ISSN 0032-9460, Problems of Information Transmission, 2012, Vol. 48, No. 2, pp. 154–172.
Pleiades Publishing, Inc., 2012.
Original Russian Text
M.A. Raskin, 2012, published in Problemy Peredachi Informatsii, 2012, Vol. 48, No. 2, pp. 79–99.
Toom’s Partial Order Is Transitive
M. A. Raskin
Faculty of Mathematics and Mechanics, Lomonosov Moscow State University
Received April 11, 2011; in ﬁnal form, January 16, 2012
Abstract—We prove that the partial order on measures on biinﬁnite sequences proposed by
Toom is transitive. This partial order was introduced as a possible tool for proving nonergodicity
of some cellular automata.
The paper investigates some particular probabilistic 1-D cellular automata with the possibility of
deleting a position (neighbors of a deleted position become neighbors of each other). Such automata
were introduced by Toom in . Components of these cellular automata have two possible states,
⊕ and . Thus, a state of a cellular automaton at each time step is a sequence of states of its
component cells, i.e., a biinﬁnite sequence of symbols ⊕, .
One of operators that we consider is deﬁned by two real numbers, ε and δ,lyingintheinterval
from zero to one; it acts as follows. Let its state at time step t be a sequence s. Then the state at the
next step t + 1 is obtained from s by successive application of two operators: “disturbance,” which
changes the state of each component into the opposite state with probability δ (diﬀerent components
are changed independently of each other), and “correction,” which deletes each occurrence of ⊕in
the sequence of component states with probability ε (diﬀerent occurrences are deleted independently
of each other). There is a diﬃculty related to enumeration of components; this question will be
considered below, when giving formal deﬁnitions of operators corresponding to cellular automata.
Hereafter we call the operator on probability distributions corresponding to this description the
symmetric Toom operator;wedenoteitbyT
In , the following (Toom’s) conjecture was stated.
Conjecture. Assume that the initial state (the state at step zero) of the process generated by
the symmetric Toom operator consists only of minuses. Then for any positive constants τ and ε
there exists a (small enough) positive δ satisfying the following condition: For every given time
step and every given position, the probability of ﬁnding ⊕ in this position at this step is less than τ .
Clearly, this is equivalent to a symmetric conjecture obtained from it by the interchange ⊕↔.
Let us explain what is meant by a given position in this setting. A state of the process at
each time step is a probability distribution on the set of biinﬁnite sequences of pluses and minuses.
We consider uniform measures only, i.e., those unchanged under a translation of indices of symbols
in the sequence. For probability distributions on the set of biinﬁnite sequences of pluses and
minuses, it makes sense to speak about the probability of a plus (or minus) in a “given position”;
this means the probability of the event the i-th symbol of the sequence is plus. Under the uniformity
condition, this probability is independent of i.
Supported in part by the Russian Foundation for Basic Research, project no. 10-01-93109-CNRS-a.