# Notes

## Section 11: Traditional Mathematics and Mathematical Formulas

History [of Boolean functions]

Logic has been used as an abstraction of arguments in ordinary language since antiquity. Its serious mathematical formulation began with the work of George Boole in the mid-1800s. (See page 1151.) Concepts of Boolean algebra were applied to electronic switching circuits by Claude Shannon in 1937, and became a standard part of electronic design methodology by the 1950s. DNF had been introduced as part of the development of mathematical logic in the early 1900s, but became particularly popular in the 1970s with the advent of programmable logic arrays (PLAs) used in application-specific integrated circuits (ASICs). Diagrammatic and mechanical methods for minimizing simple logic expressions have existed since at least medieval times. More systematic methods for minimizing complex expressions began to be developed in the early 1950s, but until well into the 1980s a diagrammatic method known as a Karnaugh map was the most commonly used in practice. In the late 1970s there began to be computer programs for large-scale Boolean minimization—the best known being Espresso. Only in the 1990s, however, did exact minimization of complex DNF expressions become common. Minimization of Boolean expressions with depth larger than 2 has been considered off and on since the late 1950s, and became popular in the 1990s in connection with the BDDs discussed above. Various forms of Boolean minimization have routinely been used in chip and circuit design since the late 1980s, though often physical and geometrical constraints are now more important than pure logical ones. In addition, theoretical studies of minimal Boolean circuits became increasingly popular starting in the 1980s, as discussed on page 1148.

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