Notes

Chapter 12: The Principle of Computational Equivalence

Section 8: Undecidability and Intractability


History [of computational complexity theory]

Ideas of characterizing problems by growth rates in the computational resources needed to solve them were discussed in the 1950s, notably in the context of operation counts for numerical calculations, sizes of circuits for switching and other applications, and theoretical lengths of proofs. In the 1960s such ideas were increasingly formalized, particularly for execution times on Turing machines, and in 1965 the suggestion was made that one should consider computations feasible if they take times that grow like polynomials in their input size. NP completeness (see below) was introduced by Stephen Cook in 1971 and Leonid Levin around the same time. And over the course of the 1970s a great many well-known problems were shown to be NP-complete. A variety of additional classes of computations—notably ones like NC with various kinds of parallelism, ones based on circuits and ones based on algebraic operations—were defined in the 1970s and 1980s, and many detailed results about them were found. In the 1980s much work was also done on the average difficulty of solving NP-complete problems—both exactly and approximately (see page 985). When computational complexity theory was at its height in the early 1980s it was widely believed that if a problem could be shown, for example, to be NP-complete then there was little chance of being able to work with it in a practical situation. But increasingly it became clear that general asymptotic results are often quite irrelevant in typical problems of reasonable size. And certainly pattern matching with __ in Mathematica, as well as polynomial manipulation functions like GroebnerBasis, routinely deal with problems that are formally NP-complete.


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