5 research outputs found
Computability and Complexity
We study the uniform computational content of the Vitali Covering Theorem for
intervals using the tool of Weihrauch reducibility. We show that a more
detailed picture emerges than what a related study by Giusto, Brown, and
Simpson has revealed in the setting of reverse mathematics. In particular,
different formulations of the Vitali Covering Theorem turn out to have
different uniform computational content. These versions are either computable
or closely related to uniform variants of Weak Weak K\H{o}nig's Lemma.Comment: 13 page
Connected Choice and the Brouwer Fixed Point Theorem
We study the computational content of the Brouwer Fixed Point Theorem in the
Weihrauch lattice. Connected choice is the operation that finds a point in a
non-empty connected closed set given by negative information. One of our main
results is that for any fixed dimension the Brouwer Fixed Point Theorem of that
dimension is computably equivalent to connected choice of the Euclidean unit
cube of the same dimension. Another main result is that connected choice is
complete for dimension greater than or equal to two in the sense that it is
computably equivalent to Weak K\H{o}nig's Lemma. While we can present two
independent proofs for dimension three and upwards that are either based on a
simple geometric construction or a combinatorial argument, the proof for
dimension two is based on a more involved inverse limit construction. The
connected choice operation in dimension one is known to be equivalent to the
Intermediate Value Theorem; we prove that this problem is not idempotent in
contrast to the case of dimension two and upwards. We also prove that Lipschitz
continuity with Lipschitz constants strictly larger than one does not simplify
finding fixed points. Finally, we prove that finding a connectedness component
of a closed subset of the Euclidean unit cube of any dimension greater or equal
to one is equivalent to Weak K\H{o}nig's Lemma. In order to describe these
results, we introduce a representation of closed subsets of the unit cube by
trees of rational complexes.Comment: 36 page
Le direzioni della logica in Italia: la reverse mathematics e l'analisi computazionale
Nelle conversazioni tra matematici non \ue8 infrequente sentire affermazioni del tipo \u201ci teoremi \u3a6 e \u3a8 sono equivalenti\u201d, oppure \u201cil teorema \u3a6 \ue8 pi\uf9 forte del teorema \u3a8\u201d. Dato che \u3a6 e \u3a8 (essendo teoremi) sono entrambi dimostrabili, prendendo alla lettera le due affermazioni abbiamo che la prima \ue8 banalmente vera e la seconda banalmente falsa. Sappiamo tutti per\uf2 che queste affermazioni hanno un altro significato, molto meno banale, e c\u2019\ue8 quindi una ragione per cui vengono fatte. Negli ultimi decenni la logica matematica ha sviluppato alcuni strumenti in grado di rendere precise, e suscettibili di dimostrazione o refutazione, affermazioni come le precedenti. In particolare ci riferiamo alla reverse mathematics e all\u2019analisi computazionale. Questi sono due programmi di ricerca di origine diverse che nell\u2019ultimo decennio, anche grazie al contributo di alcuni ricercatori italiani, hanno trovato significativi punti di contatto. In questo lavoro presenteremo i due programmi, con particolare riferimento alle loro aree di contatto. Evidenzieremo in particolare i contributi dei ricercatori italiani attivi in queste aree, e concluderemo indicando alcune prospettive di sviluppo su cui anche in Italia si sta cercando di lavorare