Computability and Physical Theories

Abstract

The familiar theories of physics have the feature that the application of the theory to make predictions in specific circumstances can be done by means of an algorithm. We propose a more precise formulation of this feature -one based on the issue of whether or not the physically measurable numbers predicted by the theory are computable in the mathematical sense. Applying this formulation to one approach to a quantum theory of gravity, there are found indications that there may exist no such algorithms in this case. Finally, we discuss the issue of whether the existence of an algorithm to implement a theory should be adopted as a criterion for acceptable physical theories.

On a Possibly Pure Set-Theoretic Contribution to Black Hole Entropy

“…Motivated in various ways by [7,12,13] we interpret this quite surprising mathematical fact from our viewpoint as follows: Taking into account that the only known property of a generic real number which fully characterizes it is its existing (decimal) expansion, but the Kolmogorov complexity of this expansion hence the expansion itself generally is not fully determinable (by proving theorems on it in an axiomatic system), there is in general no way, using standard mathematical tools in the broadest sense, to "sharply pick" any element from the arithmetical continuum. Consequently, from the viewpoint of "effective mathematical activity", the structure of the arithmetical continuum i.e.…”

The Ghost in the Quantum Turing Machine

“…On the other hand, if F ′ = F , then we have two independent measurements of f to f 's future, violating the No-Cloning Theorem. 40 Note that this argument wouldn't have worked if f had been a macrofact, since macroscopic information can be measured many times independently.…”

“…In Shadows [68], Penrose seems to admit that this is a weak link in his argument. As evidence for uncomputability, the best he can offer is a theorem of Markov that the 4-manifold homeomorphism problem is undecidable (indeed, equivalent to the halting problem) [60], and a speculation of Geroch and Hartle [40] that maybe that fact has something to do with quantum gravity, since some attempted formulations of quantum gravity involve sums over 4-manifolds.…”

Counting steps: a finitist approach to objective probability in physics

“…These assumptions are working hypotheses in the framework from which our interpretation of probability stems, namely, physical computational complexity. In this framework (Geroch & Hartle, 1986;Pitowsky, 1990;Pitowsky, 1996), the performance of physical systems is analyzed with notions and concepts that originate in computational complexity theory, by approximating dynamical evolutions with a discrete set of computational steps to an arbitrary degree of accuracy. These assumptions help us delineate the two probability spaces in our models: the space of physically allowable states, and the space of physically allowable dynamical evolutions.…”

“…One could still observe chaotic behavior in this context, 13 but this would require redefining notions such as "sensitivity to initial conditions" or "dynamical instability" to fit the discrete background, and would also require a careful analysis of time scales. 14 But even such a demonstration would still fall short of supporting the common lore. The standard notion of probability concerns a sequence of 13 There is no compelling reason to associate chaos only with the cardinality of the reals.…”