# Some problems in automata theory which depend on the models of set theory

RAIRO - Theoretical Informatics and Applications (2012)

- Volume: 45, Issue: 4, page 383-397
- ISSN: 0988-3754

## Access Full Article

top## Abstract

top## How to cite

topFinkel, Olivier. "Some problems in automata theory which depend on the models of set theory." RAIRO - Theoretical Informatics and Applications 45.4 (2012): 383-397. <http://eudml.org/doc/222073>.

@article{Finkel2012,

abstract = {We prove that some fairly basic questions on automata reading infinite words depend on the models of the axiomatic system ZFC. It is known that there are only three possibilities for the cardinality of the complement of an ω-language \hbox\{$L(\mathcal\{A\})$\}L(𝒜) accepted by a Büchi 1-counter automaton \hbox\{$\mathcal\{A\}$\}𝒜. We prove the following surprising result: there exists a 1-counter Büchi automaton \hbox\{$\mathcal\{A\}$\}𝒜 such that the cardinality of the complement \hbox\{$L(\mathcal\{A\})^-$\}L(𝒜) − of the ω-language \hbox\{$L(\mathcal\{A\})$\}L(𝒜) is not determined by ZFC: (1) There is a model V1 of ZFC in which \hbox\{$L(\mathcal\{A\})^-$\}L(𝒜) − is countable. (2) There is a model V2 of ZFC in which \hbox\{$L(\mathcal\{A\})^-$\}L(𝒜) − has cardinal 2ℵ0. (3) There is a model V3 of ZFC in which \hbox\{$L(\mathcal\{A\})^-$\}L(𝒜) − has cardinal ℵ1 with ℵ0 < ℵ1 < 2ℵ0. We prove a very similar result for the complement of an infinitary rational relation accepted by a 2-tape Büchi automaton ℬ. As a corollary, this proves that the continuum hypothesis may be not satisfied for complements of 1-counter ω-languages and for complements of infinitary rational relations accepted by 2-tape Büchi automata. We infer from the proof of the above results that basic decision problems about 1-counter ω-languages or infinitary rational relations are actually located at the third level of the analytical hierarchy. In particular, the problem to determine whether the complement of a 1-counter ω-language (respectively, infinitary rational relation) is countable is in Σ1312 ∪ Σ12).
This is rather surprising if compared to the fact that it is decidable whether an infinitary rational relation is countable (respectively, uncountable). },

author = {Finkel, Olivier},

journal = {RAIRO - Theoretical Informatics and Applications},

keywords = {Automata and formal languages; logic in computer science; computational complexity; infinite words; ω-languages; 1-counter automaton; 2-tape automaton; cardinality problems; decision problems; analytical hierarchy; largest thin effective coanalytic set; models of set theory; independence from the axiomatic system ZFC; formal languages; -languages; axiomatic system ZFC; independence from ZFC},

language = {eng},

month = {1},

number = {4},

pages = {383-397},

publisher = {EDP Sciences},

title = {Some problems in automata theory which depend on the models of set theory},

url = {http://eudml.org/doc/222073},

volume = {45},

year = {2012},

}

TY - JOUR

AU - Finkel, Olivier

TI - Some problems in automata theory which depend on the models of set theory

JO - RAIRO - Theoretical Informatics and Applications

DA - 2012/1//

PB - EDP Sciences

VL - 45

IS - 4

SP - 383

EP - 397

AB - We prove that some fairly basic questions on automata reading infinite words depend on the models of the axiomatic system ZFC. It is known that there are only three possibilities for the cardinality of the complement of an ω-language \hbox{$L(\mathcal{A})$}L(𝒜) accepted by a Büchi 1-counter automaton \hbox{$\mathcal{A}$}𝒜. We prove the following surprising result: there exists a 1-counter Büchi automaton \hbox{$\mathcal{A}$}𝒜 such that the cardinality of the complement \hbox{$L(\mathcal{A})^-$}L(𝒜) − of the ω-language \hbox{$L(\mathcal{A})$}L(𝒜) is not determined by ZFC: (1) There is a model V1 of ZFC in which \hbox{$L(\mathcal{A})^-$}L(𝒜) − is countable. (2) There is a model V2 of ZFC in which \hbox{$L(\mathcal{A})^-$}L(𝒜) − has cardinal 2ℵ0. (3) There is a model V3 of ZFC in which \hbox{$L(\mathcal{A})^-$}L(𝒜) − has cardinal ℵ1 with ℵ0 < ℵ1 < 2ℵ0. We prove a very similar result for the complement of an infinitary rational relation accepted by a 2-tape Büchi automaton ℬ. As a corollary, this proves that the continuum hypothesis may be not satisfied for complements of 1-counter ω-languages and for complements of infinitary rational relations accepted by 2-tape Büchi automata. We infer from the proof of the above results that basic decision problems about 1-counter ω-languages or infinitary rational relations are actually located at the third level of the analytical hierarchy. In particular, the problem to determine whether the complement of a 1-counter ω-language (respectively, infinitary rational relation) is countable is in Σ1312 ∪ Σ12).
This is rather surprising if compared to the fact that it is decidable whether an infinitary rational relation is countable (respectively, uncountable).

LA - eng

KW - Automata and formal languages; logic in computer science; computational complexity; infinite words; ω-languages; 1-counter automaton; 2-tape automaton; cardinality problems; decision problems; analytical hierarchy; largest thin effective coanalytic set; models of set theory; independence from the axiomatic system ZFC; formal languages; -languages; axiomatic system ZFC; independence from ZFC

UR - http://eudml.org/doc/222073

ER -

## References

top- J. Castro and F. Cucker, Nondeterministic ω-computations and the analytical hierarchy. J. Math. Logik Grundl. Math.35 (1989) 333–342.
- R.S. Cohen and A.Y. Gold, ω-computations on Turing machines. Theoret. Comput. Sci.6 (1978) 1–23.
- O. Finkel, Borel ranks and Wadge degrees of omega context free languages. Math. Structures Comput. Sci.16 (2006) 813–840.
- O. Finkel, 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.
- O. Finkel, The complexity of infinite computations in models of set theory. Log. Meth. Comput. Sci.5 (2009) 1–19.
- O. Finkel, Highly undecidable problems for infinite computations. RAIRO – Theor. Inf. Appl.43 (2009) 339–364.
- O. Finkel, Decision problems for recognizable languages of infinite pictures, in Studies in Weak Arithmetics, Proceedings of the International Conference 28th Weak Arithmetic Days, 2009, Publications of the Center for the Study of Language and Information. Lect. Notes196 (2010) 127–151.
- F. Gire, Relations rationnelles infinitaires. Ph.D. thesis, Université Paris VII (1981).
- F. Gire and M. Nivat, Relations rationnelles infinitaires. CalcoloXXI (1984) 91–125.
- E. Grädel, W. Thomas and W. Wilke Eds., Automata, Logics, and Infinite Games : A Guide to Current Research [outcome of a Dagstuhl seminar, February 2001]. Lect. Notes Comput. Sci.2500 (2002).
- Y. Gurevich, M. Magidor and S. Shelah, The monadic theory of ω2. J. Symbolic Logic48 (1983) 387–398.
- J.E. Hopcroft, R. Motwani and J.D. Ullman, Introduction to automata theory, languages, and computation. Addison-Wesley Publishing Co., Reading, Mass. Addison-Wesley Series in Computer Science (2001).
- T. Jech, Set Theory, 3rd edition. Springer (2002).
- A. Kanamori, The Higher Infinite. Springer-Verlag (1997).
- A.S. Kechris, The theory of countable analytical sets. Trans. Amer. Math. Soc.202 (1975) 259–297.
- D. Kuske and M. Lohrey, First-order and counting theories of omega-automatic structures. J. Symbolic Logic73 (2008) 129–150.
- L.H. Landweber, Decision problems for ω-automata. Math. Syst. Theor.3 (1969) 376–384.
- H. Lescow and W. Thomas, Logical specifications of infinite computations, in A Decade of Concurrency, J.W. de Bakker, W.P. de Roever and G. Rozenberg, Eds. Lect. Notes Comput. Sci.803 (1994) 583–621.
- Y.N. Moschovakis, Descriptive set theory. North-Holland Publishing Co., Amsterdam (1980).
- I. Neeman, Finite state automata and monadic definability of singular cardinals. J. Symbolic Logic73 (2008) 412–438.
- D. Niwinski, On the cardinality of sets of infinite trees recognizable by finite automata, in Proceedings of the International Conference MFCS. Lect. Notes Comput. Sci.520 (1991) 367–376.
- D. Perrin and J.-E. Pin, Infinite words, automata, semigroups, logic and games. Pure Appl. Math.141 (2004).
- H. Rogers, Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967).
- L. Staiger, Hierarchies of recursive ω-languages. Elektronische Informationsverarbeitung und Kybernetik22 (1986) 219–241.
- L. Staiger, ω-languages, in Handbook of formal languages3. Springer, Berlin (1997) 339–387.
- W. Thomas, Automata on infinite objects, in Handbook of Theoretical Computer ScienceB, Formal models and semantics. J. van Leeuwen, Ed. Elsevier (1990) 135–191.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.