1,192 research outputs found
Conjugacy of one-dimensional one-sided cellular automata is undecidable
Two cellular automata are strongly conjugate if there exists a
shift-commuting conjugacy between them. We prove that the following two sets of
pairs of one-dimensional one-sided cellular automata over a full shift
are recursively inseparable: (i) pairs where has strictly larger
topological entropy than , and (ii) pairs that are strongly conjugate and
have zero topological entropy.
Because there is no factor map from a lower entropy system to a higher
entropy one, and there is no embedding of a higher entropy system into a lower
entropy system, we also get as corollaries that the following decision problems
are undecidable: Given two one-dimensional one-sided cellular automata and
over a full shift: Are and conjugate? Is a factor of ? Is
a subsystem of ? All of these are undecidable in both strong and weak
variants (whether the homomorphism is required to commute with the shift or
not, respectively). It also immediately follows that these results hold for
one-dimensional two-sided cellular automata.Comment: 12 pages, 2 figures, accepted for SOFSEM 201
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
Physics Without Physics: The Power of Information-theoretical Principles
David Finkelstein was very fond of the new information-theoretic paradigm of
physics advocated by John Archibald Wheeler and Richard Feynman. Only recently,
however, the paradigm has concretely shown its full power, with the derivation
of quantum theory (Chiribella et al., Phys. Rev. A 84:012311, 2011; D'Ariano et
al., 2017) and of free quantum field theory (D'Ariano and Perinotti, Phys. Rev.
A 90:062106, 2014; Bisio et al., Phys. Rev. A 88:032301, 2013; Bisio et al.,
Ann. Phys. 354:244, 2015; Bisio et al., Ann. Phys. 368:177, 2016) from
informational principles. The paradigm has opened for the first time the
possibility of avoiding physical primitives in the axioms of the physical
theory, allowing a refoundation of the whole physics over logically solid
grounds. In addition to such methodological value, the new
information-theoretic derivation of quantum field theory is particularly
interesting for establishing a theoretical framework for quantum gravity, with
the idea of obtaining gravity itself as emergent from the quantum information
processing, as also suggested by the role played by information in the
holographic principle (Susskind, J. Math. Phys. 36:6377, 1995; Bousso, Rev.
Mod. Phys. 74:825, 2002). In this paper I review how free quantum field theory
is derived without using mechanical primitives, including space-time, special
relativity, Hamiltonians, and quantization rules. The theory is simply provided
by the simplest quantum algorithm encompassing a countable set of quantum
systems whose network of interactions satisfies the three following simple
principles: homogeneity, locality, and isotropy. The inherent discrete nature
of the informational derivation leads to an extension of quantum field theory
in terms of a quantum cellular automata and quantum walks. A simple heuristic
argument sets the scale to the Planck one, and the observed regime is that of
small wavevectors ...Comment: 34 pages, 8 figures. Paper for in memoriam of David Finkelstei
Undecidable Properties of Limit Set Dynamics of Cellular Automata
Cellular Automata (CA) are discrete dynamical systems and an abstract model
of parallel computation. The limit set of a cellular automaton is its maximal
topological attractor. A well know result, due to Kari, says that all
nontrivial properties of limit sets are undecidable. In this paper we consider
properties of limit set dynamics, i.e. properties of the dynamics of Cellular
Automata restricted to their limit sets. There can be no equivalent of Kari's
Theorem for limit set dynamics. Anyway we show that there is a large class of
undecidable properties of limit set dynamics, namely all properties of limit
set dynamics which imply stability or the existence of a unique subshift
attractor. As a consequence we have that it is undecidable whether the cellular
automaton map restricted to the limit set is the identity, closing, injective,
expansive, positively expansive, transitive
The algebraic entropy of one-dimensional finitary linear cellular automata
The aim of this paper is to present one-dimensional finitary linear cellular
automata on from an algebraic point of view. Among various
other results, we:
(i) show that the Pontryagin dual of is a classical
one-dimensional linear cellular automaton on ;
(ii) give several equivalent conditions for to be invertible with inverse
a finitary linear cellular automaton;
(iii) compute the algebraic entropy of , which coincides with the
topological entropy of by the so-called Bridge Theorem.
In order to better understand and describe the entropy we introduce the
degree and of and .Comment: 21 page
Linear And Nonlinear Arabesques: A Study Of Closed Chains Of Negative 2-Element Circuits
In this paper we consider a family of dynamical systems that we call
"arabesques", defined as closed chains of 2-element negative circuits. An
-dimensional arabesque system has 2-element circuits, but in addition,
it displays by construction, two -element circuits which are both positive
vs one positive and one negative, depending on the parity (even or odd) of the
dimension . In view of the absence of diagonal terms in their Jacobian
matrices, all these dynamical systems are conservative and consequently, they
can not possess any attractor. First, we analyze a linear variant of them which
we call "arabesque 0" or for short "A0". For increasing dimensions, the
trajectories are increasingly complex open tori. Next, we inserted a single
cubic nonlinearity that does not affect the signs of its circuits (that we call
"arabesque 1" or for short "A1"). These systems have three steady states,
whatever the dimension is, in agreement with the order of the nonlinearity. All
three are unstable, as there can not be any attractor in their state-space. The
3D variant (that we call for short "A1\_3D") has been analyzed in some detail
and found to display a complex mixed set of quasi-periodic and chaotic
trajectories. Inserting cubic nonlinearities (one per equation) in the same
way as above, we generate systems "A2\_D". A2\_3D behaves essentially as
A1\_3D, in agreement with the fact that the signs of the circuits remain
identical. A2\_4D, as well as other arabesque systems with even dimension, has
two positive -circuits and nine steady states. Finally, we investigate and
compare the complex dynamics of this family of systems in terms of their
symmetries.Comment: 22 pages, 12 figures, accepted for publication at Int. J. Bif. Chao
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