Real-time generation of primes by a one-dimensional cellular automaton with 11 states

Korec, Ivan

doi:10.1007/bfb0029979

Cited by 18 publications

(17 citation statements)

References 3 publications

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“…Our eight-state implementation is an improvement over a nine-state prime generator presented in Korec [1998]. Those two constructions on cellular automata with different communication models are the smallest realizations in the number of states, at present.…”

Section: Discussionmentioning

confidence: 99%

“…Note that any signal cannot propagate at speed more than one cell per one step. Arisawa [1971], Fischer [1965], Korec [1997Korec [ , 1998] and Mazoyer and Terrier [1999] have considered the sequence generation problem on the cellular automata model. Umeo and Kamikawa [2002] showed that infinite non-regular sequences such as {2 n | n = 1, 2, 3,..}, {n 2 | n = 1, 2, 3,..} and Fibonacci sequences can be generated in real-time and the prime sequence in twice real-time by CA 1−bit .…”

Section: Sequence Generation Problemmentioning

confidence: 99%

“…In addition, we show that an infinite prime sequence can be generated in real-time by an eight-state cellular automaton with constant-bit communications. Both the algorithms presented are based on the classical sieve of Eratosthenes, and our eight-state implementation is an improvement over a nine-state prime generator developed by Korec [1998]. Those two implementations on cellular automata with different communication models are the smallest realizations in the number of states, at present.…”

Section: Appendixmentioning

confidence: 99%

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Automata, Universality, Computation

Adamatzky¹

2015

Emergence, Complexity and Computation

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About this SeriesThe Emergence, Complexity and Computation (ECC) series publishes new developments, advancements and selected topics in the fields of complexity, computation and emergence. The series focuses on all aspects of reality-based computation approaches from an interdisciplinary point of view especially from applied sciences, biology, physics, or Chemistry. It presents new ideas and interdisciplinary insight on the mutual intersection of subareas of computation, complexity and emergence and its impact and limits to any computing based on physical limits (thermodynamic and quantum limits, Bremermann's limit, Seth Lloyd limits...) as well as algorithmic limits (Gödel's proof and its impact on calculation, algorithmic complexity, the Chaitin's Omega number and Kolmogorov complexity, non-traditional calculations like Turing machine process and its consequences,...) and limitations arising in artificial intelligence field. The topics are (but not limited to) membrane computing, DNA computing, immune computing, quantum computing, swarm computing, analogic computing, chaos computing and computing on the edge of chaos, computational aspects of dynamics of complex systems (systems with self-organization, multiagent systems, cellular automata, artificial life,...), emergence of complex systems and its computational aspects, and agent based computation. The main aim of this series it to discuss the above mentioned topics from an interdisciplinary point of view and present new ideas coming from mutual intersection of classical as well as modern methods of computation. Within the scope of the series are monographs, lecture notes, selected contributions from specialized conferences and workshops, special contribution from international experts.More information about this series at

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Section: Discussionmentioning

confidence: 99%

Section: Sequence Generation Problemmentioning

confidence: 99%

Section: Appendixmentioning

confidence: 99%

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Automata, Universality, Computation

Adamatzky¹

2015

Emergence, Complexity and Computation

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“…We study a sequence generation problem on the CA. Arisawa [1], Fischer [2] and Korec [3] studied generation of a class of natural numbers on CA. In this paper, we study the sequence generation power of CA with 2 internal states.…”

Section: Introductionmentioning

confidence: 99%

A note on sequence generation power o two-states cellular automata

Kamikawa

Umeo

2008

2008 SICE Annual Conference

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Cellular automaton (CA) are considered to be a non-linear model of complex systems in which an infinite one-dimensional array of finite state machines (cells) updates itself in a synchronous manner according to a uniform local rule. It is studied in many fields such as complex systems. We study a sequence generation problem on the CA. Arisawa, Fischer and Korec studied generation of a class of natural numbers on CA. In this paper, we study the sequence generation power of CA with 2 internal states.

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“…On the other hand, Arisawa [5], Fischer [6], Korec [7], Umeo and Kamikawa [8], [9], and Kamikawa and Umeo [10]- [12] studied a similar sequence generation problem in different settings, where a leftmost cell of the array generates a state sequence indicated by an internal state set. The time series in which the leftmost cell falls into a special designated state as the sequence generated by the CA is described.…”

Section: Introductionmentioning

confidence: 99%

A Study on Sequence Generation Powers of Small Cellular Automata

Kamikawa

Umeo

2012

SICE Journal of Control, Measurement, and System Integration

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A model of cellular automata (CA) is considered to be a well-studied non-linear model of complex systems in which an infinite one-dimensional array of finite state machines (cells) updates itself in a synchronous manner according to a uniform local rule. A sequence generation problem on the CAs has been studied and many scholars proposed several real-time sequence generation algorithms for a variety of non-regular sequences such as prime, Fibonacci, and {2 n | n = 1, 2, 3, . . .} sequences etc. The paper describes the sequence generation powers of CAs having a small number of states, focusing on the CAs with one, two, and three internal states, respectively. The authors enumerate all of the sequences generated by two-state CAs and present several non-regular sequences that can be generated in real-time by three-state CAs, but not generated by any two-state CA. It is shown that there exists a sequence generation gap among the powers of those small CAs.as the authors know, no work, except for this paper, has been done concerning the study of generation powers of CAs with a small number of states.This paper extends the study of Kamikawa and Umeo [12] exhaustively, study sequence generation powers of CAs with a small number of states, and give formal proofs of all algorithms presented. The sequence generation powers of small CAs, focusing on the CAs with one, two, and three internal states is investigeted. The authors enumerate all of the sequences generated by two-state CAs and present several non-regular sequences that can be generated in real-time by three-state CAs, but not generated by any two-state CAs. It is shown that there exists a sequence generation gap among the powers of those small CAs.

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Real-time generation of primes by a one-dimensional cellular automaton with 11 states

Cited by 18 publications

References 3 publications

Automata, Universality, Computation

Automata, Universality, Computation

A note on sequence generation power o two-states cellular automata

A Study on Sequence Generation Powers of Small Cellular Automata

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