case of tiling pictures studied earlier by
Giammarresi et al.Comment: arXiv admin note: substantial text overlap with arXiv:0904.245
Subshifts, MSO Logic, and Collapsing Hierarchies
We use monadic second-order logic to define two-dimensional subshifts, or
sets of colorings of the infinite plane. We present a natural family of
quantifier alternation hierarchies, and show that they all collapse to the
third level. In particular, this solves an open problem of [Jeandel & Theyssier
2013]. The results are in stark contrast with picture languages, where such
hierarchies are usually infinite.Comment: 12 pages, 5 figures. To appear in conference proceedings of TCS 2014,
published by Springe
Decidability and Universality in Symbolic Dynamical Systems
Many different definitions of computational universality for various types of
dynamical systems have flourished since Turing's work. We propose a general
definition of universality that applies to arbitrary discrete time symbolic
dynamical systems. Universality of a system is defined as undecidability of a
model-checking problem. For Turing machines, counter machines and tag systems,
our definition coincides with the classical one. It yields, however, a new
definition for cellular automata and subshifts. Our definition is robust with
respect to initial condition, which is a desirable feature for physical
realizability.
We derive necessary conditions for undecidability and universality. For
instance, a universal system must have a sensitive point and a proper
subsystem. We conjecture that universal systems have infinite number of
subsystems. We also discuss the thesis according to which computation should
occur at the `edge of chaos' and we exhibit a universal chaotic system.Comment: 23 pages; a shorter version is submitted to conference MCU 2004 v2:
minor orthographic changes v3: section 5.2 (collatz functions) mathematically
improved v4: orthographic corrections, one reference added v5:27 pages.
Important modifications. The formalism is strengthened: temporal logic
replaced by finite automata. New results. Submitte
Enumeration Reducibility in Closure Spaces with Applications to Logic and Algebra
In many instances in first order logic or computable algebra, classical
theorems show that many problems are undecidable for general structures, but
become decidable if some rigidity is imposed on the structure. For example, the
set of theorems in many finitely axiomatisable theories is nonrecursive, but
the set of theorems for any finitely axiomatisable complete theory is
recursive. Finitely presented groups might have an nonrecursive word problem,
but finitely presented simple groups have a recursive word problem. In this
article we introduce a topological framework based on closure spaces to show
that many of these proofs can be obtained in a similar setting. We will show in
particular that these statements can be generalized to cover arbitrary
structures, with no finite or recursive presentation/axiomatization. This
generalizes in particular work by Kuznetsov and others. Examples from first
order logic and symbolic dynamics will be discussed at length
The large scale geometry of strongly aperiodic subshifts of finite type
A subshift on a group G is a closed, G-invariant subset of A^G, for some
finite set A. It is said to be a subshift of finite type (SFT) if it is defined
by a finite collection of 'forbidden patterns', to be strongly aperiodic if all
point stabilizers are trivial, and weakly aperiodic if all point stabilizers
are infinite index in G. We show that groups with at least 2 ends have a
strongly aperiodic SFT, and that having such an SFT is a QI invariant for
finitely presented torsion free groups. We show that a finitely presented
torsion free group with no weakly aperiodic SFT must be QI-rigid. The domino
problem on G asks whether the SFT specified by a given set of forbidden
patterns is empty. We show that decidability of the domino problem is a QI
invariant.Comment: 23 pages, 6 figures. The proof of the main theorem has been
simplified and some new corollaries deduce
Subshifts as Models for MSO Logic
We study the Monadic Second Order (MSO) Hierarchy over colourings of the discrete plane, and draw links between classes of formula and classes of subshifts. We give a characterization of existential MSO in terms of projections of tilings, and of universal sentences in terms of combinations of ''pattern counting'' subshifts. Conversely, we characterise logic fragments corresponding to various classes of subshifts (subshifts of finite type, sofic subshifts, all subshifts). Finally, we show by a separation result how the situation here is different from the case of tiling pictures studied earlier by Giammarresi et al
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