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Informationtheoretic Limitations of Formal Systems
 JOURNAL OF THE ACM
, 1974
"... An attempt is made to apply informationtheoretic computational complexity to metamathematics. The paper studies the number of bits of instructions that must be a given to a computer for it to perform finite and infinite tasks, and also the amount of time that it takes the computer to perform these ..."
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Cited by 45 (7 self)
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An attempt is made to apply informationtheoretic computational complexity to metamathematics. The paper studies the number of bits of instructions that must be a given to a computer for it to perform finite and infinite tasks, and also the amount of time that it takes the computer to perform these tasks. This is applied to measuring the difficulty of proving a given set of theorems, in terms of the number of bits of axioms that are assumed, and the size of the proofs needed to deduce the theorems from the axioms.
Information and Computation: Classical and Quantum Aspects
 REVIEWS OF MODERN PHYSICS
, 2001
"... Quantum theory has found a new field of applications in the realm of information and computation during the recent years. This paper reviews how quantum physics allows information coding in classically unexpected and subtle nonlocal ways, as well as information processing with an efficiency largely ..."
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Cited by 23 (2 self)
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Quantum theory has found a new field of applications in the realm of information and computation during the recent years. This paper reviews how quantum physics allows information coding in classically unexpected and subtle nonlocal ways, as well as information processing with an efficiency largely surpassing that of the present and foreseeable classical computers. Some outstanding aspects of classical and quantum information theory will be addressed here. Quantum teleportation, dense coding, and quantum cryptography are discussed as a few samples of the impact of quanta in the transmission of information. Quantum logic gates and quantum algorithms are also discussed as instances of the improvement in information processing by a quantum computer. We provide finally some examples of current experimental
Attacking the Busy Beaver 5
 Bull EATCS
, 1990
"... Since T. Rado in 1962 defined the busy beaver game several approaches have used computer technology to search busy beavers or even compute special values of E. ..."
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Cited by 22 (0 self)
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Since T. Rado in 1962 defined the busy beaver game several approaches have used computer technology to search busy beavers or even compute special values of E.
A new Gödelian argument for hypercomputing minds based on the busy beaver problem
 Applied Mathematics and Computation, in press, doi:10.1016/j.amc.2005.09.071
"... 9.9.05 1245am NY time Do human persons hypercompute? Or, as the doctrine of computationalism holds, are they information processors at or below the Turing Limit? If the former, given the essence of hypercomputation, persons must in some real way be capable of infinitary information processing. Using ..."
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Cited by 5 (1 self)
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9.9.05 1245am NY time Do human persons hypercompute? Or, as the doctrine of computationalism holds, are they information processors at or below the Turing Limit? If the former, given the essence of hypercomputation, persons must in some real way be capable of infinitary information processing. Using as a springboard Gödel’s littleknown assertion that the human mind has a power “converging to infinity, ” and as an anchoring problem Rado’s (1963) Turinguncomputable “busy beaver ” (or Σ) function, we present in this short paper a new argument that, in fact, human persons can hypercompute. The argument is intended to be formidable, not conclusive: it brings Gödel’s intuition to a greater level of precision, and places it within a sensible case against computationalism. 1
Acceleration Techniques for Busy Beaver Candidates
, 2004
"... A busy beaver is a Turing machine of N states which, given a blank tape, halts with an equal or higher number of printed symbols than all other machines of N states. Candidates for this class of Turing machine cannot realistically be run on a traditional simulator due to their high demands on memory ..."
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Cited by 4 (0 self)
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A busy beaver is a Turing machine of N states which, given a blank tape, halts with an equal or higher number of printed symbols than all other machines of N states. Candidates for this class of Turing machine cannot realistically be run on a traditional simulator due to their high demands on memory and time. This paper describes some techniques that can be used to run these machines to completion with limited memory, in a reasonable amount of time, and without human intervention. These techniques include using kmacro tape representation and arcs, and automatic inductive proof generation and utilization.
Use of Optimisation Techniques in Determining Values for the Quadruplorum Variants of Rado’s Busy Beaver Function
 Masters thesis, Rensselaer Polytechnic Institute, 2003. Georgi Georgiev, Busy Beaver Prover, http:// skelet.ludost.net/bb/index.html
, 2003
"... Various computationallybased efforts have been made by numerous groups to conquer Rado’s Σ function (the “Busy Beaver ” problem): brute force searches, genetic algorithms, heuristic behaviour analyses, etc. These efforts have been successful only for a very small inputs to the function and they eac ..."
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Cited by 3 (0 self)
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Various computationallybased efforts have been made by numerous groups to conquer Rado’s Σ function (the “Busy Beaver ” problem): brute force searches, genetic algorithms, heuristic behaviour analyses, etc. These efforts have been successful only for a very small inputs to the function and they each have focused on only one subformulation of the problem. The present paper proposes and explores the merits of a treebased search approach to the solving the problem utilising a set of optimisation techniques extending the domain for
The busy beaver competition: a historical survey
, 2009
"... Tibor Rado defined the Busy Beaver Competition in 1962. He used Turing machines to give explicit definitions for some functions that are not computable and grow faster than any computable function. He put forward the problem of computing the values of these functions on numbers 1, 2, 3,.... More and ..."
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Tibor Rado defined the Busy Beaver Competition in 1962. He used Turing machines to give explicit definitions for some functions that are not computable and grow faster than any computable function. He put forward the problem of computing the values of these functions on numbers 1, 2, 3,.... More and more powerful computers have made possible the computation of lower bounds for these values. In 1988, Brady extended the definitions to functions on two variables. We give a historical survey of these works. The successive record holders in the Busy Beaver Competition are displayed, with their discoverers, the date they were found, and, for some of them, an analysis of their behavior.
Improved Bounds for Functions Related to Busy Beavers
, 2001
"... Consider Turing machines that use a tape infinite in both directions, with the tape alphabet {0, 1}. Rado's busy beaver function ones(n) is the maximum number of 1s such a machine, with n states, started on a blank (allzero) tape, may leave on its tape when it halts. The function ones(n) is nonco ..."
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Cited by 1 (0 self)
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Consider Turing machines that use a tape infinite in both directions, with the tape alphabet {0, 1}. Rado's busy beaver function ones(n) is the maximum number of 1s such a machine, with n states, started on a blank (allzero) tape, may leave on its tape when it halts. The function ones(n) is noncomputable; in fact, it grows faster than any computable function. Other
Internal Examiner: Dr. James Power
"... My supervisor Damien Woods deserves a special thank you. His help and guidance went far beyond the role of supervisor. He was always enthusiastic, and generous with his time. This work would not have happened without him. I would also like to thank my supervisor Paul Gibson for his advice and suppor ..."
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My supervisor Damien Woods deserves a special thank you. His help and guidance went far beyond the role of supervisor. He was always enthusiastic, and generous with his time. This work would not have happened without him. I would also like to thank my supervisor Paul Gibson for his advice and support. Thanks to the staff and postgraduates in the computer science department at NUI Maynooth for their support and friendship over the last few years. In particular, I would like to mention Niall Murphy he has always been ready to help whenever he could and would often lighten the mood in dark times with some rousing Gilbert and Sullivan. I thank the following people for their interesting discussions and/or advice:
Numerical Evaluation of Algorithmic Complexity for Short Strings A Glance into the Innermost Structure of Randomness
, 2011
"... We describe a method that combines several theoretical and experimental results to numerically approximate the algorithmic (KolmogorovChaitin) complexity of all ∑ 8 n=1 2n bit strings up to 8 bits, and some bit strings between 9 and 16 bits. This is done by an exhaustive execution of all determinis ..."
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We describe a method that combines several theoretical and experimental results to numerically approximate the algorithmic (KolmogorovChaitin) complexity of all ∑ 8 n=1 2n bit strings up to 8 bits, and some bit strings between 9 and 16 bits. This is done by an exhaustive execution of all deterministic 2symbol Turing machines with up to 4 states for which the halting times are known thanks to the busy beaver problem, that is 11019960576 machines. An output frequency distribution is then computed, from which the algorithmic probability is calculated and the algorithmic complexity evaluated by way of the (LevinChaitin) coding theorem.