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Register a new userGeneric algorithms for halting problem and optimal machines revisited
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Added by
Laurent Bienvenu
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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The halting problem is undecidable --- but can it be solved for most inputs? This natural question was considered in a number of papers, in different settings. We revisit their results and show that most of them can be easily proven in a natural framework of optimal machines (considered in algorithmic information theory) using the notion of Kolmogorov complexity. We also consider some related questions about this framework and about asymptotic properties of the halting problem. In particular, we show that the fraction of terminating programs cannot have a limit, and all limit points are Martin-Lof random reals. We then consider mass problems of finding an approximate solution of halting problem and probabilistic algorithms for them, proving both positive and negative results. We consider the fraction of terminating programs that require a long time for termination, and describe this fraction using the busy beaver function. We also consider approxima
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This note introduces a generalization to the setting of infinite-time computation of the busy beaver problem from classical computability theory, and proves some results concerning the growth rate of an associated function. In our view, these results indicate that the generalization is both natural and promising.
We formulate a property $P$ on a class of relations on the natural numbers, and formulate a general theorem on $P$, from which we get as corollaries the insolvability of Hilberts tenth problem, Godels incompleteness theorem, and Turings halting problem. By slightly strengthening the property $P$, we get Tarskis definability theorem, namely that truth is not first order definable. The property $P$ together with a Cantors diagonalization process emphasizes that all the above theorems are a variation on a theme, that of self reference and diagonalization combined. We relate our results to self referential paradoxes, including a formalisation of the Liar paradox, and fixed point theorems. We also discuss the property $P$ for arbitrary rings. We give a survey on Hilberts tenth problem for quadratic rings and for the rationals pointing the way to ongoing research in main stream mathematics involving recursion theory, definability in model theory, algebraic geometry and number theory.
We consider two matrix completion problems, in which we are given a matrix with missing entries and the task is to complete the matrix in a way that (1) minimizes the rank, or (2) minimizes the number of distinct rows. We study the parameterized complexity of the two aforementioned problems with respect to several parameters of interest, including the minimum number of matrix rows, columns, and rows plus columns needed to cover all missing entries. We obtain new algorithmic results showing that, for the bounded domain case, both problems are fixed-parameter tractable with respect to all aforementioned parameters. We complement these results with a lower-bound result for the unbounded domain case that rules out fixed-parameter tractability w.r.t. some of the parameters under consideration.
We study the complexity of quantum query algorithms that make p queries in parallel in each timestep. This model is in part motivated by the fact that decoherence times of qubits are typically small, so it makes sense to parallelize quantum algorithms as much as possible. We show tight bounds for a number of problems, specifically Theta((n/p)^{2/3}) p-parallel queries for element distinctness and Theta((n/p)^{k/(k+1)} for k-sum. Our upper bounds are obtained by parallelized quantum walk algorithms, and our lower bounds are based on a relatively small modification of the adversary lower bound method, combined with recent results of Belovs et al. on learning graphs. We also prove some general bounds, in particular that quantum and classical p-parallel complexity are polynomially related for all total functions f when p is small compared to fs block sensitivity.
Simons problem is one of the most important problems demonstrating the power of quantum computers, which achieves a large separation between quantum and classical query complexities. However, Simons discussion on his problem was limited to bounded-error setting, which means his algorithm can not always get the correct answer. Exact quantum algorithms for Simons problem have also been proposed, which deterministically solve the problem with O(n) queries. Also the quantum lower bound Omega(n) for Simons problem is known. Although these algorithms are either complicated or specialized, their results give an O(n) versus Omega(sqrt{2^{n}}) separation in exact query complexities for Simons problem (Omega(sqrt{2^{n}}) is the lower bound for classical probabilistic algorithms), but it has not been proved whether this separation is optimal. In this paper, we propose another exact quantum algorithm for solving Simons problem with O(n) queries, which is simple, concrete and does not rely on special query oracles. Our algorithm combines Simons algorithm with the quantum amplitude amplification technique to ensure its determinism. In particular, we show that Simons problem can be solved by a classical deterministic algorithm with O(sqrt{2^{n}}) queries (as we are aware, there were no classical deterministic algorithms for solving Simons problem with O(sqrt{2^{n}}) queries). Combining some previous results, we obtain the optimal separation in exact query complexities for Simons problem: Theta({n}) versus Theta({sqrt{2^{n}}}).
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