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

Olivier Finkel (2011)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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(x1d49c;) accepted by a Büchi 1-counter automaton 𝒜 x1d49c;. We prove the following surprising result: there exists a 1-counter Büchi automaton 𝒜 x1d49c; such that the cardinality of the complement L ( 𝒜 ) - L(𝒜) −  of the ω-language L ( 𝒜 ) L(𝒜) is not determined...

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

Olivier Finkel (2012)

RAIRO - Theoretical Informatics and Applications

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...

Wadge degrees of ω -languages of deterministic Turing machines

Victor Selivanov (2003)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We describe Wadge degrees of ω -languages recognizable by deterministic Turing machines. In particular, it is shown that the ordinal corresponding to these degrees is ξ ω where ξ = ω 1 CK is the first non-recursive ordinal known as the Church–Kleene ordinal. This answers a question raised in [2].

Wadge Degrees of ω-Languages of Deterministic Turing Machines

Victor Selivanov (2010)

RAIRO - Theoretical Informatics and Applications

We describe Wadge degrees of ω-languages recognizable by deterministic Turing machines. In particular, it is shown that the ordinal corresponding to these degrees is ξω where ξ = ω1CK is the first non-recursive ordinal known as the Church–Kleene ordinal. This answers a question raised in [2].

What machines can and cannot do.

Luis M. Laita, Roanes-Lozano, Luis De Ledesma Otamendi (2007)

RACSAM

In this paper, the questions of what machines cannot do and what they can do will be treated by examining the ideas and results of eminent mathematicians. Regarding the question of what machines cannot do, we will rely on the results obtained by the mathematicians Alan Turing and Kurt G¨odel. Turing machines, their purpose of defining an exact definition of computation and the relevance of Church-Turing thesis in the theory of computability will be treated in detail. The undecidability of the “Entscheidungsproblem”...

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