ISSN 0032-9460, Problems of Information Transmission, 2010, Vol. 46, No. 4, pp. 346–352.
Pleiades Publishing, Inc., 2010.
Original Russian Text
S.S. Marchenkov, 2010, published in Problemy Peredachi Informatsii, 2010, Vol. 46, No. 4, pp. 83–90.
Complete and Incomplete Boolean Degrees
S. S. Marchenkov
Faculty of Computational Mathematics and Cybernetics,
Lomonosov Moscow State University
Received May 24, 2010; in ﬁnal form, September 13, 2010
Abstract—We study the partially ordered set of Boolean P
-degrees. We introduce the notions
of complete and incomplete Boolean degrees. We show that for each complete P
exist both a countable decreasing chain of P
-degrees and a countable antichain of P
We prove that above each incomplete P
-degree there is a continuum of P
-degrees. Thus, in
total we show that in the partially ordered set of P
-degrees there are no maximal elements.
One of widely used methods of comparing “information complexity” of inﬁnite Boolean sequences
is algorithmic reducibility. Usually, one selects a suﬃciently large class O of eﬃcient operators and
establishes that a sequence α O-reduces to another sequence β if there exists an operator Φ ∈O
such that α =Φ(β). Various collections of operators can be used as the O class, from enumerating
operators  to ﬁnite automata operators .
Based on the class O of operators, we deﬁne O-degrees, sets of binary sequences that are
O-reducible to each other. O-degrees can be considered as sets of “informationally equivalent”
sequences with respect to the way of “information extraction” deﬁned by the operators of O.
On the set of all O-degrees, we introduce a partial order induced by the O-reducibility relation
between sequences. As a result, we get a partially ordered set of O-degrees L
, whose structure,
in a way, characterizes the “information complexity” of binary sequences.
Usually, the partially ordered set L
is an upper semilattice, but not a lattice. The position
of an O-degree [α]
of a sequence α in the semilattice L
can serve as a measure of “information
amount” contained in α. In most semilattices L
there is a smallest element 0. Sequences from
the O-degree of 0 should be considered as “informationally poor” with respect to the class O of
reduction operators. For instance, for the class of ﬁnite automata operators the degree of 0 consists
of all periodic sequences , while for the class of T -reducibility operators it consists of all sequences
with recursive sets of ones .
Classes of operators O are always countable. Therefore, semilattices L
do not have a largest
element. However, if we restrict our attention from the set of all binary sequences to some of its
countable subsets, maximal elements may appear in the corresponding semilattices. Degrees that
form largest elements in semilattices are called complete degrees; they play an important role in
the theory of degrees of undecidability. Sequences from a complete degree contain, in a sense,
information about all sequences from the considered countable set of sequences.
The notion of O-reducibility allows us to pose several problems about the character of “infor-
mation distribution” in a binary sequence. For example, how much information is duplicated in
Supported in part by the Federal Target Program “Research and Educational Personnel of Innovation
Russia” for 2009–2013.