In other words, any function that can be computed, is computable using the semi-Thue formalism.
In each case the simple manipulation of symbols according to explicit rules imparts upon the system the capacity to perform complex computations – in fact, given access to infinite memory and sufficient time, any computable function is calculable (i.e., any function that can be computed, is computable by the given system).
This is what Turing established, and into the bargain the remarkable fact that anything that is computable can in fact be computed by one machine, a universal Turing machine.
It was also novel in its notion of a 'Universal Machine' (now known as a universal Turing machine), with the idea that such a machine could perform the tasks of any other computation machine, or in other words, it is provably capable of computing anything that is computable.
Fundamental problems in computer science about what is computable by digital machines will change.
Equation (3), then, is computable in O md|??|) time, since it computes the comparison value for each gene in the set.
A mathematical problem is computable if it can be solved in principle by a computing device.
On the other hand, systems biologists rarely stay the course with the two book-length expositions of relational biology [ 6, 7], and may even believe that anything is computable, given sufficient time and processing power, despite the fact that computer scientists have long known this to be trivially false [ 14].
The final computation of GS follows: > As we now show, our method is computable in time O md|??|), where 𝒢 is the gene set, m is the maximum number of GO annotation terms per gene, and d is the maximum size of an ancestor set.
Or, put differently, every function that is computable at all, is computable on a Turing machine.
A computable martingale is one where d itself is computable.
