LAUSR

The authors use the phrase ‘‘solves in a polynomial number of steps’’ in a polysemous fashion. Specifically, the term ‘‘solves’’ is used in at least two different capacities—with two different types, so to speak. In the first case, the left domain of the solves predicate is physical phenomena, and in the second case it is Turing machines. The authors need to distinguish these senses, say, by indexing them as solves1 and solves2. Doing so would help to bring out the main flaw of their argument, which is this: What it means for a macroscopic physical process to solve a problem in a polynomial number of steps is very different from what it means for a digital computer—such as a Turing machine—to solve a problem in a polynomial number of steps. Accordingly, the appeal to digital physics for justifying the third premise is misplaced. The core theses of digital physics are two: (a) the universe is discrete, both spatially and temporally (e.g., at any given point in time the state of every elementary particle can be described by a finite amount of information, say, a bit) and (b) its dynamic evolution can be described by a finite set of computable rules akin to the transition rules of a cellular automaton. Even if we grant these claims, it does not follow that every macroscopic procedure A with complexity T(n) (where n is the number of elementary steps taken by A) can be carried out by a cellular automaton with the same complexity T(n), or even with complexity T(p(n)) for some polynomial p. What digital physics tells us, in short, is that the evolution of every part of the universe proceeds in accordance with some Turing machine. It says nothing about how the asymptotic complexity of that machine correlates to the complexity of macroscopic procedures such as the milking of cows or the revolutions of planets around stars. It does not follow, in other words, that the milking of n cows will