Results 1  10
of
23
ON INTERPRETING CHAITIN’S INCOMPLETENESS THEOREM
, 1998
"... The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin’s famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin’s famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good measure of the strength of the theory. I exhibit certain strong counterexamples and establish conclusively that the received view is false. Moreover, I show that the limiting constants provided by the theorem do not in any way reflect the power of formalized theories, but that the values of these constants are actually determined by the chosen coding of Turing machines, and are thus quite accidental.
Is Complexity a Source of Incompleteness?
 IS COMPLEXITY A SOURCE OF INCOMPLETENESS
, 2004
"... ..."
(Show Context)
On the unusual effectiveness of Logic in computer science
 Bulletin of Symbolic Logic
"... Effectiveness of Mathematics in the Natural Sciences [Wig60]. This paper can be construed as an examination and affirmation of Galileo’s tenet that “The book of nature is written in the language of mathematics”. To this effect, Wigner presented a large number of examples that demonstrate the effecti ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
Effectiveness of Mathematics in the Natural Sciences [Wig60]. This paper can be construed as an examination and affirmation of Galileo’s tenet that “The book of nature is written in the language of mathematics”. To this effect, Wigner presented a large number of examples that demonstrate the effectiveness of
Evolution of Mutating Software ∗
"... We propose using random walks in software space as abstract formal models of biological evolution. The goal is to shed light on biological creativity using toy models of evolution that are simple enough to prove theorems about them. We consider two models: a single mutating piece of software, and a ..."
Abstract

Cited by 5 (5 self)
 Add to MetaCart
(Show Context)
We propose using random walks in software space as abstract formal models of biological evolution. The goal is to shed light on biological creativity using toy models of evolution that are simple enough to prove theorems about them. We consider two models: a single mutating piece of software, and a population of mutating software. The fitness function is taken from a wellknown problem in computability theory that requires an unlimited amount of creativity, the Busy Beaver problem.
An algebraic characterization of the halting probability
 FUNDAMENTA INFORMATICAE
, 2007
"... Using 1947 work of Post showing that the word problem for semigroups is unsolvable, we explicitly exhibit an algebraic characterization of the bits of the halting probability Ω. Our proof closely follows a 1978 formulation of Post’s work by M. Davis. The proof is selfcontained and not very complicat ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
Using 1947 work of Post showing that the word problem for semigroups is unsolvable, we explicitly exhibit an algebraic characterization of the bits of the halting probability Ω. Our proof closely follows a 1978 formulation of Post’s work by M. Davis. The proof is selfcontained and not very complicated.
Centre for Discrete Mathematics and
, 2008
"... We propose using random walks in software space as abstract formal models of biological evolution. The goal is to shed light on biological creativity using toy models of evolution that are simple enough to prove theorems about them. We consider two models: a single mutating piece of software, and a ..."
Abstract
 Add to MetaCart
(Show Context)
We propose using random walks in software space as abstract formal models of biological evolution. The goal is to shed light on biological creativity using toy models of evolution that are simple enough to prove theorems about them. We consider two models: a single mutating piece of software, and a population of mutating software. The fitness function is taken from a wellknown problem in computability theory that requires an unlimited amount of creativity, the Busy Beaver problem.