On the existence of a connected component of a graph

  title={On the existence of a connected component of a graph},
  author={Kirill Gura and Jeffry L. Hirst and Carl Mummert},
We study the reverse mathematics and computability of countable graph theory, obtaining the following results. The principle that every countable graph has a connected component is equivalent to $\mathsf{ACA}_0$ over $\mathsf{RCA}_0$. The problem of decomposing a countable graph into connected components is strongly Weihrauch equivalent to the problem of finding a single component, and each is equivalent to its infinite parallelization. For graphs with finitely many connected components, the… 

Figures from this paper

Reverse Mathematics of Matroids

It is shown that the existence of bases for vector spaces of bounded dimension is equivalent to the induction scheme for $\Sigma^0_2$ formulas.

The uniform content of partial and linear orders

Weihrauch Reducibility and Finite-Dimensional Subspaces

In this thesis we study several principles involving subspaces and decompositions of vector spaces, matroids, and graphs from the perspective of Weihrauch reducibility. We study the problem of

Convex choice, finite choice and sorting

The main results are that choice for finite sets of cardinality i + 1 is reducible to choice for convex sets in dimension j, which in turn isredcible to sorting infinite sequences over an alphabet of size k iff i \leq j, and that sequential composition of one-dimensional convex choice is not reducible in any dimension.

Finite Choice, Convex Choice and Sorting

It is shown that choice for finite sets of cardinality \(i + 1\) is reducible to choice for convex sets in dimension j, which in turn isredcible to sorting infinite sequences over an alphabet of size k.

Measuring the Complexity of Computational Content : Weihrauch Reducibility and Reverse Analysis

This report documents the program and the outcomes of Dagstuhl Seminar 15392 “Measuring the Complexity of Computational Content: Weihrauch Reducibility and Reverse Analysis.” It includes abstracts on

A Complete Bibliography of Computability

above [CDHTM20]. admissible [Joh20]. affine [BA15]. Algebraic [DHS13]. algebras [AZ19]. algorithm [CD18, CD20]. Algorithmic [FrKHNS14, STZDG13, Muc16]. algorithmically [HTKT19]. Analog [PZ18, Mil20].

Weihrauch Complexity in Computable Analysis

A self-contained introduction into Weihrauch complexity and its applications to computable analysis and a survey on some classification results and a discussion of the relation to other approaches are provided.

qqqqqqqqqq qqqqqqqqqq " * ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

  • 2018



Connected components of graphs and reverse mathematics

The goal of this paper is to illustrate the use of reverse mathematics techniques in an accessible setting, relatively free of complicated coding, revealing an interesting connection between set comprehension axioms and induction schemes.

On the strength of the finite intersection principle

It is shown that there is a computable instance of FIP every solution of which has hyperimmune degree, and that every computable instances has a solution in every nonzero c.e. degree.

On uniform relationships between combinatorial problems

The enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic. In this setting, called reverse mathematics, one investigates which

On the (semi)lattices induced by continuous reducibilities

  • A. Pauly
  • Mathematics, Chemistry
    Math. Log. Q.
  • 2010
The order-theoretic properties of several variants of the two most important definitions of continuous reducibilities are studied, and suprema are shown to commutate with several characteristic numbers.

Weihrauch degrees, omniscience principles and weak computability

It is proved that parallelized LLPO is equivalent to Weak Kőnig's Lemma and hence to the Hahn–Banach Theorem in this new and very strong sense and any single-valued weakly computable operation is already computable in the ordinary sense.

Effective Choice and Boundedness Principles in Computable Analysis

This paper develops a number of separation techniques based on a new parallelization principle, on certain invariance properties of Weihrauch reducibility, on the Low Basis Theorem of Jockusch and Soare and based on the Baire Category Theorem.

Closed choice and a Uniform Low Basis Theorem

Computability of the Radon-Nikodym Derivative

It is proved that for every computable measurable space, RN is W-reducible to EC, and a computable measured space is constructed for which EC isW-reduced to RN.

Effective Borel measurability and reducibility of functions

A notion of Borel computability for single-valued as well as for multi-valued functions by a direct effectivization of the classical definition is introduced and this reducibility leads to a new and effective proof of the Banach-Hausdorff-Lebesgue Theorem.

Weihrauch Degrees of Finding Equilibria in Sequential Games

The degrees of non-computability of finding winning strategies in infinite sequential games with certain winning sets are considered and it is shown that as the complexity of the winning sets increases in the difference hierarchy, the difficulty of constructing winning strategies increases inThe effective Borel hierarchy.