article

Descriptional complexity of cellular automata and decidability questions

Author:

Andreas MalcherAuthors Info & Claims

Journal of Automata, Languages and Combinatorics, Volume 7, Issue 4

Pages 549 - 560

Published: 01 September 2002 Publication History

Abstract

We study the descriptional complexity of cellular automata (CA) which are a parallel model of computation. We show that between one of the simplest cellular models, the realtime one-way CA (realtime-OCA), and "classical" models like deterministic finite automata or pushdown automata, there will be savings concerning the size of description not bounded by any recursive function, so-called nonrecursive trade-offs. Furthermore, nonrecursive trade-offs are shown to exist between certain restricted classes of cellular automata. The set of valid computations of a Turing machine can be recognized by a realtime-OCA. This implies that many decidability questions are not even semidecidable for cellular automata. There is no pumping lemma and no minimization algorithm for cellular automata. Finally, we prove that the language class accepted by realtime-OCA is incomparable to many known and well-investigated language classes.

References

[1]

{1} T. BUCHHOLZ, A. KLEIN, M. KUTRIB, One guess one-way cellular arrays. In: L. BRIM, J. GRUSKA, J. ZLATUSKA (eds.), Proc. 23th Int. Symposium Mathematical Foundations of Computer Science (MFCS '98). LNCS 1450, Springer-Verlag, 1998, 807-815.]]

[2]

{2} G. BUNTROCK, Einige Bemerkungen zum Pumpen. In: M. KUTRIB, T. WORSCH (eds.), 5. Theorietag "Automatentheorie und Formale Sprachen". Bericht 9503, Arbeitsgruppe Informatik, Universität Gießen, 1995, 25-27.]]

[3]

{3} C. CHOFFRUT, K. CULIK II, On real-time cellular automata and trellis automata. Acta Informatica21 (1984), 393-407.]]

[4]

{4} K. CULIK II, J. GRUSKA, A. SALOMAA, Systolic trellis automata: Stability, decidability and complexity. Information and Control71 (1986), 218-230.]]

[5]

{5} M. DELORME, J. MAZOYER, Cellular automata as language recognizers. In: M. DELORME, J. MAZOYER (eds.), Cellular automata. Kluwer Academic Publishers, Dordrecht, 1999, 153-179.]]

[6]

{6} J. HARTMANIS, On the succinctness of different representations of languages. In: H. MAURER (ed.), Proc. 6th Int. Colloquium on Automata, Languages and Programming (ICALP '79). LNCS 71, Springer-Verlag, 1979, 282-288.]]

[7]

{7} J. GOLDSTINE, M. KAPPES, C.M.R. KINTALA, H. LEUNG, A. MALCHER, D. WOTSCHKE, Descriptional complexity of machines with limited resources. Journal of Universal Computer Science8 (2002), 193-234.]]

[8]

{8} J.E. HOPCROFT, J.D. ULLMAN, Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading MA, 1979.]]

[9]

{9} O.H. IBARRA, S.M. KIM, Characterizations and computational complexity of systolic trellis automata. Theoretical Computer Science29 (1984), 123-153.]]

[10]

{10} M. KUTRIB, Automata arrays and context-free languages. In: C. MARTÍN-VIDE, V. MITRANA (eds.), Where Mathematics, Computer Science, Linguistics and Biology Meet. Kluwer Academic Publishers, Dordrecht, 2001, 139-148.]]

[11]

{11} J. MAZOYER, V. TERRIER, Signals in one-dimensional cellular automata. Theoretical Computer Science217 (1999), 53-80.]]

[12]

{12} A.R. MEYER, M. J. FISCHER, Economy of descriptions by automata, grammars, and formal systems. In: IEEE Symposium on Foundations of Computer Science. 1971, 188-191.]]

[13]

{13} K. NAKAMURA, Real-time language recognition by one-way and two-way cellular automata. In: M. KUTYLOWSKI, L. PACHOLSKI, T. WIERZBICKI (eds.), Proc. 24th Int. Symposium on Mathematical Foundations of Computer Science (MFCS '99). LNCS 1672, Springer-Verlag, 1999, 220-230.]]

[14]

{14} S.R. SEIDEL, Language recognition and the synchronization of cellular automata. Technical Report 79-02, University of Iowa, Department of Computer Science, 1979.]]

[15]

{15} A.R. SMITH III, Cellular automata and formal languages. In: 11th Ann. IEEE Symposium on Switching and Automata Theory. 1970, 216-224.]]

[16]

{16} V. TERRIER, On real time one-way cellular arrays. Theoretical Computer Science141 (1995), 331-335.]]

Cited By

Holzer MKutrib M(2019)Non-Recursive Trade-Offs Are “Almost Everywhere”Computing with Foresight and Industry10.1007/978-3-030-22996-2_3(25-36)Online publication date: 15-Jul-2019
https://dl.acm.org/doi/10.1007/978-3-030-22996-2_3
Carton OGuillon BReiter F(2018)Counter Machines and Distributed AutomataCellular Automata and Discrete Complex Systems10.1007/978-3-319-92675-9_2(13-28)Online publication date: 20-Jun-2018
https://dl.acm.org/doi/10.1007/978-3-319-92675-9_2
Malcher AMereghetti CPalano B(2011)Descriptional complexity of two-way pushdown automata with restricted head reversalsProceedings of the 13th international conference on Descriptional complexity of formal systems10.5555/2032539.2032561(248-260)Online publication date: 25-Jul-2011
https://dl.acm.org/doi/10.5555/2032539.2032561
Show More Cited By

Index Terms

Descriptional complexity of cellular automata and decidability questions

Recommendations

Descriptional complexity of unambiguous input-driven pushdown automata

It is known that a nondeterministic input-driven pushdown automaton (IDPDA) (a.k.a. visibly pushdown automaton; a.k.a. nested word automaton) with n states can be transformed to an equivalent deterministic automaton with 2 ( n 2 ) states (B. von ...
Descriptional complexity of limited automata

A k-limited automaton is a linear bounded automaton that may rewrite each tape cell only in the first k visits, where k0 is a fixed constant. It is known that these automata accept context-free languages only. We investigate the descriptional complexity ...
Descriptional complexity of pushdown store languages

It is well known that the pushdown store language P(M) of a pushdown automaton (PDA) M- i.e., the language consisting of words occurring on the pushdown along accepting computations of M - is a regular language. Here, we design succinct nondeterministic ...

Comments

Information & Contributors

Information

Published In

cover image Journal of Automata, Languages and Combinatorics

Journal of Automata, Languages and Combinatorics Volume 7, Issue 4

Third international workshop on descriptional complexity of automata, grammars and related structures

September 2002

187 pages

ISSN:1430-189X

Issue’s Table of Contents

Publisher

Otto-von-Guericke-Universitat

Germany

Publication History

Published: 01 September 2002

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

10
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 22 Sep 2024

Other Metrics

View Author Metrics

Citations

Cited By

Holzer MKutrib M(2019)Non-Recursive Trade-Offs Are “Almost Everywhere”Computing with Foresight and Industry10.1007/978-3-030-22996-2_3(25-36)Online publication date: 15-Jul-2019
https://dl.acm.org/doi/10.1007/978-3-030-22996-2_3
Carton OGuillon BReiter F(2018)Counter Machines and Distributed AutomataCellular Automata and Discrete Complex Systems10.1007/978-3-319-92675-9_2(13-28)Online publication date: 20-Jun-2018
https://dl.acm.org/doi/10.1007/978-3-319-92675-9_2
Malcher AMereghetti CPalano B(2011)Descriptional complexity of two-way pushdown automata with restricted head reversalsProceedings of the 13th international conference on Descriptional complexity of formal systems10.5555/2032539.2032561(248-260)Online publication date: 25-Jul-2011
https://dl.acm.org/doi/10.5555/2032539.2032561
Bordihn HKutrib MMalcher A(2010)Undecidability and hierarchy results for parallel communicating finite automataProceedings of the 14th international conference on Developments in language theory10.5555/1881463.1881475(88-99)Online publication date: 17-Aug-2010
https://dl.acm.org/doi/10.5555/1881463.1881475
Holzer MKutrib MReimann J(2007)Non-recursive trade-offs for deterministic restarting automataJournal of Automata, Languages and Combinatorics10.5555/1463362.146337512:1(195-213)Online publication date: 1-Jan-2007
https://dl.acm.org/doi/10.5555/1463362.1463375
Malcher A(2005)On two-way communication in cellular automata with a fixed number of cellsTheoretical Computer Science10.1016/j.tcs.2004.04.014330:2(325-338)Online publication date: 2-Feb-2005
https://dl.acm.org/doi/10.1016/j.tcs.2004.04.014
Kutrib M(2005)On the descriptional power of heads, counters, and pebblesTheoretical Computer Science10.1016/j.tcs.2004.04.013330:2(311-324)Online publication date: 2-Feb-2005
https://dl.acm.org/doi/10.1016/j.tcs.2004.04.013
Okhotin A(2004)On the number of nonterminals in linear conjunctive grammarsTheoretical Computer Science10.1016/j.tcs.2004.03.002320:2-3(419-448)Online publication date: 14-Jun-2004
https://dl.acm.org/doi/10.1016/j.tcs.2004.03.002
Malcher A(2003)On One-Way Cellular Automata with a Fixed Number of CellsFundamenta Informaticae10.5555/2371016.237102458:3-4(355-368)Online publication date: 1-Aug-2003
https://dl.acm.org/doi/10.5555/2371016.2371024
Malcher A(2003)On one-way cellular automata with a fixed number of cellsFundamenta Informaticae10.5555/1006455.100646458:3-4(355-368)Online publication date: 1-May-2003
https://dl.acm.org/doi/10.5555/1006455.1006464

View Options

View options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Issue’s Table of Contents