4,946 research outputs found
Cellular Automata and Powers of
We consider one-dimensional cellular automata which multiply
numbers by in base for relatively prime integers and . By
studying the structure of traces with respect to we show that for
(and then as a simple corollary for ) there are arbitrarily
small finite unions of intervals which contain the fractional parts of the
sequence , () for some . To the other
direction, by studying the measure theoretical properties of , we show
that for there are finite unions of intervals approximating the unit
interval arbitrarily well which don't contain the fractional parts of the whole
sequence for any .Comment: 15 pages, 8 figures. Accepted for publication in RAIRO-IT
Cellular automata and powers of p/q
We consider one-dimensional cellular automata Fp,q which multiply numbers by p∕q in base pq for relatively prime integers p and q. By studying the structure of traces with respect to Fp,q we show that for p ≥ 2q – 1 (and then as a simple corollary for p > q > 1) there are arbitrarily small finite unions of intervals which contain the fractional parts of the sequence ξ(p∕q)n, (n = 0, 1, 2, …) for some ξ > 0. To the other direction, by studying the measure theoretical properties of Fp,q, we show that for p > q > 1 there are finite unions of intervals approximating the unit interval arbitrarily well which don’t contain the fractional parts of the whole sequence ξ(p∕q)n for any ξ > 0.</p
Quasi-Linear Cellular Automata
Simulating a cellular automaton (CA) for t time-steps into the future
requires t^2 serial computation steps or t parallel ones. However, certain CAs
based on an Abelian group, such as addition mod 2, are termed ``linear''
because they obey a principle of superposition. This allows them to be
predicted efficiently, in serial time O(t) or O(log t) in parallel.
In this paper, we generalize this by looking at CAs with a variety of
algebraic structures, including quasigroups, non-Abelian groups, Steiner
systems, and others. We show that in many cases, an efficient algorithm exists
even though these CAs are not linear in the previous sense; we term them
``quasilinear.'' We find examples which can be predicted in serial time
proportional to t, t log t, t log^2 t, and t^a for a < 2, and parallel time log
t, log t log log t and log^2 t.
We also discuss what algebraic properties are required or implied by the
existence of scaling relations and principles of superposition, and exhibit
several novel ``vector-valued'' CAs.Comment: 41 pages with figures, To appear in Physica
Dirac and Weyl Equations on a Lattice as Quantum Cellular Automata
A discretized time evolution of the wave function for a Dirac particle on a
cubic lattice is represented by a very simple quantum cellular automaton. In
each evolution step the updated value of the wave function at a given site
depends only on the values at the nearest sites, the evolution is unitary and
preserves chiral symmetry. Moreover, it is shown that the relationship between
Dirac particles and cellular automata operating on two component objects on a
lattice is indeed very close. Every local and unitary automaton on a cubic
lattice, under some natural assumptions, leads in the continuum limit to the
Weyl equation. The sum over histories is evaluated and its connection with path
integrals and theories of fermions on a lattice is outlined.Comment: 6, RevTe
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