Topic
Logical depth
About: Logical depth is a research topic. Over the lifetime, 80 publications have been published within this topic receiving 5278 citations.
Papers published on a yearly basis
Papers
Book•
1 Sep 1993TL;DR: The Journal of Symbolic Logic as discussed by the authors presents a thorough treatment of the subject with a wide range of illustrative applications such as the randomness of finite objects or infinite sequences, Martin-Loef tests for randomness, information theory, computational learning theory, the complexity of algorithms, and the thermodynamics of computing.
Abstract: The book is outstanding and admirable in many respects is necessary reading for all kinds of readers from undergraduate students to top authorities in the field Journal of Symbolic Logic Written by two experts in the field, this is the only comprehensive and unified treatment of the central ideas and their applications of Kolmogorov complexity The book presents a thorough treatment of the subject with a wide range of illustrative applications Such applications include the randomness of finite objects or infinite sequences, Martin-Loef tests for randomness, information theory, computational learning theory, the complexity of algorithms, and the thermodynamics of computing It will be ideal for advanced undergraduate students, graduate students, and researchers in computer science, mathematics, cognitive sciences, philosophy, artificial intelligence, statistics, and physics The book is self-contained in that it contains the basic requirements from mathematics and computer science Included are also numerous problem sets, comments, source references, and hints to solutions of problems New topics in this edition include Omega numbers, KolmogorovLoveland randomness, universal learning, communication complexity, Kolmogorov's random graphs, time-limited universal distribution, Shannon information and others
3,938 citations
TL;DR: This article provides an overview of several logic redundancy schemes, including von Neumann's multiplexing logic, N-tuple modular redundancy, and interwoven redundant logic, and discusses several important concepts for redundant nanoelectronic system designs based on recent results.
Abstract: This article provides an overview of several logic redundancy schemes, including von Neumann's multiplexing logic, N-tuple modular redundancy, and interwoven redundant logic. We discuss several important concepts for redundant nanoelectronic system designs based on recent results. First, we use Markov chain models to describe the error-correcting and stationary characteristics of multiple-stage multiplexing systems. Second, we show how to obtain the fundamental error bounds by using bifurcation analysis based on probabilistic models of unreliable gates. Third, we describe the notion of random interwoven redundancy. Finally, we compare the reliabilities of quadded and random interwoven structures by using a simulation-based approach. We observe that the deeper a circuit's logical depth, the more fault-tolerant the circuit tends to be for a fixed number of faults. For a constant gate failure rate, a circuit's reliability tends to reach a stationary state as its logical depth increases.
140 citations
Book•
2 Jan 1995
TL;DR: In this article, the authors define the logical depth of an object as the time required by a standard universal Turing machine to generate it from an input that is algorithmically random (i.e., Martin-Lof random).
Abstract: Some mathematical and natural objects (a random sequence, a sequence of zeros, a perfect crystal, a gas) are intuitively trivial, while others (e.g. the human body, the digits of π) contain internal evidence of a nontrivial causal history. We formalize this distinction by defining an object’s “logical depth” as the time required by a standard universal Turing machine to generate it from an input that is algorithmically random (i.e. Martin-Lof random). This definition of depth is shown to be reasonably machineindependent, as well as obeying a slow-growth law: deep objects cannot be quickly produced from shallow ones by any deterministic process, nor with much probability by a probabilistic process, but can be produced slowly. Next we apply depth to the physical problem of “self-organization,” inquiring in particular under what conditions (e.g. noise, irreversibility, spatial and other symmetries of the initial conditions and equations of motion) statistical-mechanical model systems can imitate computers well enough to undergo unbounded increase of depth in the limit of infinite space and time.
137 citations
IBM1
TL;DR: In this paper, the authors define complexity as the number of steps in the deductive or causal path connecting a thing with its plausible origin, and assess the effects of dissipation, noise, and spatial and other symmetries of the initial conditions and equations of motion on the asymptotic complexity-generating abilities of statistical-mechanical model systems.
Abstract: The observed complexity of nature is often attributed to an intrinsic propensity of matter to self-organize under certain (e.g., dissipative) conditions. In order better to understand and test this vague thesis, we define complexity as “logical depth,” a notion based on algorithmic information and computational time complexity. Informally, logical depth is the number of steps in the deductive or causal path connecting a thing with its plausible origin. We then assess the effects of dissipation, noise, and spatial and other symmetries of the initial conditions and equations of motion on the asymptotic complexity-generating abilities of statistical-mechanical model systems. We concentrate on discrete, spatially-homogeneous, locally-interacting systems such as kinetic Ising models and cellular automata.
86 citations
TL;DR: It is shown that if a Boolean formula has a nonnegligible fraction of its satisfying assignments with low depth, then the authors can find a satisfying assignment efficiently and every computable set that is reducible to a shallow set has polynomial-size circuits.
69 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 1 |
| 2020 | 8 |
| 2019 | 4 |
| 2018 | 4 |
| 2017 | 3 |
| 2016 | 8 |