Approximation systems for functions in topological and in metric spaces

A notable feature of the TTE approach to computability is the representation of the argument values and the corresponding function values by means of infinitistic names. Two ways to eliminate the using of such names in certain cases are indicated in the paper. The first one is intended for the case of topological spaces with selected indexed denumerable bases. Suppose a partial function is given from one such space into another one whose selected base has a recursively enumerable index set, and suppose that the intersection of base open sets in the first space is computable in the sense of Weihrauch-Grubba. Then the ordinary TTE computability of the function is characterized by the existence of an appropriate recursively enumerable relation between indices of base sets containing the argument value and indices of base sets containing the corresponding function value.This result can be regarded as an improvement of a result of Korovina and Kudinov. The second way is applicable to metric spaces with selected indexed denumerable dense subsets. If a partial function is given from one such space into another one, then, under a semi-computability assumption concerning these spaces, the ordinary TTE computability of the function is characterized by the existence of an appropriate recursively enumerable set of quadruples. Any of them consists of an index of element from the selected dense subset in the first space, a natural number encoding a rational bound for the distance between this element and the argument value, an index of element from the selected dense subset in the second space and a natural number encoding a rational bound for the distance between this element and the function value. One of the examples in the paper indicates that the computability of real functions can be characterized in a simple way by using the first way of elimination of the infinitistic names.Comment: 21 pages, published in Logical Methods in Computer Scienc

Contemporary Mathematical Approaches to Computability Theory

In this paper, I present an introduction to computability theory and adopt contemporary mathematical definitions of computable numbers and computable functions to prove important theorems in computability theory. I start by exploring the history of computability theory, as well as Turing Machines, undecidability, partial recursive functions, computable numbers, and computable real functions. I then prove important theorems in computability theory, such that the computable numbers form a field and that the computable real functions are continuous

Neural-Symbolic Recursive Machine for Systematic Generalization

Despite the tremendous success, existing machine learning models still fall short of human-like systematic generalization -- learning compositional rules from limited data and applying them to unseen combinations in various domains. We propose Neural-Symbolic Recursive Machine (NSR) to tackle this deficiency. The core representation of NSR is a Grounded Symbol System (GSS) with combinatorial syntax and semantics, which entirely emerges from training data. Akin to the neuroscience studies suggesting separate brain systems for perceptual, syntactic, and semantic processing, NSR implements analogous separate modules of neural perception, syntactic parsing, and semantic reasoning, which are jointly learned by a deduction-abduction algorithm. We prove that NSR is expressive enough to model various sequence-to-sequence tasks. Superior systematic generalization is achieved via the inductive biases of equivariance and recursiveness embedded in NSR. In experiments, NSR achieves state-of-the-art performance in three benchmarks from different domains: SCAN for semantic parsing, PCFG for string manipulation, and HINT for arithmetic reasoning. Specifically, NSR achieves 100% generalization accuracy on SCAN and PCFG and outperforms state-of-the-art models on HINT by about 23%. Our NSR demonstrates stronger generalization than pure neural networks due to its symbolic representation and inductive biases. NSR also demonstrates better transferability than existing neural-symbolic approaches due to less domain-specific knowledge required

Logic and the Challenge of Computer Science

https://deepblue.lib.umich.edu/bitstream/2027.42/154161/1/39015099114889.pd

Toward an Epistemic Web

In the beginning knowledge was local. With the development of more complex forms of economic organization knowledge began to travel. The Library of Alexandria was the fulfillment - however partial and transitory - of a vision to bring together all the knowledge of the world. But to obtain the knowledge one had to go to Alexandria. Today the World Wide Web promises to make universally accessible the knowledge of a world grown larger. To be sure, much work remains to be done: many documents need to be made available (i.e. digitized if they are not already, and freed from restrictive access controls); and various biases (economic, legal, linguistic, social, technological) need to be overcome. But what do we do with this knowledge? Is it enough to create a digital library of Alexandria, with (perhaps) improved finding aids? We propose that the crucial question is how to structure knowledge on the Web to facilitate the construction of new knowledge, knowledge that will be critical in addressing the challenges of the emerging global society. We begin by asking three questions about the Web and its future. In the remainder of the paper we explore the possibility of an Epistemic Web in the context of a more general discussion of knowledge representation technologies, technologies used for storing, manipulating and spreading knowledge

Toward an Epistemic Web

In the beginning knowledge was local. With the development of more complex forms of economic organization knowledge began to travel. The Library of Alexandria was the fulfillment - however partial and transitory - of a vision to bring together all the knowledge of the world. But to obtain the knowledge one had to go to Alexandria. Today the World Wide Web promises to make universally accessible the knowledge of a world grown larger. To be sure, much work remains to be done: many documents need to be made available (i.e. digitized if they are not already, and freed from restrictive access controls); and various biases (economic, legal, linguistic, social, technological) need to be overcome. But what do we do with this knowledge? Is it enough to create a digital library of Alexandria, with (perhaps) improved finding aids? We propose that the crucial question is how to structure knowledge on the Web to facilitate the construction of new knowledge, knowledge that will be critical in addressing the challenges of the emerging global society. We begin by asking three questions about the Web and its future. In the remainder of the paper we explore the possibility of an Epistemic Web in the context of a more general discussion of knowledge representation technologies, technologies used for storing, manipulating and spreading knowledge