220,419 research outputs found
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
Descriptive Complexity for Counting Complexity Classes
Descriptive Complexity has been very successful in characterizing complexity
classes of decision problems in terms of the properties definable in some
logics. However, descriptive complexity for counting complexity classes, such
as FP and #P, has not been systematically studied, and it is not as developed
as its decision counterpart. In this paper, we propose a framework based on
Weighted Logics to address this issue. Specifically, by focusing on the natural
numbers we obtain a logic called Quantitative Second Order Logics (QSO), and
show how some of its fragments can be used to capture fundamental counting
complexity classes such as FP, #P and FPSPACE, among others. We also use QSO to
define a hierarchy inside #P, identifying counting complexity classes with good
closure and approximation properties, and which admit natural complete
problems. Finally, we add recursion to QSO, and show how this extension
naturally captures lower counting complexity classes such as #L
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Understanding analogical reasoning : viewpoints from psychology and related disciplines
Analogy and metaphor have a long history of study in linguistics, education, philosophy and psychology. Consensus over what analogy is or how analogy functions in language and thought, however, has been elusive. This paper, the first in a two part series, examines these various research traditions, attempting to bring out major lines of agreement over the role of analogy in individual human experience. As well as being a general literature review which may be helpful for newcomers to the study of analogy, this paper attempts to extract from these literatures existing theories, models and concepts which may be interesting or useful for computational studies of analogical reasoning
Levels of discontinuity, limit-computability, and jump operators
We develop a general theory of jump operators, which is intended to provide
an abstraction of the notion of "limit-computability" on represented spaces.
Jump operators also provide a framework with a strong categorical flavor for
investigating degrees of discontinuity of functions and hierarchies of sets on
represented spaces. We will provide a thorough investigation within this
framework of a hierarchy of -measurable functions between arbitrary
countably based -spaces, which captures the notion of computing with
ordinal mind-change bounds. Our abstract approach not only raises new questions
but also sheds new light on previous results. For example, we introduce a
notion of "higher order" descriptive set theoretical objects, we generalize a
recent characterization of the computability theoretic notion of "lowness" in
terms of adjoint functors, and we show that our framework encompasses ordinal
quantifications of the non-constructiveness of Hilbert's finite basis theorem
The SP theory of intelligence: benefits and applications
This article describes existing and expected benefits of the "SP theory of
intelligence", and some potential applications. The theory aims to simplify and
integrate ideas across artificial intelligence, mainstream computing, and human
perception and cognition, with information compression as a unifying theme. It
combines conceptual simplicity with descriptive and explanatory power across
several areas of computing and cognition. In the "SP machine" -- an expression
of the SP theory which is currently realized in the form of a computer model --
there is potential for an overall simplification of computing systems,
including software. The SP theory promises deeper insights and better solutions
in several areas of application including, most notably, unsupervised learning,
natural language processing, autonomous robots, computer vision, intelligent
databases, software engineering, information compression, medical diagnosis and
big data. There is also potential in areas such as the semantic web,
bioinformatics, structuring of documents, the detection of computer viruses,
data fusion, new kinds of computer, and the development of scientific theories.
The theory promises seamless integration of structures and functions within and
between different areas of application. The potential value, worldwide, of
these benefits and applications is at least $190 billion each year. Further
development would be facilitated by the creation of a high-parallel,
open-source version of the SP machine, available to researchers everywhere.Comment: arXiv admin note: substantial text overlap with arXiv:1212.022
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The effect of multiple knowledge sources on learning and teaching
Current paradigms for machine-based learning and teaching tend to perform their task in isolation from a rich context of existing knowledge. In contrast, the research project presented here takes the view that bringing multiple sources of knowledge to bear is of central importance to learning in complex domains. As a consequence teaching must both take advantage of and beware of interactions between new and existing knowledge. The central process which connects learning to its context is reasoning by analogy, a primary concern of this research. In teaching, the connection is provided by the explicit use of a learning model to reason about the choice of teaching actions. In this learning paradigm, new concepts are incrementally refined and integrated into a body of expertise, rather than being evaluated against a static notion of correctness. The domain chosen for this experimentation is that of learning to solve "algebra story problems." A model of acquiring problem solving skills in this domain is described, including: representational structures for background knowledge, a problem solving architecture, learning mechanisms, and the role of analogies in applying existing problem solving abilities to novel problems. Examples of learning are given for representative instances of algebra story problems. After relating our views to the psychological literature, we outline the design of a teaching system. Finally, we insist on the interdependence of learning and teaching and on the synergistic effects of conducting both research efforts in parallel
The Inhuman Overhang: On Differential Heterogenesis and Multi-Scalar Modeling
As a philosophical paradigm, differential heterogenesis offers us a novel descriptive vantage with which to inscribe Deleuze’s virtuality within the terrain of “differential becoming,” conjugating “pure saliences” so as to parse economies, microhistories, insurgencies, and epistemological evolutionary processes that can be conceived of independently from their representational form. Unlike Gestalt theory’s oppositional constructions, the advantage of this aperture is that it posits a dynamic context to both media and its analysis, rendering them functionally tractable and set in relation to other objects, rather than as sedentary identities. Surveying the genealogy of differential heterogenesis with particular interest in the legacy of Lautman’s dialectic, I make the case for a reading of the Deleuzean virtual that departs from an event-oriented approach, galvanizing Sarti and Citti’s dynamic a priori vis-à -vis Deleuze’s philosophy of difference. Specifically, I posit differential heterogenesis as frame with which to examine our contemporaneous epistemic shift as it relates to multi-scalar computational modeling while paying particular attention to neuro-inferential modes of inductive learning and homologous cognitive architecture. Carving a bricolage between Mark Wilson’s work on the “greediness of scales” and Deleuze’s “scales of reality”, this project threads between static ecologies and active externalism vis-à -vis endocentric frames of reference and syntactical scaffolding
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