Mathematical model on which current computers are built

Question

It is said that "The Turing machine is not intended as practical computing technology, but rather as a hypothetical device representing a computing machine. Turing machines help computer scientists understand the limits of mechanical computation." [Wikipedia]

So on which model current machines are built?

Answer 1

The one closest to typical CPUs is probably the register machine or random access machine (RAM). A RAM has

• an infinite number of registers, each of which stores an arbitrarily large number,
• a set of operations on these registers (typically $\{+ 1, = 0\}$),
• a programming language including these operations as well as control structures for looping/branching (until/if some register holds $0$) and
• a program counter pointing to the next operation (in some program).

Real CPUs are quite similar, with some changes:

• There are only finitely many registers (which may exist only virtually), and each stores only numbers of bounded size.
• There are more operations.

Apart from that, it's very close indeed. It is common to extend the RAM model to account for memory hierarchy, which makes results a lot more applicable.

Answer 2

Von Neumann machine, and if you prefer something more mathematical, look instead at RAM machine.