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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 30 (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.
Computing a glimpse of randomness
 Experimental Mathematics
"... A Chaitin Omega number is the halting probability of a universal Chaitin (selfdelimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In ..."
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Cited by 18 (9 self)
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A Chaitin Omega number is the halting probability of a universal Chaitin (selfdelimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly noncomputable. The aim of this paper is to describe a procedure, which combines Java programming and mathematical proofs, for computing the exact values of the first 63 bits of a Chaitin Omega: 000000100000010000100000100001110111001100100111100010010011100. Full description of programs and proofs will be given elsewhere. 1
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.
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.
Number: Preliminary Version
, 2000
"... A Chaitin Omega number is the halting probability of a universal Chaitin (selfdelimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In ..."
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A Chaitin Omega number is the halting probability of a universal Chaitin (selfdelimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly noncomputable. The aim of this paper is to describe a procedure, which combines Java and Perl programming and mathematical proofs, for computing the exact values of the first 80 bits of a Chaitin Omega: 0.0000000000000000000010000001000000100000010000010000011100100111000101 0001010000. Full description of programs and proofs will be given elsewhere. 1
Incompleteness, Complexity, Randomness and Beyond
, 2001
"... The Library is composed of an... infinite number of hexagonal galleries... [it] includes all verbal structures, all variations permitted by the twentyfive orthographical symbols, but not a single example of absolute nonsense.... These phrases, at first glance incoherent, can no doubt be justified i ..."
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The Library is composed of an... infinite number of hexagonal galleries... [it] includes all verbal structures, all variations permitted by the twentyfive orthographical symbols, but not a single example of absolute nonsense.... These phrases, at first glance incoherent, can no doubt be justified in a cryptographical or allegorical manner; such a justification is verbal and, ex hypothesi, already figures in the Library.... The certitude that some shelf in some hexagon held precious books and that these precious books were inaccessible seemed almost intolerable. A blasphemous sect suggested that... all men should juggle letters and symbols until they constructed, by an improbable gift of chance, these canonical books... but the Library is... useless, incorruptible, secret. Jorge Luis Borges, “The Library of Babel” Gödel’s Incompleteness Theorems have the same scientific status as Einstein’s principle of relativity, Heisenberg’s uncertainty principle, and Watson and Crick’s double helix model of DNA. Our aim is to discuss some new faces of the incompleteness phenomenon unveiled by an informationtheoretic approach to randomness and recent developments in quantum computing.
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"... We first show that the Halting Function (the noncomputable function that solves the Halting Problem) has explicit expressions in the language of calculus. Out of that fact we elaborate on the possible meaning of hypercomputation theory within the setting of formal mathematical theories. ..."
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We first show that the Halting Function (the noncomputable function that solves the Halting Problem) has explicit expressions in the language of calculus. Out of that fact we elaborate on the possible meaning of hypercomputation theory within the setting of formal mathematical theories.
2 Some thoughts on hypercomputation q
, 2005
"... 7 We first show that the Halting Function (the noncomputable function that solves the Halting Problem) has explicit 8 expressions in the language of calculus. Out of that fact we elaborate on the possible meaning of hypercomputation theory 9 within the setting of formal mathematical theories. ..."
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7 We first show that the Halting Function (the noncomputable function that solves the Halting Problem) has explicit 8 expressions in the language of calculus. Out of that fact we elaborate on the possible meaning of hypercomputation theory 9 within the setting of formal mathematical theories.
Abstract Analog computation beyond the Turing limit
"... The main purpose of this paper is quite uncontroversial. First, we recall some models of analog computations (including these allowed to perform Turing uncomputable tasks). Second, we support the suggestions that such hypercomputable capabilities of such systems can be explained by the use of infini ..."
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The main purpose of this paper is quite uncontroversial. First, we recall some models of analog computations (including these allowed to perform Turing uncomputable tasks). Second, we support the suggestions that such hypercomputable capabilities of such systems can be explained by the use of infinite limits. Additionally, the inner restrictions of analog models of computations are indicated.