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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
Exact Approximations of Omega Numbers
, 2006
"... A Chaitin Omega number is the halting probability of a universal prefixfree Turing machine. Every Omega number is simultaneously computably enumerable (the limit of a computable, increasing, converging sequence of rationals), and algorithmically random (its binary expansion is an algorithmic random ..."
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A Chaitin Omega number is the halting probability of a universal prefixfree Turing machine. Every Omega number is simultaneously computably enumerable (the limit of a computable, increasing, converging sequence of rationals), and algorithmically random (its binary expansion is an algorithmic random sequence), hence uncomputable. The value of an Omega number is highly machinedependent. In general, no more than finitely many scattered bits of the binary expansion of an Omega number can be exactly computed; but, in some cases, it is possible to prove that no bit can be computed. In this paper we will simplify and improve both the method and its correctness proof proposed in an earlier paper, and we will compute the exact approximations of two Omega numbers of the same prefixfree Turing machine, which is universal when used with data in base 16 or base 2: we compute 43 exact bits for the base 16 machine and 40 exact bits for the base 2 machine.
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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Cited by 1 (0 self)
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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
Simplicity via Provability for Universal Prefixfree Turing Machines
, 2008
"... Universality is one of the most important ideas in computability theory. There are various criteria of simplicity for universal Turing machines. Probably the most popular one is to count the number of states/symbols. This criterion is more complex than it may appear at a first glance. In this note w ..."
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Universality is one of the most important ideas in computability theory. There are various criteria of simplicity for universal Turing machines. Probably the most popular one is to count the number of states/symbols. This criterion is more complex than it may appear at a first glance. In this note we review recent results in Algorithmic Information Theory and propose three new criteria of simplicity for universal prefixfree Turing machines. These criteria refer to the possibility of proving various natural properties of such a machine (its universality, for example) in a formal theory, PA or ZFC. In all cases some, but not all, machines are simple.