Hostname: page-component-7bb8b95d7b-cx56b Total loading time: 0 Render date: 2024-09-15T05:23:39.306Z Has data issue: false hasContentIssue false
  • English
  • Français

Highly Undecidable Problems For Infinite Computations

Published online by Cambridge University Press:  14 February 2009

Olivier Finkel
Olivier Finkel*
Affiliation:
Equipe de Logique Mathématique, CNRS et Université Paris 7, France; finkel@logique.jussieu.fr
Get access

Abstract

We show that many classical decision problems about 1-counter ω-languages, context free ω-languages, or infinitary rational relations, are Π½ -complete, hence located at the second level of the analytical hierarchy, and “highly undecidable”. In particular, the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, and the unambiguity problem are all Π½ -complete for context-free ω-languages or for infinitary rational relations. Topological and arithmetical properties of 1-counter ω-languages, context free ω-languages, or infinitary rational relations, are also highly undecidable. These very surprising results provide the first examples of highly undecidable problems about the behaviour of very simple finite machines like 1-counter automata or 2-tape automata.

Keywords

Infinite computations1-counter-automata2-tape automatadecision problemsarithmetical hierarchyanalytical hierarchycomplete setshighly undecidable problems.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 43 , Issue 2 , April 2009 , pp. 339 - 364
Copyright
© EDP Sciences, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

J.-M. Autebert, J. Berstel and L. Boasson, Context free languages and pushdown automata, in Handbook of formal languages, Vol. 1. Springer-Verlag (1996).
Alur, R. and Dill, D.L., A theory of timed automata. Theoret. Comput. Sci. 126 (1994) 183–235. CrossRef
Altenbernd, J.-H., Thomas, W. and Wöhrle, S., Tiling systems over infinite pictures and their acceptance conditions, in Proceedings of the 6th International Conference Developments in Language Theory, DLT 2002. Lect. Notes Comput. Sci. 2450 (2003) 297–306. CrossRef
J. Berstel, Transductions and context free languages. Teubner Studienbücher Informatik (1979).
Castro, J. and Cucker, F., Nondeterministic ω-computations and the analytical hierarchy. Z. Math. Logik Grundlagen Math 35 (1989) 333–342. CrossRef
Cohen, R.S. and Gold, A.Y., Theory of ω-languages, parts one and two. J. Comput. System. Sci. 15 (1977) 169–208. CrossRef
Cohen, R.S. and Gold, A.Y., ω-computations on deterministic pushdown machines. J. Comput. System. Sci. 16 (1978) 275–300. CrossRef
Cohen, R.S. and Gold, A.Y., ω-computations on Turing machines. Theoret. Comput. Sci. 6 (1978) 1–23. CrossRef
Darondeau, P. and Yoccoz, S., Proof systems for infinite behaviours. Inform. Comput. 99 (1992) 178–191. CrossRef
Engelfriet, J. and Hoogeboom, H.J., X-automata on ω-words. Theoret. Comput. Sci. 110 (1993) 1–51. CrossRef
Finkel, O., Topological properties of omega context free languages. Theoret. Comput. Sci. 262 (2001) 669–697. CrossRef
Finkel, O., Ambiguity in omega context free languages. Theoret. Comput. Sci. 301 (2003) 217–270. CrossRef
Finkel, O., Borel hierarchy and omega context free languages. Theoret. Comput. Sci. 290 (2003) 1385–1405. CrossRef
Finkel, O., On the topological complexity of infinitary rational relations. RAIRO-Theor. Inf. Appl. 37 (2003) 105–113. CrossRef
Finkel, O., Undecidability of topological and arithmetical properties of infinitary rational relations. RAIRO-Theor. Inf. Appl. 37 (2003) 115–126. CrossRef
Finkel, O., On recognizable languages of infinite pictures. Int. J. Found. Comput. Sci. 15 (2004) 823–840. CrossRef
Finkel, O., Borel ranks and Wadge degrees of omega context free languages. Math. Struct. Comput. Sci. 16 (2006) 813–840. CrossRef
Finkel, O., On the accepting power of two-tape Büchi automata, in Proceedings of the 23rd International Symposium on Theoretical Aspects of Computer Science, STACS 2006. Lect. Notes Comput. Sci. 3884 (2006) 301–312. CrossRef
Finkel, O., Undecidable problems about timed automata, in Proceedings of the 4th International Conference on Formal Modeling and Analysis of Timed Systems, FORMATS 2006. Lect. Notes Comput. Sci. 4202 (2006) 187–199. CrossRef
O. Finkel, Highly undecidable problems about recognizability by tiling systems. Fundamenta Informaticae, 2009. Special Issue on Machines, Computations and Universality (to appear).
O. Finkel and D. Lecomte. Classical and effective descriptive complexities of omega-powers (2007). Preprint http://fr.arxiv.org/abs/0708.4176.
Frougny, C. and Sakarovitch, J., Synchronized rational relations of finite and infinite words. Theoret. Comput. Sci. 108 (1993) 45–82. CrossRef
Finkel, O. and Simonnet, P., Topology and ambiguity in omega context free languages. Bull. Belg. Math. Soc. 10 (2003) 707–722.
F. Gire, Relations rationnelles infinitaires. Ph.D. thesis, Université Paris VII (1981).
Gire, F., Une extension aux mots infinis de la notion de transduction rationelle, in Theoretical Computer Science, 6th GI-Conference, Dortmund, Germany, January 5–7, 1983, Proceedings. Lect. Notes Comput. Sci. 145 (1983) 123–139. CrossRef
F. Gire and M. Nivat, Relations rationnelles infinitaires. Calcolo (1984) 91–125.
T. Harju and J. Karhumäki. The equivalence problem of multitape finite automata. Theoret. Comput. Sci. 78 (1991) 347–355.
J.E. Hopcroft, R. Motwani and J.D. Ullman, Introduction to automata theory, languages, and computation. Addison-Wesley Publishing Co., Reading, Mass. (2001). Addison-Wesley Series in Computer Science.
A.S. Kechris, Classical descriptive set theory. Springer-Verlag, New York (1995).
Landweber, L.H., Decision problems for ω-automata. Math. Syst. Theor. 3 (1969) 376–384. CrossRef
Linna, M., On ω-words and ω-computations. Ann. Univ. Turku. Ser A I 168 (1975) 53.
H. Lescow and W. Thomas, Logical specifications of infinite computations, in A Decade of Concurrency, edited by J.W. de Bakker, Willem P. de Roever and Grzegorz Rozenberg. Lect. Notes Comput. Sci. 803 (1994) 583–621.
Y.N. Moschovakis, Descriptive set theory. North-Holland Publishing Co., Amsterdam (1980).
Nivat, M., Mots infinis engendrés par une grammaire algébrique. RAIRO-Inf. Théor. Appl. 11 (1977) 311–327. CrossRef
Nivat, M., Sur les ensembles de mots infinis engendrés par une grammaire algébrique. RAIRO-Inf. Théor. Appl. 12 (1978) 259–278. CrossRef
P.G. Odifreddi, Classical Recursion Theory, Vol I, Studies in Logic and the Foundations of Mathematics, Vol. 125. North-Holland Publishing Co., Amsterdam (1989).
P.G. Odifreddi, Classical Recursion Theory, Vol II, Studies in Logic and the Foundations of Mathematics, Vol. 143. North-Holland Publishing Co., Amsterdam (1999).
D. Perrin and J.-E. Pin, Infinite words, automata, semigroups, logic and games, Pure and Applied Mathematics, Vol. 141. Elsevier (2004).
Prasad Sistla, A., On verifying that a concurrent program satisfies a nondeterministic specification. Inform. Process. Lett. 32 (1989) 17–23. CrossRef
H. Rogers, Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967).
Sénizergues, G., L(A) = L(B)? decidability results from complete formal systems. Theoret. Comput. Sci. 251 (2001) 1–166. CrossRef
P. Simonnet, Automates et théorie descriptive. Ph.D. thesis, Université Paris VII (1992).
Staiger, L., Hierarchies of recursive ω-languages. Elektronische Informationsverarbeitung und Kybernetik 22 (1986) 219–241.
Staiger, L., Research in the theory of ω-languages. J. Inf. Process. Cybernetics 23 (1987) 415–439. Mathematical aspects of informatics (Mägdesprung, 1986).
L. Staiger, ω-languages, in Handbook of formal languages, Vol. 3. Springer, Berlin (1997) 339–387.
W. Thomas, Automata on infinite objects, in Handbook of Theoretical Computer Science, Vol. B, Formal models and semantics, edited by J. van Leeuwen. Elsevier (1990) 135–191.