Related papers: Dense computability, upper cones, and minimal pair…
A relatively new topic in computability theory is the study of notions of computation that are robust against mistakes on some kind of small set. However, despite the recent popularity of this topic relatively foundational questions about…
Generic computability has been studied in group theory and we now study it in the context of classical computability theory. A set A of natural numbers is generically computable if there is a partial computable function f whose domain has…
We study connections between classical asymptotic density and c.e. sets. We prove that a c.e. Turing degree d is not low if and only if d contains a c.e. set A of density 1 which has no computable subsets of density 1, giving a natural…
For $r \in [0,1]$ we say that a set $A \subseteq \omega$ is \emph{coarsely computable at density} $r$ if there is a computable set $C$ such that $\{n : C(n) = A(n)\}$ has lower density at least $r$. Let $\gamma(A) = \sup \{r : A \hbox{ is…
In a previous paper, the author introduced the idea of intrinsic density --- a restriction of asymptotic density to sets whose density is invariant under computable permutation. We prove that sets with well-defined intrinsic density (and…
A coarse description of a subset A of omega is a subset D of omega such that the symmetric difference of A and D has asymptotic density 0. We study the extent to which noncomputable information can be effectively recovered from all coarse…
Partiality is a natural phenomenon in computability that we cannot get around. So, the question is whether we can give the areas where partiality occurs, that is, where non-termination happens, more structure. In this paper we consider…
Recent work in computability theory has focused on various notions of asymptotic computability, which capture the idea of a set being "almost computable." One potentially upsetting result is that all four notions of asymptotic computability…
A generic computation of a subset $A$ of $\mathbb{N}$ is a computation which correctly computes most of the bits of $A$, but which potentially does not halt on all inputs. The motivation for this concept is derived from complexity theory,…
In computable analysis, sequences of rational numbers which effectively converge to a real number x are used as the (rho-) names of x. A real number x is computable if it has a computable name, and a real function f is computable if there…
The coarse similarity class $[A]$ of $A$ is the set of all $B$ whose symmetric difference with $A$ has asymptotic density 0. There is a natural metric $\delta$ on the space $\mathcal{S}$ of coarse similarity classes defined by letting…
Regular functions from infinite words to infinite words can be equivalently specified by MSO-transducers, streaming $\omega$-string transducers as well as deterministic two-way transducers with look-ahead. In their one-way restriction, the…
We show that \emph{efficient Turing computability} at any fixed input/output precision implies the existence of \emph{compositionally sparse} (bounded-fan-in, polynomial-size) DAG representations and of corresponding neural approximants…
In this paper, we investigate the problem of synthesizing computable functions of infinite words over an infinite alphabet (data $\omega$-words). The notion of computability is defined through Turing machines with infinite inputs which can…
In this article we survey the development of generic and coarse computability and the main results on how classical asymptotic density interacts with the theory of computability.
Motivated by results on generic-case complexity in group theory, we apply the ideas of effective Baire category and effective measure theory to study complexity classes of functions which are "fractionally computable" by a partial…
Computational pseudorandomness studies the extent to which a random variable $\bf{Z}$ looks like the uniform distribution according to a class of tests $\cal{F}$. Computational entropy generalizes computational pseudorandomness by studying…
Measurable cones, with linear and measurable functions as morphisms, are a model of intuitionistic linear logic and of call-by-name probabilistic PCF which accommodates "continuous data types" such as the real line. So far however, they…
This is the first of a series of papers in which we study deep computations (ultracomputations) and deep iterates, formalizing the ideas of "asymptotic limit" of computations and compositional iterates, respectively. In this first paper of…
A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal…