Notes

Chapter 12: The Principle of Computational Equivalence

Section 2: Outline of the Principle


[History of] Church's Thesis

The idea that any computation that can be done at all can be done by a universal system such as a universal Turing machine is often referred to as Church's Thesis. Following the introduction of so-called primitive recursive functions (see page 907) in the 1880s, there had by the 1920s emerged the idea that perhaps any reasonable function could be computed using the small set of operations on which primitive recursive functions are based. This notion was supported by the fact that certain modifications to these operations were found to allow only the exact same set of functions. But the discovery of the Ackermann function in the late 1920s (see page 906) showed that there are reasonable functions that are not primitive recursive. The proof of Gödel's Theorem in 1931 made use of so-called general recursive functions (see page 1121) as a way to represent possible functions in arithmetic. And in the early 1930s the two basic idealizations used in foundational studies of mathematical processes were then general recursive functions and lambda calculus (see page 1121). By 1934 these were known to be equivalent, and in 1935 Alonzo Church suggested that either of them could be used to do any mathematical calculation which could effectively be done. (It had been noted that many specific kinds of calculations could be done within such systems—and that processes like diagonalization led to operations of a seemingly rather different character.) In 1936 Alan Turing then introduced the idea of Turing machines, and argued that any mathematical process that could be carried out in practice, say by a person, could be carried out by a Turing machine. Turing proved that his machines were exactly equivalent in their computational capabilities to lambda calculus. By the 1940s Emil Post had shown that the string rewriting systems he had studied were also equivalent, and as electronic computers began to be developed it became quite firmly established that Turing machines provided an appropriate idealization for what computations could be done. From the 1940s to 1960s many different types of systems—almost all mentioned at some point or another in this book—were shown to be equivalent in their computational capabilities. (Starting in the 1970s, as discussed on page 1143, emphasis shifted to studies not of overall equivalence but instead equivalence with respect to classes of transformations such as polynomial time.)

When textbooks of computer science began to be written some confusion developed about the character of Church's Thesis: was it something that could somehow be deduced, or was it instead essentially just a definition of computability? Turing and Post seem to have thought of Church's Thesis as characterizing the "mathematicizing power" of humans, and Turing at least seems to have thought that it might not apply to continuous processes in physics. Kurt Gödel privately discussed the question of whether the universe could be viewed as following Church's Thesis and being "mechanical". And starting in the 1950s a few physicists, notably Richard Feynman, asked about fundamental comparisons between computational and physical processes. But it was not until the 1980s—perhaps particularly following some of my work—that it began to be more widely realized that Church's Thesis should best be considered a statement about nature and about the kinds of computations that can be done in our universe. The validity of Church's Thesis has long been taken more or less for granted by computer scientists, but among physicists there are still nagging doubts, mostly revolving around the perfect continua assumed in space and quantum mechanics in the traditional formalism of theoretical physics (see page 730). Such doubts will in the end only be put to rest by the explicit construction of a discrete fundamental theory along the lines I discuss in Chapter 9.


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