# Notes

## Section 1: Computation as a Framework

Practical computers

At the lowest level the hardware of a practical computer consists of digital electronic circuits. In these circuits, lumps of electric charge (in 2001 about half a million electrons each) flow through channels which cross to form various kinds of gates. Each gate performs a simple logic operation; for example, letting charge pass in one channel only if charge is present in the other channel. From circuits containing millions of such gates are built the two main elements of the computer: the processor which actually performs computations, and the memory which stores data. The memory consists of an array of cells, with the presence or absence of a lump of charge at gates in each cell representing a 1 or 0 value for the bit of data associated with that cell.

One of the crucial ideas of a general-purpose computer is that sequences of such bits of data in memory can represent information of absolutely any kind. Numbers for example are typically represented in base 2 by sequences of 32 or more bits. Similarly, characters of text are usually represented by sequences of 8 or more bits. (The character "a" is typically 01100001.) Images are usually represented by bitmaps containing thousands or millions of bits, with each bit specifying for example whether a pixel at a particular location should, say, be black or white. Every possible location in memory has a definite address, independent of its contents. The address is typically represented as a number which itself can be stored in memory.

What makes possible essential universality in a practical computer is that the data which is stored in memory can be a program. At the lowest level, a program consists of a sequence of instructions to be executed by the processor. Any particular kind of processor is built to support a certain fixed set of possible kinds of instructions, each represented by a specific number or opcode. There are typically a few tens of possible instructions, each executed by a certain part of the circuit in the processor. A typical one of these instructions might add two numbers together; a program would specify which numbers to add by giving their addresses in memory.

What practical computers always basically do is to repeat millions of times a second a simple cycle, in which the processor fetches an instruction from memory, then executes the instruction. The address of the instruction to be fetched at each point is specified by the current value of the program counter—a number stored in memory that is incremented by the processor, or can be modified by instruction in the program. At any given time, there are usually several programs stored in the memory of a computer, all organized by an operating system program which determines when other programs should run. Devices like keyboards, mice and microphones convert input into data that is inserted into memory at certain fixed locations. The operating system periodically checks these locations, and if necessary runs programs to respond to the input that is given.

A crucial achievement in practical computing over the past several decades has been the creation of more and more sophisticated software. Often the programs that make up this software are several million instructions long. They usually contain many subprograms that perform parts of their task. Some programs are set up to perform very specific applications, say word processing. But an important class of programs are languages. A language provides a fixed set of constructs that allow one to specify computations. The set of instructions performed by the processor in a computer constitutes a low-level "machine" language. In practice, however, programs are rarely written at such a low level. More often, languages like C, Fortran, Java or Mathematica are used. In these languages, each construct represents what is often a large number of machine instructions. There are two basic ways that languages can operate: compiled or interpreted. In a compiled language like C or Fortran, the source code of the program must always first be translated by a compiler program into object code that essentially consists of machine instructions. Once compiled, a program can be executed any number of times. In an interpreted language, each piece of input effectively causes a fixed subprogram to be executed to perform an operation specified by that input.

From Stephen Wolfram: A New Kind of Science [citation]