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Unique decipherability in formal languages

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journal contribution
posted on 2019-11-21, 12:06 authored by Paul Bell, Daniel Reidenbach, Jeffrey O Shallit
We consider several language-theoretic aspects of various notions of unique decipherability (or unique factorization) in formal languages. Given a language L at some position within the Chomsky hierarchy, we investigate the language of words UD(L) in Lthat have unique factorization over L. We also consider similar notions for weaker forms of unique decipherability, such as numerically decipherable words ND(L), multiset decipherable words MSD(L) and set decipherable words SD(L). Although these notions of unique factorization have been considered before, it appears that the languages of words having these properties have not been positioned in the Chomsky hierarchy up until now.
We show that UD(L), ND(L), MSD(L) and SD(L) need not be context-free if L is context-free. In fact ND(L) and MSD(L) need not be context-free even if L is finite, although UD(L) and SD(L) are regular in this case. We show that if L is context-sensitive, then so are UD(L), ND(L), MSD(L) and SD(L). We also prove that the membership problem (resp., emptiness problem) for these
classes is PSPACE-complete (resp., undecidable). We finally determine upper and lower bounds on the length of the shortest word of L not having the various forms of unique decipherability into elements of L.

History

School

  • Science

Department

  • Computer Science

Published in

Theoretical Computer Science

Volume

804

Pages

149-160

Publisher

Elsevier

Version

  • AM (Accepted Manuscript)

Rights holder

© Elsevier B.V.

Publisher statement

This paper was accepted for publication in the journal Theoretical Computer Science and the definitive published version is available at https://doi.org/10.1016/j.tcs.2019.11.022.

Acceptance date

2019-11-16

Publication date

2019-11-22

Copyright date

2019

ISSN

0304-3975

Language

  • en

Depositor

Dr Daniel Reidenbach. Deposit date: 20 November 2019

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