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 .

HOW TO SIMULATE TURING MACHINES BY INVERTIBLE ONE-DIMENSIONAL CELLULAR AUTOMATA

1995 ◽  
Vol 06 (04) ◽  
pp. 395-402 ◽  
Author(s):  
JEAN-CHRISTOPHE DUBACQ
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Turing Machine ◽  
Turing Machines ◽  
One Dimensional

The issue of testing invertibility of cellular automata has been often discussed. Constructing invertible automata is very useful for simulating invertible dynamical systems, based on local rules. The computation universality of cellular automata has long been positively resolved, and by showing that any cellular automaton could be simulated by an invertible one having a superior dimension, Toffoli proved that invertible cellular automaton of dimension d≥2 were computation-universal. Morita proved that any invertible Turing Machine could be simulated by a one-dimensional invertible cellular automaton, which proved computation-universality of invertible cellular automata. This article shows how to simulate any Turing Machine by an invertible cellular automaton with no loss of time and gives, as a corollary, an easier proof of this result.

scholarly journals Sublinear-Time Language Recognition and Decision by One-Dimensional Cellular Automata

2021 ◽  
pp. 1-19
Author(s):  
Augusto Modanese
Keyword(s):  
Cellular Automata ◽  
Parallel Computation ◽  
Time Complexity ◽  
Regular Languages ◽  
Constant Time ◽  
Language Recognition ◽  
One Dimensional ◽  
Acceptance Condition

After an apparent hiatus of roughly 30 years, we revisit a seemingly neglected subject in the theory of (one-dimensional) cellular automata: sublinear-time computation. The model considered is that of ACAs, which are language acceptors whose acceptance condition depends on the states of all cells in the automaton. We prove a time hierarchy theorem for sublinear-time ACA classes, analyze their intersection with the regular languages, and, finally, establish strict inclusions in the parallel computation classes [Formula: see text] and (uniform) [Formula: see text]. As an addendum, we introduce and investigate the concept of a decider ACA (DACA) as a candidate for a decider counterpart to (acceptor) ACAs. We show the class of languages decidable in constant time by DACAs equals the locally testable languages, and we also determine [Formula: see text] as the (tight) time complexity threshold for DACAs up to which no advantage compared to constant time is possible.

scholarly journals AUTOMATIC AND POLYNOMIAL-TIME ALGEBRAIC STRUCTURES

2019 ◽  
Vol 84 (4) ◽  
pp. 1630-1669 ◽  
Author(s):  
NIKOLAY BAZHENOV ◽  
MATTHEW HARRISON-TRAINOR ◽  
ISKANDER KALIMULLIN ◽  
ALEXANDER MELNIKOV ◽  
KENG MENG NG
Keyword(s):  
Polynomial Time ◽  
Regular Languages ◽  
Decision Problems ◽  
Turing Machines ◽  
Algebraic Structures ◽  
Automatic Structure ◽  
Image Position

AbstractA structure is automatic if its domain, functions, and relations are all regular languages. Using the fact that every automatic structure is decidable, in the literature many decision problems have been solved by giving an automatic presentation of a particular structure. Khoussainov and Nerode asked whether there is some way to tell whether a structure has, or does not have, an automatic presentation. We answer this question by showing that the set of Turing machines that represent automata-presentable structures is ${\rm{\Sigma }}_1^1 $-complete. We also use similar methods to show that there is no reasonable characterisation of the structures with a polynomial-time presentation in the sense of Nerode and Remmel.

CNN Genes for One-Dimensional Cellular Automata: A Multi-Nested Piecewise-Linear Approach

1998 ◽  
Vol 08 (10) ◽  
pp. 1987-2001 ◽  
Author(s):  
Radu Dogaru ◽  
Leon O. Chua
Keyword(s):  
Cellular Automata ◽  
Boolean Functions ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Truth Table ◽  
Local Rule ◽  
Mathematical Formula ◽  
One Dimensional ◽  
Linear Approach ◽  
Piecewise Linear Approach

This paper introduces a novel CNN cell which guarantees the implementation of any local rule on three variables defined by a Boolean truth table. Moreover, since the output of the cell is completely specified by a simple mathematical formula, it is possible to develop a systematic theory for locating those regions in the CNN genes parameter space where complex behaviors may occur. The output cell formula is a simple piecewise-linear function, and for the case of a one-dimensional CNN the entire set of 256 CNN genes associated with the corresponding local Boolean functions are listed in this paper.

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.

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