The lettericity of a graph G=(V,E) is defined as the smallest size of an alphabet such that there is a word w_1 … w_|V|∈ ^* and a decoder 𝒟⊆ ^2 with the property that G is isomorphic to the letter graph G(𝒟, w) , that is, the graph with vertex set {1, … , n} and edge set {ij | 1≤ i < j ≤ n, w_iw_j ∈𝒟} . Note that G(𝒟, w) can be seen as a graph with inherent coloring χ :V(G) → . It is unknown whether the lettericity of a given graph can be computed in polynomial time. The problem to determine the lettericity of a given graph is called the lettericity problem. As a step towards answering the complexity of this problem, we investigate the following retrieval problems: given a graph G together with two of the three solution-objects (word w, decoder 𝒟 , and coloring χ ), the goal is to compute the third solution-object. We show that word retrieval and decoder retrieval are solvable in polynomial time, while coloring retrieval is equivalent to the graph isomorphism problem. Beyond this, we introduce symmetric lettericity which is a restricted version of lettericity where each decoder needs to be symmetrical ( ab∈𝒟 if and only if ba∈𝒟 ). As we show, the symmetric lettericity of a graph always equals the neighborhood diversity of the graph, which in fact can be computed in linear time.
In this paper we provide an easy proof of Barmpalias–Lewis-Pye result saying that all computable increasing sequences converging to random reals converge at the same speed by noting that it easily follows from Bishop’s upcrossing inequality. We also provide a simple derivation of this inequality.
We investigate spaces of spaces in the category of represented spaces and type-2 computable maps. Concretely, such spaces of spaces are given by bundles whose bases are names for spaces. We give natural examples of such bundles for Polish and compact Polish spaces and show they are essentially equivalent. We then propose definitions of genericity and computable categoricity, according to which the Cantor space is computably categorical and generic as a compact Polish space. We also show that the degree of categoricity of 𝒮^1 is the Weihrauch degree lim .
The paper studies learnability from positive data for families of existentially definable subsets in a given computable structure 𝒮 . While provided larger and larger pieces of a subset V of the domain of 𝒮 , a learner tries to guess the Gödel number of an ∃ -formula which defines V in 𝒮 . In this setting, we consider several classical learning criteria. For a computable structure 𝒮 , we work with the class ^0_1(𝒮) containing all ∃ -definable (without parameters) subsets of 𝒮 . We focus on some familiar classes of structures 𝒮 , including equivalence structures and Boolean algebras. We establish the following results. If the ∀∃ -theory of a structure 𝒮 is decidable, then the notions of explanatory learnability and behaviorally correct learnability for families ℱ⊆ ^0_1(𝒮) coincide. If 𝒮 is a computable equivalence structure, then the notions of vacillatory learnability and behaviorally correct learnability for such families ℱ coincide. We build a computable equivalence structure ℰ such that ^0_1(ℰ) is vacillatorily learnable but not explanatorily learnable. We prove that every computable Boolean algebra has a computable isomorphic copy 𝒞 such that ^0_1(𝒞) is confidently learnable. We construct a computable Heyting algebra ℋ such that for any computable copy 𝒞 of ℋ , the family ^0_1(𝒞) is not behaviorally correctly learnable.
