On the Complexity of Infinite Advice Strings
We investigate in this paper a notion of comparison between infinite strings. In a general way, if M is a computation model (e.g. Turing machines) and C a class of objects (e.g. languages), the complexity of an infinite word alpha can be measured with respect to the amount of objects from C that are presentable with machines from M using alpha as an oracle.
In our case, the model M is finite automata and the objects C are either recognized languages or presentable structures, known respectively as advice regular languages and advice automatic structures. This leads to several different classifications of infinite words that are studied in detail; we also derive logical and computational equivalent measures. Our main results explore the connections between classes of advice automatic structures, MSO-transductions and two-way transducers. They suggest a closer study of the resulting hierarchy over infinite words.
infinite words
advice automata
automatic structures
transducers
Theory of computation~Automata over infinite objects
122:1-122:13
Regular Paper
https://arxiv.org/abs/1801.04908
Gaëtan
Douéneau-Tabot
Gaëtan Douéneau-Tabot
École Normale Supérieure Paris-Saclay, Université Paris-Saclay, Cachan, France
This work was partially done during a stay of the author in RWTH Aachen University.
10.4230/LIPIcs.ICALP.2018.122
Faried Abu Zaid. Algorithmic solutions via model theoretic interpretations. PhD thesis, RWTH Aachen University, 2016. URL: http://dx.doi.org/10.18154/RWTH-2017-07663.
http://dx.doi.org/10.18154/RWTH-2017-07663
Faried Abu Zaid, Erich Grädel, and Frederic Reinhardt. Advice Automatic Structures and Uniformly Automatic Classes. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017), 2017.
Jean-Paul Allouche and Jeffrey Shallit. Automatic sequences: theory, applications, generalizations. Cambridge university press, 2003.
Rajeev Alur, Emmanuel Filiot, and Ashutosh Trivedi. Regular transformations of infinite strings. In Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science, pages 65-74. IEEE Computer Society, 2012.
Robert M. Baer and Edwin H. Spanier. Referenced automata and metaregular families. Journal of Computer and System Sciences, 3(4):423-446, 1969.
Vince Bárány. A hierarchy of automatic ω-words having a decidable MSO theory. RAIRO-Theoretical Informatics and Applications, 42(3):417-450, 2008.
Aleksandrs Belovs. Some algebraic properties of machine poset of infinite words. RAIRO-Theoretical Informatics and Applications, 42(3):451-466, 2008.
Achim Blumensath and Erich Grädel. Automatic structures. In Logic in Computer Science, 2000. Proceedings. 15th Annual IEEE Symposium on, pages 51-62. IEEE, 2000.
Gaëtan Douéneau-Tabot. Comparing the power of advice strings: a notion of complexity for infinite words. arXiv preprint arXiv:1801.04908, 2018.
Jörg Endrullis, Jan Willem Klop, Aleksi Saarela, and Markus Whiteland. Degrees of transducibility. In International Conference on Combinatorics on Words, pages 1-13. Springer, 2015.
Joost Engelfriet and Hendrik Jan Hoogeboom. MSO definable string transductions and two-way finite-state transducers. ACM Transactions on Computational Logic (TOCL), 2(2):216-254, 2001.
Bakhadyr Khoussainov, André Nies, Sasha Rubin, and Frank Stephan. Automatic structures: richness and limitations. In Logic in Computer Science, 2004. Proceedings of the 19th Annual IEEE Symposium on, pages 44-53. IEEE, 2004.
Alex Kruckman, Sasha Rubin, John Sheridan, and Ben Zax. A Myhill-Nerode theorem for automata with advice. In GandALF, pages 238-246, 2012.
Christof Löding and Thomas Colcombet. Transforming structures by set interpretations. Logical Methods in Computer Science, 3, 2007.
Alexander Rabinovich. A proof of Kamp’s theorem. Logical Methods in Computer Science, 10(1), 2014.
Frédéric Reinhardt. Automatic structures with parameters. Master’s thesis, Rheinisch-Westfälischen Technischen Hochschule Aachen, 2013.
Arto Salomaa. On finite automata with a time-variant structure. Information and Control, 13(2):85-98, 1968.
Dana Scott. Some definitional suggestions for automata theory. Journal of Computer and System Sciences, 1(2):187-212, 1967.
Aleksei Lvovich Semenov. Logical theories of one-place functions on the set of natural numbers. Mathematics of the USSR-Izvestiya, 22(3):587, 1984.
Todor Tsankov. The additive group of the rationals does not have an automatic presentation. The Journal of Symbolic Logic, 76(04):1341-1351, 2011.
Gaëtan Douéneau-Tabot
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode