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Tien D. Kieu

Publications and source records attributed to Tien D. Kieu.

25 records · Page 2Linked to original sources
arXiv2004

Using biased coins as oracles

While it is well known that a Turing machine equipped with the ability to flip a fair coin cannot compute more that a standard Turing machine, we show that this is not true for a biased coin. Indeed, any oracle set $X$ may be coded as a probability $p_{X}$ such that if a Turing machine is given a coin which lands heads with probability $p_{X}$ it can compute any function recursive in $X$ with arbitrarily high probability. We also show how the assumption of a non-recursive bias can be weakened by using a sequence of increasingly accurate recursive biases or by choosing the bias at random from a distribution with a non-recursive mean. We conclude by briefly mentioning some implications regarding the physical realisability of such methods.

cs.OH↗
arXiv2003

Quantum adiabatic algorithm for Hilbert's tenth problem: I. The algorithm

We review the proposal of a quantum algorithm for Hilbert's tenth problem and provide further arguments towards the proof that: (i) the algorithm terminates after a finite time for any input of Diophantine equation; (ii) the final ground state which contains the answer for the Diophantine equation can be identified as the component state having better-than-even probability to be found by measurement at the end time--even though probability for the final ground state in a quantum adiabatic process need not monotonically increase towards one in general. Presented finally are the reasons why our algorithm is outside the jurisdiction of no-go arguments previously employed to show that Hilbert's tenth problem is recursively non-computable.

quant-ph↗
arXiv2003

On the existence of a new family of Diophantine equations for $\bf Ω$

We show how to determine the $k$-th bit of Chaitin's algorithmically random real number $Ω$ by solving $k$ instances of the halting problem. From this we then reduce the problem of determining the $k$-th bit of $Ω$ to determining whether a certain Diophantine equation with two parameters, $k$ and $N$, has solutions for an odd or an even number of values of $N$. We also demonstrate two further examples of $Ω$ in number theory: an exponential Diophantine equation with a parameter $k$ which has an odd number of solutions iff the $k$-th bit of $Ω$ is 1, and a polynomial of positive integer variables and a parameter $k$ that takes on an odd number of positive values iff the $k$-th bit of $Ω$ is 1.

math.NT↗
arXiv2003

Computing the Noncomputable

We explore in the framework of Quantum Computation the notion of computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm that exploits the quantum adiabatic processes is considered for the Hilbert's tenth problem, which is equivalent to the Turing halting problem and known to be mathematically noncomputable. Generalised quantum algorithms are also considered for some other mathematical noncomputables in the same and of different noncomputability classes. The key element of all these algorithms is the measurability of both the values of physical observables and of the quantum-mechanical probability distributions for these values. It is argued that computability, and thus the limits of Mathematics, ought to be determined not solely by Mathematics itself but also by physical principles.

quant-ph↗
arXiv2003

The Diagonal Method and Hypercomputation

The diagonal method is often used to show that Turing machines cannot solve their own halting problem. There have been several recent attempts to show that this method also exposes either contradiction or arbitrariness in other theoretical models of computation that claim to be able to solve the halting problem for Turing machines. We show that such arguments are flawed -- a contradiction only occurs if a type of machine can compute its own diagonal function. We then demonstrate why such a situation does not occur for the methods of hypercomputation under attack and why it is unlikely to occur in any other serious methods.

math.LO↗
arXiv2003

Representations of Omega in Number Theory: Finitude Versus Parity

We present a new method for expressing Chaitin's random real, Omega, through Diophantine equations. Where Chaitin's method causes a particular quantity to express the bits of Omega by fluctuating between finite and infinite values, in our method this quantity is always finite and the bits of Omega are expressed in its fluctuations between odd and even values, allowing for some interesting developments. We then use exponential Diophantine equations to simplify this result and finally show how both methods can also be used to create polynomials which express the bits of Omega in the number of positive values they assume.

math.NT↗