Related papers: Discrete Transfinite Computation
The architecture of neural Turing machines is differentiable end to end and is trainable with gradient descent methods. Due to their large unfolded depth Neural Turing Machines are hard to train and because of their linear access of…
We investigate infinitary wellfounded systems for linear logic with fixed points, with transfinite branching rules indexed by some closure ordinal $\alpha$ for fixed points. Our main result is that provability in the system for some…
The Turing Machine has two implicit properties that depend on its underlying notion of computing: the format is fully determinate and computations are information preserving. Distributed representations lack these properties and cannot be…
This work exposes which mechanisms and procesess in the Nature of evolution compute a function not computable by Turing machine. The computer with intelligence that is not higher than one bacteria population could have, but with efficency…
Recent advances in depth-recurrent language models show that recurrence can decouple train-time compute and parameter count from test-time compute. In this work, we study how to convert existing pretrained non-recurrent language models into…
Length generalization refers to the ability to extrapolate from short training sequences to long test sequences and is a challenge for current large language models. While prior work has proposed some architecture or data format changes to…
We derive computationally tractable methods to select a small subset of experiment settings from a large pool of given design points. The primary focus is on linear regression models, while the technique extends to generalized linear models…
We develop theory and applications of forward characteristic processes in discrete time following a seminal paper of Jan Kallsen and Paul Kr\"uhner. Particular emphasis is placed on the dynamics of volatility surfaces which can be easily…
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…
The calculus of Dependent Object Types (DOT) has enabled a more principled and robust implementation of Scala, but its support for type-level computation has proven insufficient. As a remedy, we propose $F^\omega_{..}$, a rigorous…
Leivant's ramified recurrence is one of the earliest examples of an implicit characterization of the polytime functions as a subalgebra of the primitive recursive functions. Leivant's result, however, is originally stated and proved only…
We discuss some claims that certain UCOMP devices can perform hypercomputation (compute Turing-uncomputable functions) or perform super-Turing computation (solve NP-complete problems in polynomial time). We discover that all these claims…
One of the fundamental results in computability is the existence of well-defined functions that cannot be computed. In this paper we study the effects of data representation on computability; we show that, while for each possible way of…
We define a model of advised computation by finite automata where the advice is provided on a separate tape. We consider several variants of the model where the advice is deterministic or randomized, the input tape head is allowed…
We define an on-line (incremental) algorithm that, given a (possibly infinite) pseudo-transitive oriented graph, produces a transitive reorientation. This implies that a theorem of Ghouila-Houri is provable in RCA_0 and hence is computably…
Generalized inverses of tensors play increasingly important roles in computational mathematics and numerical analysis. It is appropriate to develop the theory of generalized inverses of tensors within the algebraic structure of a ring. In…
Recursive calls over recursive data are useful for generating probability distributions, and probabilistic programming allows computations over these distributions to be expressed in a modular and intuitive way. Exact inference is also…
A type system is introduced for a generic Object Oriented programming language in order to infer resource upper bounds. A sound andcomplete characterization of the set of polynomial time computable functions is obtained. As a consequence,…
In this note we consider a new variant of network of splicing processors which simplifies the general model such that filters remain associated with nodes but the input and output filters of every node coincide. This variant is called {\it…
We propose a type system to analyze the time consumed by multi-threaded imperative programs with a shared global memory, which delineates a class of safe multi-threaded programs. We demonstrate that a safe multi-threaded program runs in…