Olivier Finkel
We give in this paper an example of infinitary rational relation, accepted by a 2-tape Büchi automaton, which is Pi^0_3-complete in the Borel hierarchy. Moreover the example of infinitary rational relation given in this paper has a very simple structure and can be easily described by its sections.
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We prove in this paper that there exists some infinitary rational relations which are analytic but non Borel sets, giving an answer to a question of Simonnet [Automates et Théorie Descriptive, Ph. D. Thesis, Université Paris 7, March 1992].
We prove in this paper that there exists some infinitary rational relations which are Sigma^0_3-complete Borel sets and some others which are Pi^0_3-complete. This implies that there exists some infinitary rational relations which are Delta^0_4-sets but not (Sigma^0_3U Pi^0_3)-sets. These results give additional answers to questions of Simonnet and of Lescow and Thomas.
We show that, from the topological point of view, 2-tape Büchi automata have the same accepting power as Turing machines equipped with a Büchi acceptance condition. The Borel and the Wadge hierarchies of the class RAT_omega of infinitary rational relations accepted by 2-tape Büchi automata are equal to the Borel and the Wadge hierarchies of omega-languages accepted by real-time Büchi 1-counter automata or by Büchi Turing machines. In particular, for every non-null recursive ordinal $α$, there exist some $Σ^0_α$-complete and some $Π^0_α$-complete infinitary rational relations. And the supremum …
We show that many classical decision problems about 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are $Π_2^1$-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 $Π_2^1$-complete for context-free omega-languages or for infinitary rational relations. Topological and arithmetical properties of 1-counter omega-languages, c…
We consider model-theoretic properties related to the expressive power of three analogues of $L_{ω_1, ω}$ for metric structures. We give an example showing that one of these infinitary logics is strictly more expressive than the other two, but also show that all three have the same elementary equivalence relation for complete separable metric structures. We then prove that a continuous function on a complete separable metric structure is automorphism invariant if and only if it is definable in the more expressive logic. Several of our results are related to the existence of Scott sentences for…
We prove the following surprising result: there exist a 1-counter Büchi automaton and a 2-tape Büchi automaton such that the ω-language of the first and the infinitary rational relation of the second in one model of ZFC are π_2^0-sets, while in a different model of ZFC both are analytic but non Borel sets. This shows that the topological complexity of an ω-language accepted by a 1-counter Büchi automaton or of an infinitary rational relation accepted by a 2-tape Büchi automaton is not determined by the axiomatic system ZFC. We show that a similar result holds for the class of languages of infi…
Given a weakly compact cardinal $κ$, we give an axiomatization of intuitionistic first-order logic over $\mathcal{L}_{κ^+, κ}$ and prove it is sound and complete with respect to Kripke models. As a consequence we get the disjunction and existence properties for that logic. This generalizes the work of Nadel for intuitionistic logic over $\mathcal{L}_{ω_1, ω}$. When $κ$ is a regular cardinal such that $κ^{<κ}=κ$, we deduce, by an easy modification of the proof, a complete axiomatization of intuitionistic first-order logic over $\mathcal{L}_{κ^+, κ, κ}$, the language with disjunctions of at most…
It was noticed by Harel in [Har86] that "one can define $Σ_1^1$-complete versions of the well-known Post Correspondence Problem". We first give a complete proof of this result, showing that the infinite Post Correspondence Problem in a regular $ω$-language is $Σ_1^1$-complete, hence located beyond the arithmetical hierarchy and highly undecidable. We infer from this result that it is $Π_1^1$-complete to determine whether two given infinitary rational relations are disjoint. Then we prove that there is an amazing gap between two decision problems about $ω$-rational functions realized by finite …
We prove that the determinacy of Gale-Stewart games whose winning sets are infinitary rational relations accepted by 2-tape Büchi automata is equivalent to the determinacy of (effective) analytic Gale-Stewart games which is known to be a large cardinal assumption. Then we prove that winning strategies, when they exist, can be very complex, i.e. highly non-effective, in these games. We prove the same results for Gale-Stewart games with winning sets accepted by real-time 1-counter Büchi automata, then extending previous results obtained about these games. Then we consider the strenghs of determi…
A note on restriction rules
本文研究限制规则，证明在仅需推导给定模型完全理论时，可省略可数性假设，并将巴怀斯定理推广至任意基数词汇表及超紧基数下的无穷逻辑。
The class of abelian $p$-groups are an example of some very interesting phenomena in computable structure theory. We will give an elementary first-order theory $T_p$ whose models are each bi-interpretable with the disjoint union of an abelian $p$-group and a pure set (and so that every abelian $p$-group is bi-interpretable with a model of $T_p$) using computable infinitary formulas. This answers a question of Knight by giving an example of an elementary first-order theory of "Ulm type": Any two models, low for $ω_1^{CK}$, and with the same computable infinitary theory, are isomorphic. It also …
We show that, from a topological point of view, 2-tape Büchi automata have the same accepting power than Turing machines equipped with a Büchi acceptance condition. In particular, we show that for every non null recursive ordinal alpha, there exist some Sigma^0_alpha-complete and some Pi^0_alpha-complete infinitary rational relations accepted by 2-tape Büchi automata. This very surprising result gives answers to questions of W. Thomas [Automata and Quantifier Hierarchies, in: Formal Properties of Finite automata and Applications, Ramatuelle, 1988, LNCS 386, Springer, 1989, p.104-119], of P. Si…
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