Related papers: Computing in the Limit
Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…
We exhibit a sound and complete implicit-complexity formalism for functions feasibly computable by structural recursions over inductively defined data structures. Feasibly computable here means that the structural-recursive definition runs…
It is often useful to perform integration over learned functions represented by neural networks. However, this integration is usually performed numerically, as analytical integration over learned functions (especially neural networks) is…
We extend the classical notion of solvability to a lambda-calculus equipped with pattern matching. We prove that solvability can be characterized by means of typability and inhabitation in an intersection type system P based on…
Identity theorem for analytic complex functions says that a function is uniquely defined by its values on a set that contains a density point. The paper presents sufficient conditions for classes of real analytic functions that ensures…
Challenging the standard notion of totality in computable functions, one has that, given any sufficiently expressive formal axiomatic system, there are total functions that, although computable and "intuitively" understood as being total,…
D-finite functions and P-recursive sequences are defined in terms of linear differential and recurrence equations with polynomial coefficients. In this paper, we introduce a class of numbers closely related to D-finite functions and…
This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for…
Some type-based approaches to termination use sized types: an ordinal bound for the size of a data structure is stored in its type. A recursive function over a sized type is accepted if it is visible in the type system that recursive calls…
We study computable topological spaces and semicomputable and computable sets in these spaces. In particular, we investigate conditions under which semicomputable sets are computable. We prove that a semicomputable compact manifold $M$ is…
Claims about recursive self-improvement in AI often slide from repeated internal revision to the possibility of qualitatively stronger capability without clearly distinguishing the underlying computational regimes. This paper gives a formal…
We prove a computable version of de Finetti's theorem on exchangeable sequences of real random variables. As a consequence, exchangeable stochastic processes expressed in probabilistic functional programming languages can be automatically…
This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of Lp-computable functions (computable Lebesgue integrable functions) with a size notion, by introducing…
It is well known that limits can be computed by restricting along an initial functor, and that this often simplifies limit computation. We systematically study the algorithmic implications of this idea for diagrams indexed by a finite…
For a positive integer n, let f(n) denote the smallest non-negative integer b such that for each system S \subseteq {x_k=1,x_i+x_j=x_k,x_i*x_j=x_k: i,j,k \in {1,...,n}} with a solution in non-negative integers x_1,...,x_n, there exists a…
This paper describes a sequence of natural numbers that grows faster than any Turing computable function. This sequence is generated from a version of the tiling problem, called a coloring system. In our proof that generates the sequence,…
Computation is an input-output process, where a program encoding a problem to be solved is inserted into a machine that outputs a solution. Quantum computation conventionally relies on classical, external control outside the quantum…
To date, work on formalizing connectionist computation in a way that is at least Turing-complete has focused on recurrent architectures and developed equivalences to Turing machines or similar super-Turing models, which are of more…
In recent years we observed rapid and significant advancements in artificial intelligence (A.I.). So much so that many wonder how close humanity is to developing an A.I. model that can achieve human level of intelligence, also known as…
Whenever P is a proper definable forcing for adding a real, the countable support iteration of P has all the preservation properties it can possibly have, within a wide syntactically identified class of properties.