article

Free access

Universality of Tag Systems with P = 2

Authors:

John Cocke,

Marvin MinskyAuthors Info & Claims

Journal of the ACM (JACM), Volume 11, Issue 1

Pages 15 - 20

https://doi.org/10.1145/321203.321206

Published: 01 January 1964 Publication History

PDF eReader

Abstract

By a simple direct construction it is shown that computations done by Turing machines can be duplicated by a very simple symbol manipulation process. The process is described by a simple form of Post canonical system with some very strong restrictions.

This system is monogenic: each formula (string of symbols) of the system can be affected by one and only one production (rule of inference) to yield a unique result. Accordingly, if we begin with a single axiom (initial string) the system generates a simply ordered sequence of formulas, and this operation of a monogenic system brings to mind the idea of a machine.

The Post canonical system is further restricted to the “Tag” variety, described briefly below. It was shown in [1] that Tag systems are equivalent to Turing machines. The proof in [1] is very complicated and uses lemmas concerned with a variety of two-tape nonwriting Turing machines. The proof here avoids these otherwise interesting machines and strengthens the main result; obtaining the theorem with a best possible deletion number P = 2.

Also, the representation of the Turing machine in the present system has a lower degree of exponentiation, which may be of significance in applications.

These systems seem to be of value in establishing unsolvability of combinatorial problems.

References

[1]

MINsKY, M. lecursive unsolvability of Post's probleln of Tag and other topics in theory of Turig moehirms. Ann. Math. 7g, 3 (Nov. 1961), 437-455.

Google Scholar

[2]

For further results along these lines, see: WANG, HAO. Tag systems and Lag systems. To apper.

Google Scholar

[3]

MINgKY, M, Size and sgrtwture of universal Turing machines using Tag systems: a 4-symbol 7-share machine. In Proc. Symposium a Recursive FuncZion Theory, Am. Math. See., Providence, R, I., 1962.

Google Scholar

[4]

SHHEDSON, J. C., AD STUaIs, H. E. Coputatiopnaly of recursie fmctions. J. ACM, 10 (Apr. 1963) 217-255.

Crossref

Google Scholar

[5]

WANG, HAo. A variant of Turing's theory of computing machines. J. ACM 4 (Apr. 1957), 63.

Crossref

Google Scholar

Cited By

View all

Alhazov AFreund RIvanov S(2024)On the spectrum between reaction systems and string rewritingNatural Computing: an international journal10.1007/s11047-024-09986-123:2(159-175)Online publication date: 11-Apr-2024
https://dl.acm.org/doi/10.1007/s11047-024-09986-1
Fazekas SSeki S(2023)Freezing 1-Tag Systems with StatesElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.386.8386(82-95)Online publication date: 3-Sep-2023
https://doi.org/10.4204/EPTCS.386.8
Fernau H(2021)Parsimonious Computational CompletenessDevelopments in Language Theory10.1007/978-3-030-81508-0_2(12-26)Online publication date: 16-Aug-2021
https://dl.acm.org/doi/10.1007/978-3-030-81508-0_2
Show More Cited By

Index Terms

Universality of Tag Systems with P = 2
1. Theory of computation
  1. Models of computation
    1. Computability
      1. Turing machines

Recommendations

Reversibility and Universality: Essays Presented to Kenichi Morita on the Occasion of his 70th Birthday
Universality Analysis for One-Clock Timed Automata
Fundamentals of Software Engineering 2007: Selected Contributions

This paper is concerned with the universality problem for timed automata: given a timed automaton A, does A accept all timed words? Alur and Dill have shown that the universality problem is undecidable if A has two clocks, but they left open the status ...
Universality Analysis for One-Clock Timed Automata
Fundamentals of Software Engineering 2007: Selected Contributions

This paper is concerned with the universality problem for timed automata: given a timed automaton A, does A accept all timed words? Alur and Dill have shown that the universality problem is undecidable if A has two clocks, but they left open the status ...

Comments

Information & Contributors

Information

Published In

Journal of the ACM Volume 11, Issue 1

Jan. 1964

116 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/321203

Issue’s Table of Contents

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 January 1964

Published in JACM Volume 11, Issue 1

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

54
Total Citations
View Citations
1,232
Total Downloads

Downloads (Last 12 months)235
Downloads (Last 6 weeks)41

Reflects downloads up to 01 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

View all

Alhazov AFreund RIvanov S(2024)On the spectrum between reaction systems and string rewritingNatural Computing: an international journal10.1007/s11047-024-09986-123:2(159-175)Online publication date: 11-Apr-2024
https://dl.acm.org/doi/10.1007/s11047-024-09986-1
Fazekas SSeki S(2023)Freezing 1-Tag Systems with StatesElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.386.8386(82-95)Online publication date: 3-Sep-2023
https://doi.org/10.4204/EPTCS.386.8
Fernau H(2021)Parsimonious Computational CompletenessDevelopments in Language Theory10.1007/978-3-030-81508-0_2(12-26)Online publication date: 16-Aug-2021
https://dl.acm.org/doi/10.1007/978-3-030-81508-0_2
Gómez-Canaval SMitrana VPăun MSanchez Martín JSánchez Couso J(2020)Networks of Uniform Splicing Processors: Computational Power and SimulationMathematics10.3390/math80812178:8(1217)Online publication date: 24-Jul-2020
https://doi.org/10.3390/math8081217
Morita K(2018)Reversible Cellular AutomataEncyclopedia of Complexity and Systems Science10.1007/978-3-642-27737-5_455-7(1-25)Online publication date: 31-Jan-2018
https://doi.org/10.1007/978-3-642-27737-5_455-7
Neary TCook M(2018)Generalized Tag SystemsReachability Problems10.1007/978-3-030-00250-3_8(103-116)Online publication date: 30-Aug-2018
https://doi.org/10.1007/978-3-030-00250-3_8
Morita K(2018)Reversible Cellular AutomataCellular Automata10.1007/978-1-4939-8700-9_455(105-128)Online publication date: 28-Nov-2018
https://doi.org/10.1007/978-1-4939-8700-9_455
Morita K(2017)Reversible Cellular AutomataEncyclopedia of Complexity and Systems Science10.1007/978-3-642-27737-5_455-6(1-25)Online publication date: 26-Sep-2017
https://doi.org/10.1007/978-3-642-27737-5_455-6
Morita K(2016)Two Small Universal Reversible Turing MachinesAdvances in Unconventional Computing10.1007/978-3-319-33924-5_10(221-237)Online publication date: 19-Jul-2016
https://doi.org/10.1007/978-3-319-33924-5_10
Morita K(2015)Universal Reversible Turing Machines with a Small Number of Tape SymbolsFundamenta Informaticae10.5555/2756686.2756689138:1-2(17-29)Online publication date: 1-Jan-2015
https://dl.acm.org/doi/10.5555/2756686.2756689
Show More Cited By

View Options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Login options

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

Abstract

References

Cited By

Index Terms

Recommendations

Reversibility and Universality: Essays Presented to Kenichi Morita on the Occasion of his 70th Birthday

Universality Analysis for One-Clock Timed Automata

Universality Analysis for One-Clock Timed Automata

Comments

Information

Published In

Publisher

Publication History

Permissions

Check for updates

Qualifiers

Contributors

Other Metrics

Bibliometrics

Article Metrics

Other Metrics

Citations

Cited By

View options

PDF

eReader

Login options

Full Access

Figures

Other

Share

Share this Publication link

Share on social media

Affiliations