We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 07 May 2010
Abstract
We consider how the languages of G-automata compare with other formal language classes. We prove that if the word problem of G is accepted by a machine in the class then the language of any G-automaton is in the class. It follows that the so called counter languages (languages of ℤn-automata) are context-sensitive, and further that counter languages are indexed if and only if the word problem for ℤn is indexed.
AMS Classification: 20F65, 20F10, 68Q45
Keywords: G-automaton; counter language; word problem for groups; Chomsky hierarchy
Introduction
In this article we compare the languages of G-automata, which include the set of counter languages, with the formal language classes of context-sensitive, indexed, context-free and regular. We prove in Theorem 6 that if the word problem of G is accepted by a machine in the class M then the language of any G-automaton is in the class M. It follows that the counter languages (languages of ℤn-automata) are context-sensitive. Moreover it follows that counter languages are indexed if and only if the word problem for ℤn is indexed.
The article is organized as follows. In Section 2 we define G-automata, linearly bounded automata, nested stack, stack, and pushdown automata, and the word problem for a finitely generated group. In Section 3 we prove the main theorem, and give the corollary that counter languages are indexed if and only if the word problem for ℤn is indexed for all n.
To save this book to your Kindle, first ensure no-reply@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.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
