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Creators/Authors contains: "VALENTI, MANLIO"

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  1. ; ; (, The Journal of Symbolic Logic)
    Abstract We study versions of the tree pigeonhole principle,$$\mathsf {TT}^1$$, in the context of Weihrauch-style computable analysis. The principle has previously been the subject of extensive research in reverse mathematics, an outstanding question of which investigation is whether$$\mathsf {TT}^1$$is$$\Pi ^1_1$$-conservative over the ordinary pigeonhole principle,$$\mathsf {RT}^1$$. Using the recently introduced notion of the first-order part of an instance-solution problem, we formulate the analog of this question for Weihrauch reducibility, and give an affirmative answer. In combination with other results, we use this to show that unlike$$\mathsf {RT}^1$$, the problem$$\mathsf {TT}^1$$is not Weihrauch requivalent to any first-order problem. Our proofs develop new combinatorial machinery for constructing and understanding solutions to instances of$$\mathsf {TT}^1$$. 
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    Free, publicly-accessible full text available February 11, 2026
  2. ; ; ; ; (, Proceedings of the American Mathematical Society)
    In this paper, we study the existence of minimal covers and strong minimal covers in the Weihrauch degrees. We characterize when a problem  f f is a minimal cover or strong minimal cover of a problem  h h . We show that strong minimal covers only exist in the cone below  i d id and that the Weihrauch lattice above  i d id is dense. From this, we conclude that the degree of i d id is first-order definable in the Weihrauch degrees and that the first-order theory of the Weihrauch degrees is computably isomorphic to third-order arithmetic. 
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