Related papers: -Generic Computability, Turing Reducibility and As…
Computational feasibility is a widespread concern that guides the framing and modeling of biological and artificial intelligence. The specification of cognitive system capacities is often shaped by unexamined intuitive assumptions about the…
Depth is a complexity measure for natural systems of the kind studied in statistical physics and is defined in terms of computational complexity. Depth quantifies the length of the shortest parallel computation required to construct a…
We give a survey of the known connections between regularity conditions and amenability conditions in the setting of uniform algebras. For a uniform algebra $A$ we consider the set, $A_{lc}$, of functions in $A$ which are locally constant…
Algorithmic randomness theory starts with a notion of an individual random object. To be reasonable, this notion should have some natural properties; in particular, an object should be random with respect to image distribution if and only…
Estimation of density functions supported on general domains arises when the data is naturally restricted to a proper subset of the real space. This problem is complicated by typically intractable normalizing constants. Score matching…
We construct an increasing $\omega$-sequence $(a_n)$ of Turing degrees which forms an initial segment of the Turing degrees, and such that each~$a_{n+1}$ is diagonally noncomputable relative to $a_n$. It follows that the~$\mathsf{DNR}$…
We define a class of functions termed "Computable in the Limit", based on the Machine Learning paradigm of "Identification in the Limit". A function is Computable in the Limit if it defines a property P_p of a recursively enumerable class A…
In this survey we discuss work of Levin and V'yugin on collections of sequences that are non-negligible in the sense that they can be computed by a probabilistic algorithm with positive probability. More precisely, Levin and V'yugin…
We study the position of the computable setting in the "common theory of locality" developed in arXiv:2106.02066 and arXiv:2204.09329 for local problems on $\Delta$-regular trees, $\Delta \in \omega$. We show that such a problem admits a…
In this paper we extend the approach of M. Cavaleri to effective amenability to the class of computably enumerable groups, i.e. in particular we do not assume that groups are finitely generated. In the case of computable groups we also…
The common idea behind complexity reduction in physical systems is separation of information into "physically meaningful" and "safely ignorable". Here we consider a generic notion of such separation -- implemented by coarse-graining the…
Suppose you have an uncomputable set $X$ and you want to find a set $A$, all of whose infinite subsets compute $X$. There are several ways to do this, but all of them seem to produce a set $A$ which is fairly sparse. We show that this is…
Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence in terms of relatively simple invariants. Where…
In this article, we will show that uncomputability is a relative property not only of oracle Turing machines, but also of subrecursive classes. We will define the concept of a Turing submachine, and a recursive relative version for the Busy…
Taking the view that computation is after all physical, we argue that physics, particularly quantum physics, could help extend the notion of computability. Here, we list the important and unique features of quantum mechanics and then…
We investigate the computational properties of basic mathematical notions pertaining to $\mathbb{R}\rightarrow \mathbb{R}$-functions and subsets of $\mathbb{R}$, like finiteness, countability, (absolute) continuity, bounded variation,…
This short survey of recent work in tile self-assembly discusses the use of simulation to classify and separate the computational and expressive power of self-assembly models. The journey begins with the result that there is a single…
We discuss the possibility of constructing a function that validates the definition or not definition of the partial recursive functions of one variable. This is a topic in computability theory, which was first approached by Alan M. Turing…
The word problem for discrete groups is well-known to be undecidable by a Turing Machine; more precisely, it is reducible both to and from and thus equivalent to the discrete Halting Problem. The present work introduces and studies a real…
In this paper we shall relate computational complexity to the principle of natural selection. We shall do this by giving a philosophical account of complexity versus universality. It seems sustainable to equate universal systems to complex…