The Bolzano–Weierstrass Theorem is the jump of Weak Kőnig’s Lemma

Abstract

We classify the computational content of the Bolzano-Weierstraß Theorem and variants thereof in the Weihrauch lattice. For this purpose we first introduce the concept of a derivative or jump in this lattice and we show that it has some properties similar to the Turing jump. Using this concept we prove that the derivative of closed choice of a computable metric space is the cluster point problem of that space. By specialization to sequences with a relatively compact range we obtain a characterization of the Bolzano-Weierstraß Theorem as the derivative of compact choice. In particular, this shows that the Bolzano-Weierstraß Theorem on real numbers is the jump of Weak Kőnig's Lemma. Likewise, the Bolzano-Weierstraß Theorem on the binary space is the jump of the lesser limited principle of omniscience LLPO and the Bolzano-Weierstraß Theorem on natural numbers can be characterized as the jump of the idempotent closure of LLPO (which is the jump of the finite parallelization of LLPO). We also introduce the compositional product of two Weihrauch degrees f and g as the supremum of the composition of any two functions below f and g, respectively. Using this concept we can express the main result such that the Bolzano-Weierstraß Theorem is the compositional product of Weak Kőnig's Lemma and the Monotone Convergence Theorem. We also study the class of weakly limit computable functions, which are functions that can be obtained by composition of weakly computable functions with limit computable functions. We prove that the Bolzano-Weierstraß Theorem on real numbers is complete for this class. Likewise, the unique cluster point problem on real numbers is complete for the class of functions that are limit computable with finitely many mind changes. We also prove that the Bolzano-Weierstraß Theorem on real numbers and, more generally, the unbounded cluster point problem on real numbers is uniformly low limit computable. Finally, we also provide some separation techniques that allow to prove non-reducibilities between certain variants of the Bolzano-Weierstraß Theorem.

Game Characterizations and Lower Cones in the Weihrauch Degrees

“…One of the reasons for their relevance is that they induce endofunctors on the category of represented spaces, which in turn can characterize function classes in DST [PdB13]. The term transparent was coined in [BGM12]. Our extension of the concept to multi-valued functions between represented spaces is rather straightforward, but requires the use of the notion of tightening.…”

“…Our extension of the concept to multi-valued functions between represented spaces is rather straightforward, but requires the use of the notion of tightening. Note that if f : ⊆ X ⇒ Y is transparent, then for every y ∈ Y there is some x ∈ dom(f ) with f (x) = {y}, i.e., f is slim in the terminology of [BGM12,Definition 2.7].…”

Addendum to: “The Bolzano–Weierstrass theorem is the jump of weak Kőnig's lemma” [Ann. Pure Appl. Logic 163 (6) (2012) 623–655]

“…The reduction BWT X ≤ sW K ′ X has been proved in [1], so we focus on the reduction K ′ X ≤ sW BWT X . Let (X, d, α) be a computable metric space and let K ⊆ X be a nonempty compact set given by a κ ′ − -name p i i .…”

“…✷ We mention that it is well known that a subset of a metric space is totally bounded if and only if any sequence in it has a Cauchy subsequence [2, Exercise 4.3.A (a)]. Now we use the previous two lemmas to complete the proof of [1,Theorem 11.2]. Within the proof we use the canonical completionX of a computable metric space.…”

“…While the reduction BWT X ≤ sW K ′ X was proved correctly in [1], the proof provided for K ′ X ≤ sW BWT X contains a gap and is only correct for the special case of compact X as it stands. This fact was pointed out by one of us (M. Schröder) and is due to the fact that in general the closure of L −1 X (K) is not compact.…”