scholarly journals From NP-Completeness to DP-Completeness: A Membrane Computing Perspective

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Luis Valencia-Cabrera ◽  
David Orellana-Martín ◽  
Miguel Á. Martínez-del-Amor ◽  
Ignacio Pérez-Hurtado ◽  
Mario J. Pérez-Jiménez
Keyword(s):  
Lower Bound ◽  
Polynomial Time ◽  
Time Complexity ◽  
Membrane Computing ◽  
Decision Problems ◽  
Membrane Systems ◽  
Np Completeness ◽  
Computing Model ◽  
Complete Problems ◽  
Computing Models

Presumably efficient computing models are characterized by their capability to provide polynomial-time solutions for NP-complete problems. Given a class ℛ of recognizer membrane systems, ℛ denotes the set of decision problems solvable by families from ℛ in polynomial time and in a uniform way. PMCℛ is closed under complement and under polynomial-time reduction. Therefore, if ℛ is a presumably efficient computing model of recognizer membrane systems, then NP ∪ co-NP ⊆ PMCℛ. In this paper, the lower bound NP ∪ co-NP for the time complexity class PMCℛ is improved for any presumably efficient computing model ℛ of recognizer membrane systems verifying some simple requirements. Specifically, it is shown that DP ∪ co-DP is a lower bound for such PMCℛ, where DP is the class of differences of any two languages in NP. Since NP ∪ co-NP ⊆ DP ∩ co-DP, this lower bound for PMCℛ delimits a thinner frontier than that with NP ∪ co-NP.

scholarly journals Tuning Frontiers of Efficiency in Tissue P Systems with Evolutional Communication Rules

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
David Orellana-Martín ◽  
Luis Valencia-Cabrera ◽  
Bosheng Song ◽  
Linqiang Pan ◽  
Mario J. Pérez-Jiménez
Keyword(s):  
Polynomial Time ◽  
Efficient Solution ◽  
Efficient Solutions ◽  
Affirmative Answer ◽  
P Systems ◽  
Membrane Systems ◽  
Complete Problems ◽  
Np Complete ◽  
Np Problem ◽  
Computing Models

Over the last few years, a new methodology to address the P versus NP problem has been developed, based on searching for borderlines between the nonefficiency of computing models (only problems in class P can be solved in polynomial time) and the presumed efficiency (ability to solve NP-complete problems in polynomial time). These borderlines can be seen as frontiers of efficiency, which are crucial in this methodology. “Translating,” in some sense, an efficient solution in a presumably efficient model to an efficient solution in a nonefficient model would give an affirmative answer to problem P versus NP. In the framework of Membrane Computing, the key of this approach is to detect the syntactic or semantic ingredients that are needed to pass from a nonefficient class of membrane systems to a presumably efficient one. This paper deals with tissue P systems with communication rules of type symport/antiport allowing the evolution of the objects triggering the rules. In previous works, frontiers of efficiency were found in these kinds of membrane systems both with division rules and with separation rules. However, since they were not optimal, it is interesting to refine these frontiers. In this work, optimal frontiers of the efficiency are obtained in terms of the total number of objects involved in the communication rules used for that kind of membrane systems. These optimizations could be easier to translate, if possible, to efficient solutions in a nonefficient model.

scholarly journals Complexity-theoretic aspects of expanding cellular automata

2020 ◽  
Author(s):  
Augusto Modanese
Keyword(s):  
Cellular Automata ◽  
Polynomial Time ◽  
Time Complexity ◽  
Truth Table ◽  
Decision Problems ◽  
Turing Machines ◽  
One Dimensional ◽  
Alternative Characterization ◽  
Time Class ◽  
Theoretical Standpoint

Abstract The expanding cellular automata (XCA) variant of cellular automata is investigated and characterized from a complexity-theoretical standpoint. An XCA is a one-dimensional cellular automaton which can dynamically create new cells between existing ones. The respective polynomial-time complexity class is shown to coincide with $${\le _{tt}^p}(\textsf {NP})$$ ≤ tt p ( NP ) , that is, the class of decision problems polynomial-time truth-table reducible to problems in $$\textsf {NP}$$ NP . An alternative characterization based on a variant of non-deterministic Turing machines is also given. In addition, corollaries on select XCA variants are proven: XCAs with multiple accept and reject states are shown to be polynomial-time equivalent to the original XCA model. Finally, XCAs with alternative acceptance conditions are considered and classified in terms of $${\le _{tt}^p}(\textsf {NP})$$ ≤ tt p ( NP ) and the Turing machine polynomial-time class $$\textsf {P}$$ P .

SIMULATIONS BY TIME-BOUNDED COUNTER MACHINES

2011 ◽  
Vol 22 (02) ◽  
pp. 395-409 ◽  
Author(s):  
HOLGER PETERSEN
Keyword(s):  
Lower Bound ◽  
Real Time ◽  
Polynomial Time ◽  
Time Complexity ◽  
Linear Time ◽  
Fixed Number ◽  
Exponential Time ◽  
New Family ◽  
Counter Machines ◽  
Time Bounds

We investigate the efficiency of simulations of storages by several counters. A simulation of a pushdown store is described which is optimal in the sense that reducing the number of counters of a simulator leads to an increase in time complexity. The lower bound also establishes a tight counter hierarchy in exponential time. Then we turn to simulations of a set of counters by a different number of counters. We improve and generalize a known simulation in polynomial time. Greibach has shown that adding s + 1 counters increases the power of machines working in time ns. Using a new family of languages we show here a tight hierarchy result for machines with the same polynomial time-bound. We also prove hierarchies for machines with a fixed number of counters and with growing polynomial time-bounds. For machines with one counter and an additional "store zero" instruction we establish the equivalence of real-time and linear time. If at least two counters are available, the classes of languages accepted in real-time and linear time can be separated.

Membrane Computing Model Design with Quantum-Inspired Evolutionary Algorithms

2013 ◽  
Vol 655-657 ◽  
pp. 1761-1764 ◽  
Author(s):  
Hai Na Rong ◽  
Xiao Li Huang
Keyword(s):  
Membrane Computing ◽  
Design Method ◽  
Main Idea ◽  
Natural Computing ◽  
Automatic Design ◽  
Model Design ◽  
Evolution Rule ◽  
Computing Model ◽  
Rule Set ◽  
Computing Models

As a branch of natural computing, membrane computing has attracted much attention in various disciplines. But the programmability of membrane computing models is an ongoing and challenging issue in this area. This paper develops the automatic design of membrane computing models through predefining the membrane structure and initial objects and introducing a modified quantum-inspired evolutionary algorithm with a local disturbance to select an appropriate subset from a redundant evolution rule set. The main idea of the presented method is that multiple membrane computing models, instead of only one model like in the literature, can be designed by applying one redundant evolution rule set. The effectiveness of the design method is verified by the experiments.

Simple Neural-Like P Systems for Maximal Independent Set Selection

2013 ◽  
Vol 25 (6) ◽  
pp. 1642-1659 ◽  
Author(s):  
Lei Xu ◽  
Peter Jeavons
Keyword(s):  
Distributed Computing ◽  
Membrane Computing ◽  
Independent Set ◽  
Living Cells ◽  
P Systems ◽  
Maximal Independent Set ◽  
Membrane Systems ◽  
New Class ◽  
The Individual ◽  
Computing Models

Membrane systems (P systems) are distributed computing models inspired by living cells where a collection of processors jointly achieves a computing task. The problem of maximal independent set (MIS) selection in a graph is to choose a set of nonadjacent nodes to which no further nodes can be added. In this letter, we design a class of simple neural-like P systems to solve the MIS selection problem efficiently in a distributed way. This new class of systems possesses two features that are attractive for both distributed computing and membrane computing: first, the individual processors do not need any information about the overall size of the graph; second, they communicate using only one-bit messages.

scholarly journals Solving Vertex Cover Problem by Means of Tissue P Systems with Cell Separation

2010 ◽  
Vol 5 (4) ◽  
pp. 540 ◽  
Author(s):  
Chun Lu ◽  
Xingyi Zhang
Keyword(s):  
Cell Separation ◽  
Membrane Computing ◽  
Linear Time ◽  
Vertex Cover ◽  
P Systems ◽  
Computing Model ◽  
Vertex Cover Problem ◽  
Complete Problems ◽  
Np Complete ◽  
Cover Problem

Tissue P systems is a computing model in the framework of membrane computing inspired from intercellular communication and cooperation between neurons. Many different variants of this model have been proposed. One of the most important models is known as tissue P systems with cell separation. This model has the ability of generating an exponential amount of workspace in linear time, thus it allows us to design cellular solutions to NP-complete problems in polynomial time. In this paper, we present a solution to the Vertex Cover problem via a family of such devices. This is the first solution to this problem in the framework of tissue P systems with cell separation.

scholarly journals Average polynomial time complexity of some NP-complete problems

1986 ◽  
Vol 46 ◽  
pp. 219-237 ◽  
Author(s):  
Phan Dinh Dieu ◽  
Le Cong Thanh ◽  
Le Tuan Hoa
Keyword(s):  
Polynomial Time ◽  
Time Complexity ◽  
Polynomial Time Complexity ◽  
Complete Problems ◽  
Np Complete

The P versus NP Problem from the Membrane Computing View

2014 ◽  
Vol 22 (1) ◽  
pp. 18-33 ◽  
Author(s):  
Mario J. Pérez-Jiménez
Keyword(s):  
Membrane Computing ◽  
Efficient Solutions ◽  
Significant Advance ◽  
Natural Computing ◽  
Complexity Classes ◽  
Membrane Systems ◽  
Hard Problems ◽  
Young Branch ◽  
Np Problem ◽  
Computing Models

In the last few decades several computing models using powerful tools from Nature have been developed (because of this, they are known as bio-inspired models). Commonly, the space-time trade-off method is used to develop efficient solutions to computationally hard problems. According to this, implementation of such models (in biological, electronic, or any other substrate) would provide a significant advance in the practical resolution of hard problems. Membrane Computing is a young branch of Natural Computing initiated by Gh. Păun at the end of 1998. It is inspired by the structure and functioning of living cells, as well as from the organization of cells in tissues, organs, and other higher order structures. The devices of this paradigm, called P systems or membrane systems, constitute models for distributed, parallel and non-deterministic computing. In this paper, a computational complexity theory within the framework of Membrane Computing is introduced. Polynomial complexity classes associated with different models of cell-like and tissue-like membrane systems are defined and the most relevant results obtained so far are presented. Different borderlines between efficiency and non-efficiency are shown, and many attractive characterizations of the P ≠ NP conjecture within the framework of this bio-inspired and non-conventional computing model are studied.

On the Decidability and Complexity of Autoepistemic Reasoning1

1992 ◽  
Vol 17 (1-2) ◽  
pp. 117-155
Author(s):  
Ilkka N.F. Niemelä
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Classical Logic ◽  
General Setting ◽  
Decision Problems ◽  
Consequence Relation ◽  
Autoepistemic Logic ◽  
Complete Problems ◽  
Decidability And Complexity ◽  
Polynomial Time Hierarchy

The decidability and computational complexity of autoepistemic reasoning is investigated in a general setting where the autoepistemic logic CLae built on top of a given classical logic CL is studied. Correct autoepistemic conclusions from a set of premises are defined in terms of expansions of the premises. Three classes of expansions are studied: Moore style stable expansions, enumeration based expansions, and L-hierarchic expansions. A simple finitary characterization for each type of expansions in CLae is developed. Using the characterizations conditions ensuring that a set of premises has at least one or exactly one stable expansion can be stated and an upper bound for the number of stable expansions of a set of premises can be given. With the aid of the finitary characterizations results on decidability and complexity of autoepistemic reasoning are obtained. E.g., it is shown that autoepistemic reasoning based on each of the three types of expansions is decidable if the monotonic consequence relation given by the underlying classical logic is decidable. In the propositional case decision problems related to the three classes of expansions are shown to be complete problems with respect to the second level of the polynomial time hierarchy. This implies that propositional autoepistemic reasoning is strictly harder than classical propositional reasoning unless the polynomial time hierarchy collapses.

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