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Computation in quantum spacetime can lead to a superpolynomial speedup
 in: Abstracts of the 7th Joint UTEP/NMSU Workshop on Mathematics, Computer Science, and Computational Sciences, Las Cruces
, 2010
"... In theoretical computer science, researchers usually distinguish between problems that can be solved in polynomial time (i.e., in time that is bounded by a polynomial of the length n of the input) and problems that require more computation time. Problems solvable in polynomial time are usually calle ..."
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In theoretical computer science, researchers usually distinguish between problems that can be solved in polynomial time (i.e., in time that is bounded by a polynomial of the length n of the input) and problems that require more computation time. Problems solvable in polynomial time are usually called _feasible_, while others are called _intractable_. Of course, this association is not perfect for example, an algorithm that requires 10^100 * n steps is polynomial time but not feasible but this is the best available definition of feasibility. A natural question is: can we use new physical processes, processes that have not been used in modern computers, to make computations drastically faster e.g., to make intractable problems feasible? Such a possibility would occur if a physical process provides a superpolynomial ( = faster than polynomial) speedup. In this direction, the most active research is undertaken in quantum computing. It is well known that quantum processes can speed up computations; see, e.g., (Nielsen and Chuang 2000). For example, Grover's algorithm enables us to reduce the computation time of many computations from 2^n to sqrt(2^n) = 2^(n/2). Some known quantum algorithms e.g., Shor's
If Many Physicists Are Right and No Physical Theory Is Perfect, Then by Using Physical Observations, We Can Feasibly Solve Almost All Instances of Each NPComplete Problem
"... Many reallife problems are, in general, NPcomplete, i.e., informally speaking, are difficult to solve – at least on computers based on the usual physical techniques. A natural question is: can the use of nonstandard physics speed up the solution of these problems? This question has been analyzed ..."
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Many reallife problems are, in general, NPcomplete, i.e., informally speaking, are difficult to solve – at least on computers based on the usual physical techniques. A natural question is: can the use of nonstandard physics speed up the solution of these problems? This question has been analyzed for several specific physical theories, e.g., for quantum field theory, for cosmological solutions with wormholes and/or casual anomalies, etc. However, many physicists believe that no physical theory is perfect, i.e., that no matter how many observations support a physical theory, inevitably, new observations will come which will require this theory to be updated. In this paper, we show that if such a noperfecttheory principle is true, then the use of physical data can drastically speed up the solution of NPcomplete problems: namely, we can feasibly solve almost all instances of each NPcomplete problem. 1 Formulation of the Problem
SpaceTime Assumptions Behind NPHardness of Propositional Satisfiability
"... For some problems, we know feasible algorithms for solving them. Other computational problems (such as propositional satisfiability) are known to be NPhard, which means that, unless P=NP (which most computer scientists believe to be impossible), no feasible algorithm is possible for solving all pos ..."
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For some problems, we know feasible algorithms for solving them. Other computational problems (such as propositional satisfiability) are known to be NPhard, which means that, unless P=NP (which most computer scientists believe to be impossible), no feasible algorithm is possible for solving all possible instances of the corresponding problem. Most usual proofs of NPhardness, however, use Turing machine – a very simplified version of a computer – as a computation model. While Turing machine has been convincingly shown to be adequate to describe what can be computed in principle, it is much less intuitive that these oversimplified machine are adequate for describing what can be computed effectively; while the corresponding adequacy results are known, they are not easy to prove and are, thus, not usually included in the textbooks. To make the NPhardness result more intuitive and more convincing, we provide a new proof in which, instead of a Turing machine, we use a generic computational
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"... Communicated by (xxxxxxxxxx) In theoretical computer science, researchers usually distinguish between feasible problems (that can be solved in polynomial time) and problems that require more computation time. A natural question is: can we use new physical processes, processes that have not been used ..."
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Communicated by (xxxxxxxxxx) In theoretical computer science, researchers usually distinguish between feasible problems (that can be solved in polynomial time) and problems that require more computation time. A natural question is: can we use new physical processes, processes that have not been used in modern computers, to make computations drastically faster – e.g., to make intractable problems feasible? Such a possibility would occur if a physical process provides a superpolynomial ( = faster than polynomial) speedup. In this direction, the most active research is undertaken in quantum computing. It is well known that quantum processes can speed up computations; however, the only proven quantum speedups are polynomial. Parallelization is another potential source of speedup. In Euclidean space, parallelization only leads to a polynomial speedup. We show that in quantum spacetime, parallelization can potentially leads to superpolynomial speedup of computations.
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"... • In theoretical computer science: researchers usually distinguish between – problems that can be solved in polynomial time, i.e., in time ≤ P (n) where n is input length, and – problems that require more computation time. • Terminology: – problems solvable in polynomial time are usually called feas ..."
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• In theoretical computer science: researchers usually distinguish between – problems that can be solved in polynomial time, i.e., in time ≤ P (n) where n is input length, and – problems that require more computation time. • Terminology: – problems solvable in polynomial time are usually called feasible, – while others are called intractable. • Warning: this association is not perfect. • Example: an algorithm that requires 10 100 · n steps is – polynomial time, but – not practiclaly feasible.
Designing, Understanding, and Analyzing Unconventional Computation: The Important Role of Logic and Constructive Mathematics
"... In this paper, we explain why, in our opinion, logic and constructive mathematics are playing – and should play – an important role in the design, understanding, and analysis of unconventional computation. ..."
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In this paper, we explain why, in our opinion, logic and constructive mathematics are playing – and should play – an important role in the design, understanding, and analysis of unconventional computation.
Journal of Uncertain Systems Vol.6, No.x, pp.xxxx, 2012 Online at: www.jus.org.uk Computation in Quantum SpaceTime Could Lead to a SuperPolynomial Speedup
, 2011
"... In theoretical computer science, researchers usually distinguish between feasible problems (that can be solved in polynomial time) and problems that require more computation time. A natural question is: can we use new physical processes, processes that have not been used in modern computers, to make ..."
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In theoretical computer science, researchers usually distinguish between feasible problems (that can be solved in polynomial time) and problems that require more computation time. A natural question is: can we use new physical processes, processes that have not been used in modern computers, to make computations drastically faster – e.g., to make intractable problems feasible? Such a possibility would occur if a physical process provides a superpolynomial ( = faster than polynomial) speedup. In this direction, the most active research is undertaken in quantum computing. It is well known that quantum processes can drastically speed up computations; however, there are proven superpolynomial quantum speedups of the overall computation time. Parallelization is another potential source of speedup. In Euclidean space, parallelization only leads to a polynomial speedup. We show that in quantum spacetime, parallelization could potentially lead to (provably) superpolynomial speedup of computations. c⃝2012 World Academic Press, UK. All rights reserved.
Most Important Activities in the World { As Well as Physics and Game Theory
"... In the 1970 and 1980s, logic and constructive mathematics were an important part of my life, this is what I defended by Master's thesis on, this was an important part of my PhD dissertation. I was privileged to work with the giants. I visited them in their homes. They were who I went to for adv ..."
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In the 1970 and 1980s, logic and constructive mathematics were an important part of my life, this is what I defended by Master's thesis on, this was an important part of my PhD dissertation. I was privileged to work with the giants. I visited them in their homes. They were who I went to for advise when in trouble, be it girl trouble or KGB harassment. And this is my story.