Minimum Instruction Set

But, according to the definition of computing power, my very fast processor is no more powerful than the SWAC.

Processors must control their peripheral devices, and send and receive information from them. They do this by reading and writing bit patterns on the system bus, and by reading and writing special memory addresses that are assigned to peripherals. So any processor that can read and write memory can also control hard disks and graphics boards. No special kind of processor "power" is needed for that. The multimedia instructions that have been added to recent processor chips do not add extra computing power. Of course, they do add convenience and speed.

What machine instructions must a processor absolutely have?

Important Fact: All processors that support the fundamental machine instructions of bit manipulation, conditional branching, and memory access have the same computing power. All processors have these instructions (and many more). All processors are equivalent in computing power (in the sense of the previous definition).

Arithmetic (both integer and floating point) can be done with bit manipulation instructions, so arithmetic instructions are not fundamental (but are almost always included in a processor).

Above a certain minimum set of instructions adding new instructions does not add to the computing power of a processor. (To learn more about this topic, take a course in Foundations of Computation or in Mathematical Logic).