232 research outputs found
Simulating counting oracles with cooperation
We prove that monodirectional shallow chargeless P systems with active
membranes and minimal cooperation working in polynomial time precisely characterise
P#P
k , the complexity class of problems solved in polynomial time by deterministic
Turing machines with a polynomial number of parallel queries to an oracle for a counting
problem
Subroutines in P Systems and Closure Properties of Their Complexity Classes
The literature on membrane computing describes several variants of P systems
whose complexity classes C are "closed under exponentiation", that is, they satisfy
the inclusion PC C, where PC is the class of problems solved by polynomial-time
Turing machines with oracles for problems in C. This closure automatically implies closure
under many other operations, such as regular operations (union, concatenation,
Kleene star), intersection, complement, and polynomial-time mappings, which are inherited
from P. Such results are typically proved by showing how elements of a family of
P systems can be embedded into P systems simulating Turing machines, which exploit
the elements of as subroutines. Here we focus on the latter construction, abstracting
from the technical details which depend on the speci c variant of P system, in order to
describe a general strategy for proving closure under exponentiation
Characterising the complexity of tissue P systems with fission rules
We analyse the computational efficiency of tissue P systems, a biologically-inspired computing device modelling the communication between cells. In particular, we focus on tissue P systems with fission rules (cell division and/or cell separation), where the number of cells can increase exponentially during the computation. We prove that the complexity class characterised by these devices in polynomial time is exactly P^#P, the class of problems solved by polynomial-time Turing machines with oracles for counting problems
Characterizing PSPACE with Shallow Non-Confluent P Systems
In P systems with active membranes, the question of understanding the
power of non-confluence within a polynomial time bound is still an open problem. It is
known that, for shallow P systems, that is, with only one level of nesting, non-con
uence
allows them to solve conjecturally harder problems than con
uent P systems, thus reaching PSPACE. Here we show that PSPACE is not only a bound, but actually an exact
characterization. Therefore, the power endowed by non-con
uence to shallow P systems
is equal to the power gained by con
uent P systems when non-elementary membrane
division and polynomial depth are allowed, thus suggesting a connection between the
roles of non-confluence and nesting depth
Monodirectional P Systems
We investigate the in
uence that the
ow of information in membrane systems
has on their computational complexity. In particular, we analyse the behaviour of P systems
with active membranes where communication only happens from a membrane towards
its parent, and never in the opposite direction. We prove that these \monodirectional
P systems" are, when working in polynomial time and under standard complexity-theoretic
assumptions, much less powerful than unrestricted ones: indeed, they characterise classes
of problems de ned by polynomial-time Turing machines with NP oracles, rather than
the whole class PSPACE of problems solvable in polynomial space
Characterizing PSPACE with Shallow Non-Confluent P Systems
In P systems with active membranes, the question of understanding the
power of non-confluence within a polynomial time bound is still an open problem. It is
known that, for shallow P systems, that is, with only one level of nesting, non-con
uence
allows them to solve conjecturally harder problems than con
uent P systems, thus reaching PSPACE. Here we show that PSPACE is not only a bound, but actually an exact
characterization. Therefore, the power endowed by non-con
uence to shallow P systems
is equal to the power gained by con
uent P systems when non-elementary membrane
division and polynomial depth are allowed, thus suggesting a connection between the
roles of non-confluence and nesting depth
A Toolbox for Simpler Active Membrane Algorithms
We show that recogniser P systems with active membranes can be
augmented with a priority over their set of rules and any number of membrane
charges without loss of generality, as they can be simulated by standard P systems
with active membranes, in particular using only two charges. Furthermore, we
show that more general accepting conditions, such as sending out several, possibly
contradictory results and keeping only the first one, or rejecting by halting without
output, are also equivalent to the standard accepting conditions. The simulations
we propose are always without significant loss of efficiency, and thus the results of
this paper can hopefully simplify the design of algorithms for P systems with active
membranes
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
The Computational Complexity of Tissue P Systems with Evolutional Symport/Antiport Rules
Tissue P systems with evolutional communication (symport/antiport) rules are computational models inspired by biochemical
systems consisting of multiple individuals living and cooperating in a certain environment, where objects can be modified when
moving from one region to another region. In this work, cell separation, inspired from membrane fission process, is introduced in
the framework of tissue P systems with evolutional communication rules.The computational complexity of this kind of P systems
is investigated. It is proved that only problems in class P can be efficiently solved by tissue P systems with cell separation with
evolutional communication rules of length at most (��, 1), for each natural number �� ≥ 1. In the case where that length is upper
bounded by (3, 2), a polynomial time solution to the SAT problem is provided, hence, assuming that P ̸= NP a new boundary
between tractability and NP-hardness on the basis of the length of evolutional communication rules is provided. Finally, a new
simulator for tissue P systems with evolutional communication rules is designed and is used to check the correctness of the solution
to the SAT problem
Evaluating space measures in P systems
P systems with active membranes are a variant of P systems where membranes can be created by division of existing membranes, thus creating an exponential amount of resources in a polynomial number of steps. Time and space complexity classes for active membrane systems have been introduced, to characterize classes of problems that can be solved by different membrane systems making use of different resources. In particular, space complexity classes introduced initially considered a hypothetical real implementation by means of biochemical materials, assuming that every single object or membrane requires some constant physical space (corresponding to unary notation). A different approach considered implementation of P systems in silico, allowing to store the multiplicity of each object in each membrane using binary numbers. In both cases, the elements contributing to the definition of the space required by a system (namely, the total number of membranes, the total number of objects, the types of different membranes, and the types of different objects) was considered as a whole. In this paper, we consider a different definition for space complexity classes in the framework of P systems, where each of the previous elements is considered independently. We review the principal results related to the solution of different computationally hard problems presented in the literature, highlighting the requirement of every single resource in each solution. A discussion concerning possible alternative solutions requiring different resources is presented
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