Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

CHAPTER 5A Potential Subtlety Concerningthe Distinction betweenDeterminism andNondeterminismW. Hugh Woodin5.1 The Coding of Information into TimeIt is well known that the property of randomness for finite binary sequences based oninformation content is not decidable (Li and Vitányi 1997). We produce a dramaticversion of this, but it is not our goal to simply reproduce this undecidability result;rather, our intention in this chapter is to illustrate a potentially subtle aspect of thedistinction between determinism and nondeterminism. This subtlety is the possibilityof coding arbitrary information into time in such a way that a specific deterministicprocess computes precisely that information (as additional output).Our basic argument begins as follows. Given any specific set of physical laws,we produce a Turing program e 0 with the following feature. First, the output of theprogram e 0 (by virtue of its format) must be a finite binary sequence s, but there may beno output generated. Let t be any nonempty finite binary sequence (e.g., flip a quantumcoin 10 100 times, and let t be the outcome). Then by “extending time” and preserving allthe specified physical laws, one can arrange that in the “new” universe the program e 0generates as additional output exactly the chosen string t (so that in the case in whichno output was initially generated by the program, in the new universe the total outputgenerated is exactly t; otherwise, the total output generated is exactly s appendedby the sequence t). From the perspective of the inhabitants of the universe (i.e., ourperspective), passing from the initial universe to the extension is not an observablechange because all laws have been preserved; more precisely, the intial universe isnot an observable entity in the extension. The program e 0 is explicit; it only depends(as it must) on the specification of the physical laws – if that specification is simple,then so is the program. We are implicitly assuming that time is infinite in the idealizeduniverse and so the “extension” is a nonstandard extension (which again is the onlypossibility). In fact, for the construction of e 0 we give, the program e 0 generates nooutput within our (idealized) universe. We emphasize that the property for the potentialoutput that we have indicated that the program e 0 has applies to any abstract versionof our universe that obeys the specified laws and is not an artifact of our universe: it119