4,527 research outputs found
Computing halting probabilities from other halting probabilities
The halting probability of a Turing machine is the probability that the machine will halt if it starts with a random stream written on its one-way input tape. When the machine is universal, this probability is referred to as Chaitin's omega number, and is the most well known example of a real which is random in the sense of Martin-L\"{o}f. Although omega numbers depend on the underlying universal Turing machine, they are robust in the sense that they all have the same Turing degree, namely the degree of the halting problem. In this paper we give precise bounds on the redundancy growth rate that is generally required for the computation of an omega number from another omega number. We show that for each ϵ>1, any pair of omega numbers compute each other with redundancy ϵlogn. On the other hand, this is not true for ϵ=1. In fact, we show that for each omega number there exists another omega number which is not computable from the first one with redundancy logn. This latter result improves an older result of Frank Stephan
Computing stationary probability distributions and large deviation rates for constrained random walks. The undecidability results
Our model is a constrained homogeneous random walk in a nonnegative orthant
Z_+^d. The convergence to stationarity for such a random walk can often be
checked by constructing a Lyapunov function. The same Lyapunov function can
also be used for computing approximately the stationary distribution of this
random walk, using methods developed by Meyn and Tweedie. In this paper we show
that, for this type of random walks, computing the stationary probability
exactly is an undecidable problem: no algorithm can exist to achieve this task.
We then prove that computing large deviation rates for this model is also an
undecidable problem. We extend these results to a certain type of queueing
systems. The implication of these results is that no useful formulas for
computing stationary probabilities and large deviations rates can exist in
these systems
Most Programs Stop Quickly or Never Halt
Since many real-world problems arising in the fields of compiler
optimisation, automated software engineering, formal proof systems, and so
forth are equivalent to the Halting Problem--the most notorious undecidable
problem--there is a growing interest, not only academically, in understanding
the problem better and in providing alternative solutions. Halting computations
can be recognised by simply running them; the main difficulty is to detect
non-halting programs. Our approach is to have the probability space extend over
both space and time and to consider the probability that a random -bit
program has halted by a random time. We postulate an a priori computable
probability distribution on all possible runtimes and we prove that given an
integer k>0, we can effectively compute a time bound T such that the
probability that an N-bit program will eventually halt given that it has not
halted by T is smaller than 2^{-k}. We also show that the set of halting
programs (which is computably enumerable, but not computable) can be written as
a disjoint union of a computable set and a set of effectively vanishing
probability. Finally, we show that ``long'' runtimes are effectively rare. More
formally, the set of times at which an N-bit program can stop after the time
2^{N+constant} has effectively zero density.Comment: Shortened abstract and changed format of references to match Adv.
Appl. Math guideline
Universality and programmability of quantum computers
Manin, Feynman, and Deutsch have viewed quantum computing as a kind of
universal physical simulation procedure. Much of the writing about quantum
logic circuits and quantum Turing machines has shown how these machines can
simulate an arbitrary unitary transformation on a finite number of qubits. The
problem of universality has been addressed most famously in a paper by Deutsch,
and later by Bernstein and Vazirani as well as Kitaev and Solovay. The quantum
logic circuit model, developed by Feynman and Deutsch, has been more prominent
in the research literature than Deutsch's quantum Turing machines. Quantum
Turing machines form a class closely related to deterministic and probabilistic
Turing machines and one might hope to find a universal machine in this class. A
universal machine is the basis of a notion of programmability. The extent to
which universality has in fact been established by the pioneers in the field is
examined and this key notion in theoretical computer science is scrutinised in
quantum computing by distinguishing various connotations and concomitant
results and problems.Comment: 17 pages, expands on arXiv:0705.3077v1 [quant-ph
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