5 research outputs found
AND and/or OR: Uniform Polynomial-Size Circuits
We investigate the complexity of uniform OR circuits and AND circuits of
polynomial-size and depth. As their name suggests, OR circuits have OR gates as
their computation gates, as well as the usual input, output and constant (0/1)
gates. As is the norm for Boolean circuits, our circuits have multiple sink
gates, which implies that an OR circuit computes an OR function on some subset
of its input variables. Determining that subset amounts to solving a number of
reachability questions on a polynomial-size directed graph (which input gates
are connected to the output gate?), taken from a very sparse set of graphs.
However, it is not obvious whether or not this (restricted) reachability
problem can be solved, by say, uniform AC^0 circuits (constant depth,
polynomial-size, AND, OR, NOT gates). This is one reason why characterizing the
power of these simple-looking circuits in terms of uniform classes turns out to
be intriguing. Another is that the model itself seems particularly natural and
worthy of study.
Our goal is the systematic characterization of uniform polynomial-size OR
circuits, and AND circuits, in terms of known uniform machine-based complexity
classes. In particular, we consider the languages reducible to such uniform
families of OR circuits, and AND circuits, under a variety of reduction types.
We give upper and lower bounds on the computational power of these language
classes. We find that these complexity classes are closely related to tallyNL,
the set of unary languages within NL, and to sets reducible to tallyNL.
Specifically, for a variety of types of reductions (many-one, conjunctive truth
table, disjunctive truth table, truth table, Turing) we give characterizations
of languages reducible to OR circuit classes in terms of languages reducible to
tallyNL classes. Then, some of these OR classes are shown to coincide, and some
are proven to be distinct. We give analogous results for AND circuits. Finally,
for many of our OR circuit classes, and analogous AND circuit classes, we prove
whether or not the two classes coincide, although we leave one such inclusion
open.Comment: In Proceedings MCU 2013, arXiv:1309.104
Uniformity is weaker than semi-uniformity for some membrane systems
We investigate computing models that are presented as families of finite
computing devices with a uniformity condition on the entire family. Examples of
such models include Boolean circuits, membrane systems, DNA computers, chemical
reaction networks and tile assembly systems, and there are many others.
However, in such models there are actually two distinct kinds of uniformity
condition. The first is the most common and well-understood, where each input
length is mapped to a single computing device (e.g. a Boolean circuit) that
computes on the finite set of inputs of that length. The second, called
semi-uniformity, is where each input is mapped to a computing device for that
input (e.g. a circuit with the input encoded as constants). The former notion
is well-known and used in Boolean circuit complexity, while the latter notion
is frequently found in literature on nature-inspired computation from the past
20 years or so.
Are these two notions distinct? For many models it has been found that these
notions are in fact the same, in the sense that the choice of uniformity or
semi-uniformity leads to characterisations of the same complexity classes. In
other related work, we showed that these notions are actually distinct for
certain classes of Boolean circuits. Here, we give analogous results for
membrane systems by showing that certain classes of uniform membrane systems
are strictly weaker than the analogous semi-uniform classes. This solves a
known open problem in the theory of membrane systems. We then go on to present
results towards characterising the power of these semi-uniform and uniform
membrane models in terms of NL and languages reducible to the unary languages
in NL, respectively.Comment: 28 pages, 1 figur