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

Olivier Finkel

RAIRO - Theoretical Informatics and Applications (2012)

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

Abstract

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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 L ( 𝒜 ) L(𝒜) accepted by a Büchi 1-counter automaton 𝒜 𝒜. We prove the following surprising result: there exists a 1-counter Büchi automaton 𝒜 𝒜 such that the cardinality of the complement L ( 𝒜 ) - L(𝒜) −  of the ω-language L ( 𝒜 ) L(𝒜) is not determined by ZFC: (1) There is a model V1 of ZFC in which L ( 𝒜 ) - L(𝒜) −  is countable. (2) There is a model V2 of ZFC in which L ( 𝒜 ) - L(𝒜) −  has cardinal 2ℵ0. (3) There is a model V3 of ZFC in which L ( 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).

How to cite

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Finkel, 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 -

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