80 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
Bulking II: Classifications of Cellular Automata
This paper is the second part of a series of two papers dealing with bulking:
a way to define quasi-order on cellular automata by comparing space-time
diagrams up to rescaling. In the present paper, we introduce three notions of
simulation between cellular automata and study the quasi-order structures
induced by these simulation relations on the whole set of cellular automata.
Various aspects of these quasi-orders are considered (induced equivalence
relations, maximum elements, induced orders, etc) providing several formal
tools allowing to classify cellular automata
Freezing, Bounded-Change and Convergent Cellular Automata *
This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension , and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-Kurka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension 1, but also dimension 1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension
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
Freezing, Bounded-Change and Convergent Cellular Automata *
This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension , and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-Kurka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension 1, but also dimension 1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension
Directional Dynamics along Arbitrary Curves in Cellular Automata
This paper studies directional dynamics in cellular automata, a formalism
previously introduced by the third author. The central idea is to study the
dynamical behaviour of a cellular automaton through the conjoint action of its
global rule (temporal action) and the shift map (spacial action): qualitative
behaviours inherited from topological dynamics (equicontinuity, sensitivity,
expansivity) are thus considered along arbitrary curves in space-time. The main
contributions of the paper concern equicontinuous dynamics which can be
connected to the notion of consequences of a word. We show that there is a
cellular automaton with an equicontinuous dynamics along a parabola, but which
is sensitive along any linear direction. We also show that real numbers that
occur as the slope of a limit linear direction with equicontinuous dynamics in
some cellular automaton are exactly the computably enumerable numbers
Selfsimilarity, Simulation and Spacetime Symmetries
We study intrinsic simulations between cellular automata and introduce a new
necessary condition for a CA to simulate another one. Although expressed for
general CA, this condition is targeted towards surjective CA and especially
linear ones. Following the approach introduced by the first author in an
earlier paper, we develop proof techniques to tell whether some linear CA can
simulate another linear CA. Besides rigorous proofs, the necessary condition
for the simulation to occur can be heuristically checked via simple
observations of typical space-time diagrams generated from finite
configurations. As an illustration, we give an example of linear reversible CA
which cannot simulate the identity and which is 'time-asymmetric', i.e. which
can neither simulate its own inverse, nor the mirror of its own inverse
Topological Dynamics of Cellular Automata: Dimension Matters
Topological dynamics of cellular automata (CA), inherited from classical
dynamical systems theory, has been essentially studied in dimension 1. This
paper focuses on higher dimensional CA and aims at showing that the situation
is different and more complex starting from dimension 2. The main results are
the existence of non sensitive CA without equicontinuous points, the
non-recursivity of sensitivity constants, the existence of CA having only
non-recursive equicontinuous points and the existence of CA having only
countably many equicontinuous points. They all show a difference between
dimension 1 and higher dimensions. Thanks to these new constructions, we also
extend undecidability results concerning topological classification previously
obtained in the 1D case. Finally, we show that the set of sensitive CA is only
Pi_2 in dimension 1, but becomes Sigma_3-hard for dimension 3.Comment: to appear in Theory of Computing Systems (2009
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