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HIGHLY UNDECIDABLE PROBLEMS FOR INFINITE COMPUTATIONS
 THEORETICAL INFORMATICS AND APPLICATIONS
, 2009
"... We show that many classical decision problems about 1counter ωlanguages, context free ωlanguages, or infinitary rational relations, are Π 1 2complete, hence located at the second level of the analytical hierarchy, and “highly undecidable”. In particular, the universality problem, the inclusion ..."
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We show that many classical decision problems about 1counter ωlanguages, context free ωlanguages, or infinitary rational relations, are Π 1 2complete, 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 Π 1 2complete for contextfree ωlanguages or for infinitary rational relations. Topological and arithmetical properties of 1counter ωlanguages, context free ωlanguages, 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 1counter automata or 2tape automata.
Turing Incomparability in Scott Sets
 Proceedings of the American Mathematical Society
"... Abstract. For every Scott set F and every nonrecursive set X in F, there is a Y ∈ F such that X and Y are Turing incomparable. 1. ..."
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Abstract. For every Scott set F and every nonrecursive set X in F, there is a Y ∈ F such that X and Y are Turing incomparable. 1.
Low upper bounds of ideals
"... Abstract. We show that there is a low Tupper bound for the class of Ktrivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in ∆0 2 Tdegrees for which there is a low Tupper bound. 1. ..."
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Abstract. We show that there is a low Tupper bound for the class of Ktrivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in ∆0 2 Tdegrees for which there is a low Tupper bound. 1.
Decision Problems For Turing Machines
, 909
"... Université de Picardie, I.U.T. de l’Oise, site de Creil, ..."
Empiricism, Probability, and Knowledge of Arithmetic
"... The topic of this paper is the tenability of a certain type of empiricism about our knowledge of the Peano axioms. The Peano axioms constitute the standard contemporary axiomatization of arithmetic, and they consist of two parts, a set of eight axioms called Robinson’s Q, which ensure the correctnes ..."
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The topic of this paper is the tenability of a certain type of empiricism about our knowledge of the Peano axioms. The Peano axioms constitute the standard contemporary axiomatization of arithmetic, and they consist of two parts, a set of eight axioms called Robinson’s Q, which ensure the correctness of the addition and multiplication tables, and the principle of mathematical induction, whichsaysthatifzerohasagivenpropertyandn +1hasitwhenever n has it, then all natural numbers have this property (cf. Hájek & Pudlák [HP98] p.28, or Simpson [Sim09] p. 4). The type of empiricism about the Peano axioms which I want to consider holds that arithmetical knowledge is akin to the knowledge by which we infer from the past to the future, or from the observed to the unobserved. It is not uncommon today to hold that such inductive inferences can be rationally sustained by appeal to informed judgments of probability. The goal of this paper is to evaluate an empiricism which contends that judgements of probability can help us to secure knowledge of the Peano axioms. This empiricism merits our attention primarily because standard accounts of our knowledge of the Peano axioms face difficult problems, problems going above and beyond skepticism about knowledge of abstract objects. Logicism, for example, suggests that knowledge