Abstract
In [4] Gandhi, Khoussainov, and Liu introduced and studied a generalized model of finite automata able to work over arbitrary structures. The model mimics the finite automata over finite structures but has an additional ability to perform in a restricted way operations attached to the structure under consideration. As one relevant area of investigations for this model the authors of [4] identified studying the new automata over uncountable structures such as the real numbers. This research was started in [7]. However, there it turned out that many elementary properties known from classical finite automata are lost. This refers both to structural properties of accepted languages and to decidability and computability questions. The intrinsic reason for this is that the computational abilities of the new model turn out to be too strong.
We therefore propose a restricted version of the model which we call periodic GKL automata. The new model still has certain computational abilities which, however, are restricted in that computed information is deleted again after a fixed period in time. We show that this limitation regains a lot of classical properties including the pumping lemma and many decidability results. Thus the new model seems to reflect more adequately what might be considered as a finite automata over the reals and similar structures. Though our results resemble classical properties, for proving them other techniques are necessary. One fundamental proof ingredient will be quantifier elimination over real closed fields.
A. Naif was supported by a ‘Promotionsstipendium nach Graduiertenförderungsverordnung des Landes Brandenburg GradV’. The support is gratefully acknowledged.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
A semi-algebraic set in \({\mathbb R}^n\) is a set that can be defined as a finite union of solution sets of polynomial equalities and inequalities.
- 2.
In order to make the coding unique we could order the components of any tuple in \(Q'\) according to an order of \(Q \cup \{q^*\},\) but we refrain from elaborating on this because it will likely not increase understandability.
References
Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, Berlin (1998)
Blum, L., Shub, M., Smale, S.: On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bull. Am. Math. Soc. New Ser. 21(1), 1–46 (1989)
Cucker, F., Meer, K.: Logics which capture complexity classes over the reals. J. Symb. Log. 64(1), 363–390 (1999)
Gandhi, A., Khoussainov, B., Liu, J.: Finite automata over structures. In: Agrawal, M., Cooper, S.B., Li, A. (eds.) TAMC 2012. LNCS, vol. 7287, pp. 373–384. Springer, Heidelberg (2012)
Grädel, E., Gurevich, Y.: Metafinite model theory. Inf. Comput. 140(1), 26–81 (1998)
Grädel, E., Meer, K.: Descriptive complexity theory over the real numbers. In: Leighton, F.T., Borodin, A. (eds.) Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, Las Vegas, Nevada, USA, pp. 315–324. ACM, 29 May–1 June 1995
Meer, K., Naif, A.: Generalized finite automata over real and complex numbers. Theor. Comput. Sci. 591, 85–98 (2015)
Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals, parts 1–3. J. Symb. Comput. 13(3), 255–352 (1992)
Thomas, W.: Automata on infinite objects. In: van Leeuven, J. (ed.) Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics (B), pp. 133–192. Elsevier, Amsterdam (1990)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Meer, K., Naif, A. (2016). Periodic Generalized Automata over the Reals. In: Dediu, AH., Janoušek, J., Martín-Vide, C., Truthe, B. (eds) Language and Automata Theory and Applications. LATA 2016. Lecture Notes in Computer Science(), vol 9618. Springer, Cham. https://doi.org/10.1007/978-3-319-30000-9_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-30000-9_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-29999-0
Online ISBN: 978-3-319-30000-9
eBook Packages: Computer ScienceComputer Science (R0)