Results 11  20
of
47
Informationtheoretic Incompleteness
 APPLIED MATHEMATICS AND COMPUTATION
, 1992
"... We propose an improved definition of the complexity of a formal axiomatic system: this is now taken to be the minimum size of a selfdelimiting program for enumerating the set of theorems of the formal system. Using this new definition, we show (a) that no formal system of complexity n can exhibit a ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
We propose an improved definition of the complexity of a formal axiomatic system: this is now taken to be the minimum size of a selfdelimiting program for enumerating the set of theorems of the formal system. Using this new definition, we show (a) that no formal system of complexity n can exhibit a specific object with complexity greater than n + c, and (b) that a formal system of complexity n can determine at most n + c scattered bits of the halting probability\Omega . We also present a short, selfcontained proof of (b).
Information theory, evolutionary computation, and Dembski’s “complex specified information”
, 2003
"... Intelligent design advocate William Dembski has introduced a measure of information called “complex specified information”, or CSI. He claims that CSI is a reliable marker of design by intelligent agents. He puts forth a “Law of Conservation of Information” which states that chance and natural laws ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
Intelligent design advocate William Dembski has introduced a measure of information called “complex specified information”, or CSI. He claims that CSI is a reliable marker of design by intelligent agents. He puts forth a “Law of Conservation of Information” which states that chance and natural laws are incapable of generating CSI. In particular, CSI cannot be generated by evolutionary computation. Dembski asserts that CSI is present in intelligent causes and in the flagellum of Escherichia coli, and concludes that neither have natural explanations. In this paper we examine Dembski’s claims, point out significant errors in his reasoning, and conclude that there is no reason to accept his assertions.
Kolmogorov Complexity, Circuits, and the Strength of Formal Theories of Arithmetic
"... Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorovrandom strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of prob ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorovrandom strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of problems C defined in terms of polynomialtime truthtable reducibility to RK (the set of Kolmogorovrandom strings) that lies between BPP and PSPACE [4, 3]. In this paper, we investigate improving this upper bound from PSPACE to PSPACE ∩ P/poly. More precisely, we present a collection of true statements in the language of arithmetic, (each provable in ZF) and show that if these statements can be proved in certain extensions of Peano arithmetic, then BPP ⊆C⊆PSPACE ∩ P/poly. We conjecture that C is equal to P, and discuss the possibility this might be an avenue for trying to prove the equality of BPP and P.
Lisp ProgramSize Complexity II
, 1992
"... We present the informationtheoretic incompleteness theorems that arise in a theory of programsize complexity based on something close to real LISP. The complexity of a formal axiomatic system is defined to be the minimum size in characters of a LISP definition of the proofchecking function associa ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We present the informationtheoretic incompleteness theorems that arise in a theory of programsize complexity based on something close to real LISP. The complexity of a formal axiomatic system is defined to be the minimum size in characters of a LISP definition of the proofchecking function associated with the formal system. Using this concrete and easy to understand definition, we show (a) that it is difficult to exhibit complex Sexpressions, and (b) that it is difficult to determine the bits of the LISP halting probability\Omega LISP . We also construct improved versions\Omega 0 LISP and\Omega 00 LISP of the LISP halting probability that asymptotically have maximum possible LISP complexity. Copyright c fl 1992, Elsevier Science Publishing Co., Inc., reprinted by permission. 2 G. J. Chaitin 1. Introduction The main incompleteness theorems of myAlgorithmic Information Theory monograph [1] are reformulated and proved here using a concrete and easytounderstand definition ...
What Is a Random String?
, 1995
"... Chaitin's algorithmic definition of random strings  based on the complexity induced by selfdelimiting computers  is critically discussed. One shows that Chaitin's model satisfy many natural requirements related to randomness, so it can be considered as an adequate model for nite rando ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Chaitin's algorithmic definition of random strings  based on the complexity induced by selfdelimiting computers  is critically discussed. One shows that Chaitin's model satisfy many natural requirements related to randomness, so it can be considered as an adequate model for nite random objects. It is a better model than the original (Kolmogorov) proposal. Finally, some open problems will be discussed.
Randomness in Physics: Five Questions, Some Answers
, 2009
"... Despite provable unknowables in recursion theory, indeterminism and randomness in physics is confined to conventions, subjective beliefs and preliminary evidence. The history of the issue is very briefly reviewed, and answers to five questions raised by Zenil are presented. ..."
Abstract
 Add to MetaCart
Despite provable unknowables in recursion theory, indeterminism and randomness in physics is confined to conventions, subjective beliefs and preliminary evidence. The history of the issue is very briefly reviewed, and answers to five questions raised by Zenil are presented.
Propagation of partial randomness
"... Let f be a computable function from finite sequences of 0’s and 1’s to real numbers. We prove that strong frandomness implies strong frandomness relative to a PAdegree. We also prove: if X is strongly frandom and Turing reducible to Y where Y is MartinLöf random relative to Z, then X is strongl ..."
Abstract
 Add to MetaCart
Let f be a computable function from finite sequences of 0’s and 1’s to real numbers. We prove that strong frandomness implies strong frandomness relative to a PAdegree. We also prove: if X is strongly frandom and Turing reducible to Y where Y is MartinLöf random relative to Z, then X is strongly frandom relative to Z. In addition, we prove analogous propagation results for other notions of partial randomness, including nonKtriviality and autocomplexity. We prove that frandomness relative to a PAdegree implies strong frandomness, but frandomness does not imply frandomness relative to a PAdegree. Keywords: partial randomness, effective Hausdorff dimension, MartinLöf randomness, Kolmogorov complexity, models of arithmetic.
Physical unknowables
, 2008
"... Different types of physical unknowables are discussed. Provable unknowables are derived from reduction to problems which are known to be recursively unsolvable. Recent series solutions to the nbody problem and related to it, chaotic systems, may have no computable radius of convergence. Quantum unk ..."
Abstract
 Add to MetaCart
Different types of physical unknowables are discussed. Provable unknowables are derived from reduction to problems which are known to be recursively unsolvable. Recent series solutions to the nbody problem and related to it, chaotic systems, may have no computable radius of convergence. Quantum unknowables include the random occurrence of single events, complementarity and value indefiniteness.
I. COMMUNICATING ART THROUGH ENCRYPTION AND DECRYPTION
, 2008
"... Aesthetics, among other criteria, can be statistically examined in terms of the complexity required for creating and decrypting a work of art. We propose three laws of aesthetic complexity. According to the first law of aesthetic complexity, too condensed encoding makes a decryption of a work of art ..."
Abstract
 Add to MetaCart
Aesthetics, among other criteria, can be statistically examined in terms of the complexity required for creating and decrypting a work of art. We propose three laws of aesthetic complexity. According to the first law of aesthetic complexity, too condensed encoding makes a decryption of a work of art impossible and is perceived as chaotic by the untrained mind, whereas too regular structures are perceived as monotonous, too orderly and not very stimulating. Thus a necessary condition for an artistic form or design to appear appealing is its complexity to lie within a bracket between monotony and chaos. According to the second law of aesthetic complexity, due to human predisposition, this bracket is invariably based on natural forms; with rather limited plasticity. The third law of aesthetic complexity states that aesthetic complexity trends are dominated by the available resources, and thus also by cost and scarcity. PACS numbers: 89.20.a,89.75.k,01.70.+w