Efficient simulations between models such as RAM machines and Turing machines have been concretely found, with overhead bounds of O(t(n)^3) in time and O(s(n)) in space (Slot and Emde Boas 1988).
(measuring how much extra time the simulation requires)
Mathematical notation meaning that as the input size grows, the time needed grows roughly at a rate of the cube of that size.
Overhead bounds(describing the limits of inefficiency in the simulation)
The maximum amount of extra resources (like time or memory) needed when converting from one system to another.
RAM machines(as one of the two computing models being compared)
A theoretical model of a computer that has random access memory (RAM) and can directly access any piece of stored information instantly, similar to how real computers work.
Slot and Emde Boas 1988(crediting the source of this computational result)
A reference to a research paper published in 1988 by computer scientists Slot and Emde Boas who discovered these efficiency bounds.
Turing machines(as used in computer science and logic)
A theoretical mathematical model of a simple computer invented by Alan Turing; it's used to study what kinds of problems computers can solve in principle.
In particular, a RAM machine \(A\) consists of a finite sequence of instructions (or program) \(\langle \pi_1,\ldots,\pi_n \rangle\) expressing how numerical operations (typically addition and subtraction) are to be applied to a sequence of registers \(r_1,r_2, \dots\) in which values may be stored and retrieved directly by their index. Showing that one of these models \(\mathfrak{M}_1\) determines the same class of functions as some reference model \(\mathfrak{M}_2\) (such as \(\mathfrak{T}\))