ONE DIMENSIONAL nARY DENSITY CLASSIFICATION USING TWO CELLULAR AUTOMATON RULES

The purpose of the present study is to search one-dimensional Cellular Automata (CA) rules which will solve the density classification task (DCT) perfectly. The mathematical analysis of number conserving functions over binary strings of length n gives an indication of its corresponding number conserving cellular automata rules (either uniform or non-uniform). The state transition diagrams (STDs) of number conserving CA rules have been analyzed where it has been found that these STDs can generate different DCT solutions. While studying the properties of STDs, an interesting classification of binary strings could be made where equal weight strings form a class and the cardinality of each class is same as the binomial coefficient n C k ; n is the length and k is the weight of the binary string. Apart from STDs, other deterministic methods have been proposed to obtain the exact solution of DCT. All these exact solutions of DCT using different deterministic methods can be viewed as an improvement over the soft computing techniques used earlier to obtain approximate solutions. .

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“…For n=4 and (0111) 2 =7, we have 7  11  13  14  (14,14).So d=14= 4 22  = 3 22 nn   . So the string 0111 is dense in 1.…”

Section: Illustrationmentioning

confidence: 99%

“…7. Chau, Siu and Yan (1999) [3] proved that n-ary multistate density classification is impossible using single one dimensional CA rule but it can be solved using two CA rules in succession. 8.…”

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confidence: 99%

Deterministic Computing Techniques for Perfect Density Classification

Pal

Sahoo

Nayak

2019

Int. J. Bifurcation Chaos

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“…But are there any two-rule solutions for these problems? In 1999, H.F. Chau et al constructed two-rule solution to n-ary simple majority problem [11]. Their solution, however, uses rules of rather large neighbourhood size.…”

Section: Classification Problemsmentioning

confidence: 99%

Explorations of Ternary Cellular Automata and Ternary Density Classification Problems

Fukś¹,

Procyk²

2019

Acta Phys. Pol. B Proc. Suppl.

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While binary nearest-neighbour cellar automata (CA) have been studied in detail and from many different angles, the same cannot be said about ternary (three-state) CA rules. We present some results of our explorations of a small subset of the vast space of ternary rules, namely rules possessing additive invariants. We first enumerate rules with four different additive invariants, and then we investigate if any of them could be used to construct a two-rule solution of generalized density classification problem (DCP). We show that neither simple nor absolute classification is possible with a pair of ternary rules where the first rule is all-conserving and the second one is reducible to two states. Similar negative result holds for another version of DCP we propose: symmetric interval-wise DCP. Finally we show an example of a pair of rules which solve non-symmetric interval-wise DCP for initial configurations containing at least one zero.

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“…This, however, is not a deal breaker for majority-based error correction (both classical and quantum) as long as the erroneously classified instances are rare with respect to the noise channel in question. Motivated by applications for classical error correction, there evolved a vivid field concerned with the construction of approximate density classifiers (e.g., [53,35,54,55,56]) and extensions capable of performing density classification exactly (e.g., [57,58,59,60]), see [42] for a review. This is how we address the problem of finding a local decoder for the MCQC: Lemma 1 allows us to filter the literature of one-dimensional binary CAs for self-dual density classifiers; rewritten in syndrome-delta representation, these could be directly applied as potential Majorana chain decoders.…”

Section: Density Classification In 1dmentioning

confidence: 99%

Strictly local one-dimensional topological quantum error correction with symmetry-constrained cellular automata

Lang

Büchler

2018

SciPost Phys.

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Active quantum error correction on topological codes is one of the most promising routes to long-term qubit storage. In view of future applications, the scalability of the used decoding algorithms in physical implementations is crucial. In this work, we focus on the one-dimensional Majorana chain and construct a strictly local decoder based on a selfdual cellular automaton. We study numerically and analytically its performance and exploit these results to contrive a scalable decoder with exponentially growing decoherence times in the presence of noise. Our results pave the way for scalable and modular designs of actively corrected one-dimensional topological quantum memories.

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ONE DIMENSIONAL nARY DENSITY CLASSIFICATION USING TWO CELLULAR AUTOMATON RULES

Cited by 12 publications

References 6 publications

Deterministic Computing Techniques for Perfect Density Classification

Deterministic Computing Techniques for Perfect Density Classification

Explorations of Ternary Cellular Automata and Ternary Density Classification Problems

Strictly local one-dimensional topological quantum error correction with symmetry-constrained cellular automata

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