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12
Hypercomputation: computing more than the Turing machine
, 2002
"... In this report I provide an introduction to the burgeoning field of hypercomputation – the study of machines that can compute more than Turing machines. I take an extensive survey of many of the key concepts in the field, tying together the disparate ideas and presenting them in a structure which al ..."
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Cited by 31 (5 self)
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In this report I provide an introduction to the burgeoning field of hypercomputation – the study of machines that can compute more than Turing machines. I take an extensive survey of many of the key concepts in the field, tying together the disparate ideas and presenting them in a structure which allows comparisons of the many approaches and results. To this I add several new results and draw out some interesting consequences of hypercomputation for several different disciplines. I begin with a succinct introduction to the classical theory of computation and its place amongst some of the negative results of the 20 th Century. I then explain how the ChurchTuring Thesis is commonly misunderstood and present new theses which better describe the possible limits on computability. Following this, I introduce ten different hypermachines (including three of my own) and discuss in some depth the manners in which they attain their power and the physical plausibility of each method. I then compare the powers of the different models using a device from recursion theory. Finally, I examine the implications of hypercomputation to mathematics, physics, computer science and philosophy. Perhaps the most important of these implications is that the negative mathematical results of Gödel, Turing and Chaitin are each dependent upon the nature of physics. This both weakens these results and provides strong links between mathematics and physics. I conclude that hypercomputation is of serious academic interest within many disciplines, opening new possibilities that were previously ignored because of long held misconceptions about the limits of computation.
Accelerated Turing Machines
 Minds and Machines
, 2002
"... Abstract. Accelerating Turing machines are Turing machines of a sort able to perform tasks that are commonly regarded as impossible for Turing machines. For example, they can determine whether or not the decimal representation of π contains n consecutive 7s, for any n; solve the Turingmachine halti ..."
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Cited by 26 (2 self)
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Abstract. Accelerating Turing machines are Turing machines of a sort able to perform tasks that are commonly regarded as impossible for Turing machines. For example, they can determine whether or not the decimal representation of π contains n consecutive 7s, for any n; solve the Turingmachine halting problem; and decide the predicate calculus. Are accelerating Turing machines, then, logically impossible devices? I argue that they are not. There are implications concerning the nature of effective procedures and the theoretical limits of computability. Contrary to a recent paper by Bringsjord, Bello and Ferrucci, however, the concept of an accelerating Turing machine cannot be used to shove up Searle’s Chinese room argument.
The many forms of hypercomputation
 Applied Mathematics and Computation
, 2006
"... This paper surveys a wide range of proposed hypermachines, examining the resources that they require and the capabilities that they possess. ..."
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This paper surveys a wide range of proposed hypermachines, examining the resources that they require and the capabilities that they possess.
Even Turing Machines Can Compute Uncomputable Functions
 Unconventional Models of Computation
, 1998
"... Accelerated Turing machines are Turing machines that perform tasks commonly regarded as impossible, such as computing the halting function. The existence of these notional machines has obvious implications concerning the theoretical limits of computability. 2 1. Introduction Neither Turing nor Post ..."
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Cited by 15 (3 self)
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Accelerated Turing machines are Turing machines that perform tasks commonly regarded as impossible, such as computing the halting function. The existence of these notional machines has obvious implications concerning the theoretical limits of computability. 2 1. Introduction Neither Turing nor Post, in their descriptions of the devices we now call Turing machines, made much mention of time (Turing 1936, Post 1936). 1 They listed the primitive operations that their devices perform  read a square of the tape, write a single symbol on a square of the tape (first deleting any symbol already present), move one square to the right, and so forth  but they made no mention of the duration of each primitive operation. The crucial concept is that of whether or not the machine halts after a finite number of operations. Temporal considerations are not relevant to the functioning of the devices as described, nor  so we are clearly supposed to believe  to the soundness of the proofs that Turi...
BioSteps Beyond Turing
 BIOSYSTEMS
, 2004
"... Are there `biologically computing agents' capable to compute Turing uncomputable functions? It is perhaps tempting to dismiss this question with a negative answer. Quite the opposite, for the first time in the literature on molecular computing we contend that the answer is not theoretically nega ..."
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Are there `biologically computing agents' capable to compute Turing uncomputable functions? It is perhaps tempting to dismiss this question with a negative answer. Quite the opposite, for the first time in the literature on molecular computing we contend that the answer is not theoretically negative. Our results will be formulated in the language of membrane computing (P systems). Some mathematical results presented here are interesting in themselves. In contrast with most speedup methods which are based on nondeterminism, our results rest upon some universality results proved for deterministic P systems. These results will be used for building "accelerated P systems". In contrast with the case of Turing machines, acceleration is a part of the hardware (not a quality of the environment) and it is realised either by decreasing the size of "reactors" or by speedingup the communication channels.
Computing with Cells and Atoms: After Five Years
 CDMTCS Research Report Series, CDMTCS246, 2004, available at http://www.cs.auckland.ac.nz/CDMTCS/researchreports/246cris.pdf
, 2004
"... This is the text added to the Russian edition of our book Computing with Cells and Atoms (Taylor & Francis Publishers, London, 2001) to be published by Pushchino Publishing House. The translation was done by Professor Victor Vladimirovich Ivanov and Professor Robert Polozov. ..."
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This is the text added to the Russian edition of our book Computing with Cells and Atoms (Taylor & Francis Publishers, London, 2001) to be published by Pushchino Publishing House. The translation was done by Professor Victor Vladimirovich Ivanov and Professor Robert Polozov.
SuperTasks, Accelerating Turing Machines and Uncomputability
"... Accelerating Turing machines are abstract devices that have the same computational structure as Turing machines, but can perform supertasks. I argue that performing supertasks alone does not buy more computational power, and that accelerating Turing machines do not solve the halting problem. To sh ..."
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Accelerating Turing machines are abstract devices that have the same computational structure as Turing machines, but can perform supertasks. I argue that performing supertasks alone does not buy more computational power, and that accelerating Turing machines do not solve the halting problem. To show this, I analyze the reasoning that leads to Thomson's paradox, point out that the paradox rests on a conflation of different perspectives of accelerating processes, and conclude that the same conflation underlies the claim that accelerating Turing machines can solve the halting problem.
Is Feasibility in Physics Limited by Fantasy Alone? ∗
, 2011
"... Although various limits on the predicability of physical phenomena as well as on physical knowables are commonly established and accepted, we challenge their ultimate validity. More precisely, we claim that fundamental limits arise only from our limited imagination and fantasy. To illustrate this th ..."
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Although various limits on the predicability of physical phenomena as well as on physical knowables are commonly established and accepted, we challenge their ultimate validity. More precisely, we claim that fundamental limits arise only from our limited imagination and fantasy. To illustrate this thesis we give evidence that the wellknown Turing incomputability barrier can be trespassed via quantum indeterminacy. From this algorithmic viewpoint, the “fine tuning ” of physical phenomena amounts to a “(re)programming ” of the universe. Take a few moments for some anecdotal recollections. Nuclear science has made true the ancient alchemic dream of producing gold from other elements such as mercury through nuclear reactions. A century ago, similar claims would have disqualified anybody presenting them as quack. Medical chemistry discovered antibiotics which cure Bubonic plague, tuberculosis, syphilis, bacterial pneumonia, as well as a wide range of bacterial infectious diseases which were considered untreatable only one hundred years ago. For contemporaries it is hard to imagine the kind of isolation, scarcity in international communication, entertainment and transportation most of our ancestors had to cope with.
COMPSCI 350, FC Halting Problem Revisited 2005
"... Consider the set BT M = {〈M, w 〉  M is a T M, w is a string, M accepts w or M rejects w}. We prove that BT M is not Turingdecidable. Indeed, assume that the TM S decides BT M: Construct the TM E as follows: accept, if M accepts w or M rejects w, S(〈M, w〉) = reject, otherwise. E = “On input 〈M〉, w ..."
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Consider the set BT M = {〈M, w 〉  M is a T M, w is a string, M accepts w or M rejects w}. We prove that BT M is not Turingdecidable. Indeed, assume that the TM S decides BT M: Construct the TM E as follows: accept, if M accepts w or M rejects w, S(〈M, w〉) = reject, otherwise. E = “On input 〈M〉, where M is a TM, 1. Run S on the input 〈M, 〈M〉〉 2. If s rejects, then accept.” Now, run E on 〈E〉: consequently,