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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.

### Received (Day Month Year)

"... Accepted (Day Month Year) Communicated by (xxxxxxxxxx) 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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Accepted (Day Month Year) Communicated by (xxxxxxxxxx) 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.

### 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.xx-xx, 2012 Online at: www.jus.org.uk Computation in Quantum Space-Time Could Lead to a Super-Polynomial 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 super-polynomial ( = faster than polynomial) speed-up. 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 super-polynomial 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 space-time, parallelization could potentially lead to (provably) super-polynomial speedup of computations. c⃝2012 World Academic Press, UK. All rights reserved.

### Space-Time Assumptions Behind NP-Hardness of Propositional Satisfiability

"... For some problems, we know feasible algorithms for solving them. Other computational problems (such as propositional satisfiability) are known to be NP-hard, 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 NP-hard, 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 NP-hardness, 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 NP-hardness result more intuitive and more convincing, we provide a new proof in which, instead of a Turing machine, we use a generic computational