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 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 .

ON THE ADIABATIC EVOLUTION OF ONE-DIMENSIONAL PROJECTOR HAMILTONIANS

2012 ◽  
Vol 10 (04) ◽  
pp. 1250046 ◽  
Author(s):  
JIE SUN ◽  
SONG-FENG LU
Keyword(s):  
Ground State ◽  
Time Complexity ◽  
The Other ◽  
Constant Time ◽  
Simple Analysis ◽  
Infinite Time ◽  
Adiabatic Evolution ◽  
One Dimensional ◽  
Original Time

In this paper, we discuss the adiabatic evolution of one-dimensional projector Hamiltonians. Three kinds of adiabatic algorithms for this problem are shown, in which two of them provide a quadratic speedup over the other one. But when the ground state of the initial Hamiltonian and that of the final Hamiltonian have a zero overlap, the algorithms above all fail, in the sense of infinite time complexity. A corresponding revised method for this phenomenon through adding a driving Hamiltonian is also shown, from which a constant time complexity can be gained. But by a simple analysis, we find that the original time complexity for the adiabatic evolution is shifted to implement the driving Hamiltonian, which is supported by several early works in the literature.

Parallel language recognition in constant time by cellular automata

1983 ◽  
Vol 19 (4) ◽  
pp. 397-407 ◽  
Author(s):  
R. Sommerhalder ◽  
S. C. van Westrhenen
Keyword(s):  
Cellular Automata ◽  
Constant Time ◽  
Language Recognition ◽  
Parallel Language

Synthesis of Scalable Single Length Cycle, Single Attractor Cellular Automata in Linear Time

2021 ◽  
Vol 30 (3) ◽  
pp. 415-439
Author(s):  
Bidesh Chakraborty ◽  
◽  
Mamata Dalui ◽  
Biplab K. Sikdar ◽  
◽  
...  
Keyword(s):  
Cellular Automata ◽  
Linear Time ◽  
Analytical Framework ◽  
Constant Time ◽  
M Cell ◽  
Transition Diagram ◽  
Arbitrary Length ◽  
Attractor State ◽  
One Dimensional ◽  
State Transition Diagram

This paper proposes the synthesis of single length cycle, single attractor cellular automata (SACAs) for arbitrary length. The n-cell single length cycle, single attractor cellular automaton (SACA), synthesized in linear time O(n), generates a pattern and finally settles to a point state called the single length cycle attractor state. An analytical framework is developed around the graph-based tool referred to as the next state transition diagram to explore the properties of SACA rules for three-neighborhood, one-dimensional cellular automata. This enables synthesis of an (n+1)-cell SACA from the available configuration of an n-cell SACA in constant time and an (n+m)-cell SACA from the available configuration of n-cell and m-cell SACAs also in constant time.

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