Computability

Finding all linear inequalities for entropies remains an important open question in information theory. For a long time the only known inequalities for entropies of tuples of random variables were Shannon (submodularity) inequalities. Only in 1998 Zhang and Yeung 1998 found the first inequality that cannot be represented as a convex combination of Shannon inequalities, and several other non-Shannon inequalities were found soon after that. It turned out that the class of linear inequalities for entropies is rather fundamental, since the same class can be equivalently defined in terms of subgroup sizes or projections of multidimensional sets (Chan 2001, Chan, Yeung 2002, Romashchenko, Shen, Vereshchagin 2000). The non-Shannon inequalities have interesting applications (e.g., to proofs of lower bounds for the information ratio of secret sharing schemes). Still the class of linear inequalities for entropies is not well understood, though some partial results are known (e.g., Matúš has shown in 2007 that this class cannot be generated by a finite family of inequalities).

This class also appears in algorithmic information theory: the same linear inequalities are true for Shannon entropies of tuples of random variables and Kolmogorov complexities of tuples of strings (Hammer et al., 1997). This parallelism started with the Kolmogorov–Levin formula 1968 for the complexity of pairs of strings with logarithmic precision. Longpré proved in 1986 a version of this formula for the space-bounded complexities.

In this paper we prove a stronger version of Longpré’s result with a tighter space bound, using Sipser’s trick 1980. Then, using this result, we show that every linear inequality that is true for complexities or entropies, is also true for space-bounded Kolmogorov complexities with a polynomial space overhead, thus extending the parallelism to the space-bounded algorithmic information theory.

The complexity of equivalence relations has received much attention in the recent literature. The main tool for such endeavour is the following reducibility: given equivalence relations R and S on natural numbers, R is computably reducible to S if there is a computable function f : ω → ω that induces an injective map from R-equivalence classes to S-equivalence classes. In order to compare the complexity of equivalence relations which are computable, researchers considered also feasible variants of computable reducibility, such as the polynomial-time reducibility. In this work, we explore Peq, the degree structure generated by primitive recursive reducibility on primitive recursive equivalence relations with infinitely many equivalence classes. In contrast with all other known degree structures on equivalence relations, we show that Peq has much more structure: e.g., we show that it is a dense distributive lattice. On the other hand, we also offer evidence of the intricacy of Peq, proving, e.g., that the structure is neither rigid nor homogeneous.

Inspired by the study of generic and coarse computability in computability theory, we extend such investigation to the context of computable model theory. In this paper, we continue our study initiated in the previous paper (Journal of Logic and Computation32 (2022) 581–607) , where we introduced and studied the notions of generically and coarsely computable structures and their generalizations. In this paper, we introduce the notions of generically and coarsely computable isomorphisms, and their weaker variants. We sometimes also require that the isomorphisms preserve the density structure. For example, for any coarsely computable structure A , there is a density preserving coarsely computable isomorphism from A to a computable structure. We demonstrate that each notion of generically and coarsely computable isomorphisms, density preserving or not, gives interesting insights into the structures we consider, focusing on various equivalence structures and injection structures.

In 1982, Soare and Stob proved that if A is an r.e. set which isn’t computable then there is a set of the form A ⊕ W e which isn’t of r.e. Turing degree. If we define a properly n + 1 -REA set to be an n + 1 -REA set which isn’t Turing equivalent to any n-REA set this result shows that every properly 1-REA set can be extended to a properly 2-REA set. This result was extended by Cholak and Hinman in 1994. They proved that every 2-REA set can be extended to a properly 3-REA set. This leads naturally to the hypothesis that every properly n-REA set can be extended to a properly n + 1 -REA set. Here we show this hypothesis is false and that there is a properly 3-REA set which can’t be extended to a properly 4-REA set. Moreover we show this set A can be Δ 2 0 .

In her 1990 thesis, Ahmad showed that there is a so-called “Ahmad pair”, i.e., there are incomparable Σ 2 0 -enumeration degrees  a 0 and  a 1 such that every enumeration degree x < a 0 is ⩽ a 1 . At the same time, she also showed that there is no “symmetric Ahmad pair”, i.e., there are no incomparable Σ 2 0 -enumeration degrees  a 0 and  a 1 such that every enumeration degree x 0 < a 0 is ⩽ a 1 and such that every enumeration degree x 1 < a 1 is ⩽ a 0 .

In this paper, we first present a direct proof of Ahmad’s second result. We then show that her first result cannot be extended to an “Ahmad triple”, i.e., there are no Σ 2 0 -enumeration degrees  a 0 , a 1 and  a 2 such that both ( a 0 , a 1 ) and ( a 1 , a 2 ) are an Ahmad pair. On the other hand, there is a “weak Ahmad triple”, i.e., there are pairwise incomparable Σ 2 0 -enumeration degrees  a 0 , a 1 and  a 2 such that every enumeration degree x < a 0 is also ⩽ a 1 or ⩽ a 2 ; however neither ( a 0 , a 1 ) nor ( a 0 , a 2 ) is an Ahmad pair.

In this paper, we study Hausdorff and Fourier dimension from the point of view of effective descriptive set theory and Type-2 Theory of Effectivity. Working in the hyperspace K ( X ) of compact subsets of X, with X = [ 0 , 1 ] d or X = R d , we characterize the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. This, in turn, allows us to show that family of all the closed Salem sets is Π 3 0 -complete. One of our main tools is a careful analysis of the effectiveness of a classical theorem of Kaufman. We furthermore compute the Weihrauch degree of the functions computing Hausdorff and Fourier dimension of closed sets.

We study the recently suggested effective Wadge hierarchy in effective spaces, concentrating on the non-collapse property. Along with hierarchies of sets, we study hierarchies of k-partitions which are interesting on their own. In particular, we establish sufficient conditions for the non-collapse of the effective Wadge hierarchy and apply them to some concrete spaces, including the discrete space of natural numbers, the Baire space, and the Cantor space. While the latter two cases are obtained by easy effectivization of the corresponding topological result by T. Kihara and A. Montalban, the first case needs a different technique and is certainly the main technical result of this paper.