The halting problem is decidable on a set of asymptotic probability one

Resource dependent undecidability: computability landscape of distinct Turing theories

Can a problem undecidable with classical resources be decidable with quantum ones? The answer expected is no; as both being Turing theories, they should not solve the Halting problem - a problem unsolvable by any Turing machine. Yet, we provide an affirmative answer to the aforesaid question. We come up with a novel logical structure to formulate infinitely many such problems for any pair of distinct Turing theories, including but not limited to the classical and quantum theories. Importantly, a class of other decision problems, such as the Halting one, remains unsolvable in all those theories…

Generic algorithms for halting problem and optimal machines revisited

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 Mart…

关于无限复杂性的某些观察

Some Observations on Infinitary Complexity

本文研究了Koepke的序 Turing 机（OTMs）的复杂性理论，证明了Ladner定理的类比结果，展示了Turing机的速度提升定理不适用于OTMs，并展示了某些决策问题在OTM可判定性中的时间界限差异。

Optimal asymptotic bounds on the oracle use in computations from Chaitin's Omega

Chaitin's number Omega is the halting probability of a universal prefix-free machine, and although it depends on the underlying enumeration of prefix-free machines, it is always Turing-complete. It can be observed, in fact, that for every computably enumerable (c.e.) real, there exists a Turing functional via which Omega computes it, and such that the number of bits of omega that are needed for the computation of the first n bits of the given number (i.e. the use on argument n) is bounded above by a computable function h(n) = n+o(n). We characterise the asymptotic upper bounds on the use of Ch…