Skip to main content Accessibility help
×
Home
Hostname: page-component-7f7b94f6bd-lv2sk Total loading time: 0.176 Render date: 2022-06-29T02:51:48.873Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

Rabin's uniformization problem1

Published online by Cambridge University Press:  12 March 2014

Yuri Gurevich*
Affiliation:
University of Michigan, Ann Arbor, Michigan 48109
Saharon Shelah
Affiliation:
Hebrew University, Jerusalem, Israel
*
2Department of Computer and Communication Sciences, University of Michigan, Ann Arbor, MI 48109, USA

Abstract

The set of all words in the alphabet {l, r} forms the full binary tree T. If xT then xl and xr are the left and the right successors of x respectively. We consider the monadic second-order language of the full binary tree with the two successor relations. This language allows quantification over elements of rand over arbitrary subsets of T. We prove that there is no monadic second-order formula ϕ*(X, y) such that for every nonempty subset X of T there is a unique yX that satisfies ϕ*(X, y) in T.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

1

The work was done in principal during the 1980–81 academic year when both authors were fellows in the Institute for Advanced Studies of the Hebrew University in Jerusalem.

References

Büchi, J.R. [1962], On a decision method in second order arithmetic, Logic, methodology and philosophy of science (Proceedings of the 1960 international congress), Stanford University Press, Stanford, California, pp. 111.Google Scholar
Büchi, J.R. and Landweber, L.H. [1969], Solving sequential conditions by finite-state strategies, Transactions of the American Mathematical Society, vol. 138, pp. 295311.CrossRefGoogle Scholar
Gurevich, Y. [1979], Modest theory of short chains. I, this Journal, vol. 44, pp. 481490.Google Scholar
Gurevich, Y. and Harrington, L.A. [1982], Automata, trees, and games, Proceedings of the Fourteenth ACM Symposium on the Theory of Computing (San Francisco, 05, 1982) (to appear).Google Scholar
Läuchli, H. [1968], A decision procedure for the weak second order theory of linear order, Contributions to mathematical logic(Proceedings of the logic colloquium, Hannover, 1966), North-Holland, Amsterdam, pp. 189197.CrossRefGoogle Scholar
Rabin, M.O. [1969], Decidability of second-order theories and automata on infinite trees, Transactions of the American Mathematical Society, vol. 141, pp. 135.Google Scholar
Rabin, M.O. [1972], Automata on infinite objects and Church's problem, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, no. 13, American Mathematical Society, Providence, R.I.CrossRefGoogle Scholar
Shelah, S. [1975], The monadic theory of order, Annals of Mathematics, ser. 2, vol. 102, pp. 379419.CrossRefGoogle Scholar
Shoenfield, J.R. [1971], Unramified forcing. Axiomatic set theory. Proceedings of Symposia in Pure Mathematics, vol. 13, part 1, American Mathematical Society, Providence, R.I., pp. 357381.CrossRefGoogle Scholar
32
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Rabin's uniformization problem1
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Rabin's uniformization problem1
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Rabin's uniformization problem1
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *