Abstract
We introduce a finite automata model for performing computations over an arbitrary structure \(\mathcal S\). The automaton processes sequences of elements in \(\mathcal S\). While processing the sequence, the automaton tests atomic relations, performs atomic operations of the structure \(\mathcal S\), and makes state transitions. In this setting, we study several problems such as closure properties, validation problem and emptiness problems. We investigate the dependence of deciding these problems on the underlying structures and the number of registers of our model of automata. Our investigation demonstrates that some of these properties are related to the existential first order fragments of the underlying structures.
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References
Alur, R., Černý, P., Weinstein, S.: Algorithmic Analysis of Array-Accessing Programs. In: Grädel, E., Kahle, R. (eds.) CSL 2009. LNCS, vol. 5771, pp. 86–101. Springer, Heidelberg (2009)
Blum, L., Shub, M., Smale, S.: On a Theory of Computation and Complexity over the Real Numbers: NP-completeness, Recursive Functions and Universal Machines. Bulletin of the American Mathematical Society 21(1), 1–46 (1989)
Bojanczyk, M., Muscholl, A., Schwentick, T., Segoufin, L., David, C.: Two-Variable Logic on Words with Data. In: Proceedings of LICS 2006, pp. 7–16. IEEE Computer Society (2006)
Bojanczyk, M., David, C., Muscholl, M., Schwentick, T., Segoufin, L.: Two-variable logic on data trees and XML reasoning. In: Proceedings of PODS 2006, pp. 10–19. ACM (2006)
Bournez, O., Cucker, F., Jacobé de Naurois, P., Marion, J.-Y.: Computability over an Arbitrary Structure. Sequential and Parallel Polynomial Time. In: Gordon, A.D. (ed.) FOSSACS 2003. LNCS, vol. 2620, pp. 185–199. Springer, Heidelberg (2003)
Bozga, M., Iosif, R., Lakhnech, Y.: Flat Parametric Counter Automata. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 577–588. Springer, Heidelberg (2006)
Comon, S., Jurski, Y.: Multiple Counters Automata, Safety Analysis and Presburger Arithmetic. In: Vardi, M.Y. (ed.) CAV 1998. LNCS, vol. 1427, pp. 268–279. Springer, Heidelberg (1998)
Ershov, Y., Goncharov, S., Marek, V., Nerode, A., Remmel, J.: Handbook of Recursive Mathematics: Recursive Model Theory. Studies in Logic and the Foundations of Mathematics. North-Holland (1998)
Figueira, D.: Reasoning on words and trees with data. Ph.D. Thesis, ENS Cachan, France (2010)
Ibarra, O.: Reversal-bounded multicounter machines and their decision problems. J. ACM 25(1), 116–133 (1978)
Kaminsky, M., Francez, N.: Finite memory automata. Theor. Comp. Sci. 134(2), 329–363 (1994)
Ishihara, H., Khousainov, B., Rubin, S.: Some Results on Automatic Structures. In: Proceedings of LICS 2002, p. 235. IEEE Computer Society (2002)
Leroux, J.: The general vector addition system reachability problem by presburger inductive invariants. In: Procedings of LICS 2009, pp. 4–13. IEEE Computer Society (2009)
Leroux, J., Sutre, G.: Flat Counter Automata Almost Everywhere! In: Peled, D.A., Tsay, Y.-K. (eds.) ATVA 2005. LNCS, vol. 3707, pp. 489–503. Springer, Heidelberg (2005)
Matiyasevich, Y.: Hilbert’s Tenth Problem. MIT Press, Cambridge (1993)
Minsky, M.: Recursive unsolvability of Post’s problem of “Tag” and other topics in theory of Turing machines. Annals of Math. 74(3) (1961)
Neven, F., Schwentick, T., Vianu, V.: Finite state machines for strings over infinite alphabets. ACM Tran. Comput. Logic 15(3), 403–435 (2004)
Point, F.: On Decidable Extensions of Presburger Arithmetic: From A. Bertrand Numeration Systems to Pisot Numbers. J. Symb. Log. 65(3), 1347–1374 (2000)
Segoufin, L.: Automata and Logics for Words and Trees over an Infinite Alphabet. In: Ésik, Z. (ed.) CSL 2006. LNCS, vol. 4207, pp. 41–57. Springer, Heidelberg (2006)
Segoufin, L., Torunczyk, S.: Automata based verification over linearly ordered data domains. In: Proceedings of STACS 2011. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, pp. 81–92 (2011)
Tan, T.: Graph reachability and pebble automata over infinite alphabets. In: Proceedings of LICS 2009, pp. 157–166. IEEE Computer Society (2009)
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Gandhi, A., Khoussainov, B., Liu, J. (2012). Finite Automata over Structures. In: Agrawal, M., Cooper, S.B., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2012. Lecture Notes in Computer Science, vol 7287. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29952-0_37
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DOI: https://doi.org/10.1007/978-3-642-29952-0_37
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