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Most programs stop quickly or never halt
 Adv. Appl. Math
"... The aim of this paper is to provide a probabilistic, but nonquantum, analysis of the Halting Problem. Our approach is to have the probability space extend over both space and time and to consider the probability that a random Nbit program has halted by a random time. We postulate an a priori compu ..."
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The aim of this paper is to provide a probabilistic, but nonquantum, analysis of the Halting Problem. Our approach is to have the probability space extend over both space and time and to consider the probability that a random Nbit program has halted by a random time. We postulate an a priori computable probability distribution on all possible runtimes and we prove that given an integer k> 0, we can effectively compute a time bound T such that the probability that an Nbit program will eventually halt given that it has not halted by T is smaller than 2 −k. We also show that the set of halting programs (which is computably enumerable, but not computable) can be written as a disjoint union of a computable set and a set of effectively vanishing probability. Finally, we show that “long ” runtimes are effectively rare. More formally, the set of times at which an Nbit program can stop after the time 2 N+constant has effectively zero density. 1
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 ..."
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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.
Evaluating the Complexity of Mathematical Problems. Part 1
, 2009
"... In this paper we provide a computational method for evaluating in a uniform way the complexity of a large class of mathematical problems. The method, which is inspired by NKS1, is based on the possibility to completely describe complex mathematical problems, like the Riemann hypothesis, in terms of ..."
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In this paper we provide a computational method for evaluating in a uniform way the complexity of a large class of mathematical problems. The method, which is inspired by NKS1, is based on the possibility to completely describe complex mathematical problems, like the Riemann hypothesis, in terms of (very) simple programs. The method is illustrated on a variety of examples coming from different areas of mathematics and its power and limits are studied.
DeQuantising NonHalting Programs
, 2006
"... Since many realworld problems arising in the fields of compiler optimisation, automatised software engineering, formal proof systems, and so forth are equivalent to the Halting Problem—the most notorious undecidable problem—there is a growing interest, not only academically, in understanding the pr ..."
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Since many realworld problems arising in the fields of compiler optimisation, automatised software engineering, formal proof systems, and so forth are equivalent to the Halting Problem—the most notorious undecidable problem—there is a growing interest, not only academically, in understanding the problem better and in providing alternative solutions. Halting computations can be recognised by simply running them; the main difficulty is to detect nonhalting programs. For each program length on a given machine, there is an uncomputable “critical time ” after which no more programs of that length will halt. A quantum algorithm [7, 1] has been shown to solve the halting problem to any degree of certainty less than one and various experimental studies have proposed heuristics that apply to a majority of programs [4, 15]. Is it possible to have a classical effective way to describe this phenomenon? The aim of this paper is to provide a nonquantum analysis; our approach is to have the probability space extend over both space and time and to consider the probability that a random Nbit program has halted by
Most short programs halt quickly
, 2008
"... Since many realworld problems arising in the fields of compiler optimisation, automatised software engineering, formal proof systems, and so forth are equivalent to the Halting Problem—the most notorious undecidable problem—there is a growing interest, not only academically, in understanding the pr ..."
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Since many realworld problems arising in the fields of compiler optimisation, automatised software engineering, formal proof systems, and so forth are equivalent to the Halting Problem—the most notorious undecidable problem—there is a growing interest, not only academically, in understanding the problem better and in providing alternative solutions. Halting computations can be recognised by simply running them; the main difficulty is to detect nonhalting programs. For each program length on a given machine, there is an uncomputable “critical time ” after which no more programs of that length will halt. A quantum algorithm [7, 1] has been shown to solve the halting problem to any degree of certainty less than one and various experimental studies have proposed heuristics that apply to a majority of programs [4, 15]. Is it possible to have a classical effective way to describe this phenomenon? The aim of this paper is to provide a nonquantum analysis; our approach is to have the probability space extend over both space and time and to consider the probability that a random Nbit program has halted by
Mathematical Problems. Part 1 ∗
, 2008
"... In this paper we provide a computational method for evaluating in a uniform way the complexity of a large class of mathematical problems. The method is illustrated on a variety of examples coming from different areas of mathematics and its power and limits are studied. 1 ..."
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In this paper we provide a computational method for evaluating in a uniform way the complexity of a large class of mathematical problems. The method is illustrated on a variety of examples coming from different areas of mathematics and its power and limits are studied. 1
BioSystems 77 (2004) 175–194 Biosteps beyond Turing
, 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 negativ ..."
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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. Consequently, two acceleration postulates of biological inspiration are introduced; each of them poses specific questions to biology. Finally, in a more speculative part of the paper, we will deal with Turing noncomputability activity of the brain and possible forms of (extraterrestrial) intelligence.
○c○2006 Old City Publishing, Inc. Published by license under the OCP Science imprint, a member of the Old City Publishing Group A New Measure of the Difficulty of Problems
"... Guessing the degree of difficulty of a problem before seeing its solution is notoriously hard not only for beginners, but also for the most experienced professionals. Can we develop a method to evaluate, in some objective way, the difficulty of an open problem? This note proposes such a measure whic ..."
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Guessing the degree of difficulty of a problem before seeing its solution is notoriously hard not only for beginners, but also for the most experienced professionals. Can we develop a method to evaluate, in some objective way, the difficulty of an open problem? This note proposes such a measure which can be used for a fairly large class of finitely refutable conjectures which includes, for example, Riemann Hypothesis and the Goldbach’s Conjecture. According to our measure, Riemann Hypothesis is more complex than Goldbach’s Conjecture. We also show, in a nonconstructive way, that the Collatz 3x + 1Problem is finitely refutable; consequently, our method cannot be applied, hence stronger versions of this problem are studied. 1
A ProgramSize Complexity Measure for Mathematical Problems and Conjectures
"... Abstract. Cristian Calude et al. in [5] have recently introduced the idea of measuring the degree of difficulty of a mathematical problem (even those still given as conjectures) by the length of a program to verify or refute the statement. The method to evaluate and compare problems, in some objecti ..."
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Abstract. Cristian Calude et al. in [5] have recently introduced the idea of measuring the degree of difficulty of a mathematical problem (even those still given as conjectures) by the length of a program to verify or refute the statement. The method to evaluate and compare problems, in some objective way, will be discussed in this note. For the practitioner, wishing to apply this method using a standard universal register machine language, we provide (for the first time) some “small ” core subroutines or library for dealing with array data structures. These can be used to ease the development of full programs to check mathematical problems that require more than a predetermined finite number of variables. 1