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Hard-Wired vs. Programmable Logic
There is one primary distinguishing feature of a processor that makes it more than just a highly-integrated regular hardware circuit. Ordinary electronic hardware implements a predefined set of functions; the logic is "hard wired" for a single purpose, and the logic circuits are customized to that task only. A different use would require changes to the circuitry.
Processors are programmable, that is, they can perform different functions based on instructions read from a program. Instead of being customized to one single task, a processor is designed in a more general way to perform a broad range of functions. It does this by defining a set of subtasks, or instructions that controls how it works, and then letting users write programs from these instructions. This design introduces tremendous flexibility over the hard-wired design (although it comes at the small penalty of being slightly less efficient than a customized machine.)
All modern computers use this "hardware-software" model, where hardware performs functions under the control of a program, which is of course called software. In fact, modern computers are made of many layers of software and hardware, each controlled by programs written at the next higher layer.