2,237 research outputs found
Predicting Non-linear Cellular Automata Quickly by Decomposing Them into Linear Ones
We show that a wide variety of non-linear cellular automata (CAs) can be
decomposed into a quasidirect product of linear ones. These CAs can be
predicted by parallel circuits of depth O(log^2 t) using gates with binary
inputs, or O(log t) depth if ``sum mod p'' gates with an unbounded number of
inputs are allowed. Thus these CAs can be predicted by (idealized) parallel
computers much faster than by explicit simulation, even though they are
non-linear.
This class includes any CA whose rule, when written as an algebra, is a
solvable group. We also show that CAs based on nilpotent groups can be
predicted in depth O(log t) or O(1) by circuits with binary or ``sum mod p''
gates respectively.
We use these techniques to give an efficient algorithm for a CA rule which,
like elementary CA rule 18, has diffusing defects that annihilate in pairs.
This can be used to predict the motion of defects in rule 18 in O(log^2 t)
parallel time
A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
Defect Particle Kinematics in One-Dimensional Cellular Automata
Let A^Z be the Cantor space of bi-infinite sequences in a finite alphabet A,
and let sigma be the shift map on A^Z. A `cellular automaton' is a continuous,
sigma-commuting self-map Phi of A^Z, and a `Phi-invariant subshift' is a
closed, (Phi,sigma)-invariant subset X of A^Z. Suppose x is a sequence in A^Z
which is X-admissible everywhere except for some small region we call a
`defect'. It has been empirically observed that such defects persist under
iteration of Phi, and often propagate like `particles'. We characterize the
motion of these particles, and show that it falls into several regimes, ranging
from simple deterministic motion, to generalized random walks, to complex
motion emulating Turing machines or pushdown automata. One consequence is that
some questions about defect behaviour are formally undecidable.Comment: 37 pages, 9 figures, 3 table
Entanglement Generation of Clifford Quantum Cellular Automata
Clifford quantum cellular automata (CQCAs) are a special kind of quantum
cellular automata (QCAs) that incorporate Clifford group operations for the
time evolution. Despite being classically simulable, they can be used as basic
building blocks for universal quantum computation. This is due to the
connection to translation-invariant stabilizer states and their entanglement
properties. We will give a self-contained introduction to CQCAs and investigate
the generation of entanglement under CQCA action. Furthermore, we will discuss
finite configurations and applications of CQCAs.Comment: to appear in the "DPG spring meeting 2009" special issue of Applied
Physics
- …