It appears that under this pair encoding, each new input would be applied as a two-argument function to the existing list, when we presumably want it to be fed to the last element of the list in order to continue the stream of output growing to the right. Otherwise there’s effectively still a restriction to finite input, before an infinite output can be produced.
Assuming that problem were fixed:
Under Vanessa’s distribution on terms, only finite terms will be sampled with probability one, so you’d effectively feed in an infinite sequence of finite terms.
There’s some flexibility on the input distribution preserving universality; see “Learning universal predictors” theorem 9.
If I had to guess, this is probably enough for universality, but I don’t know this.
Li and Vitanyi has a section on concrete models or some such that includes a discrete lambda calculus prior (not for induction).
some quick thoughts:
It appears that under this pair encoding, each new input would be applied as a two-argument function to the existing list, when we presumably want it to be fed to the last element of the list in order to continue the stream of output growing to the right. Otherwise there’s effectively still a restriction to finite input, before an infinite output can be produced.
Assuming that problem were fixed:
Under Vanessa’s distribution on terms, only finite terms will be sampled with probability one, so you’d effectively feed in an infinite sequence of finite terms.
There’s some flexibility on the input distribution preserving universality; see “Learning universal predictors” theorem 9.
If I had to guess, this is probably enough for universality, but I don’t know this.
Li and Vitanyi has a section on concrete models or some such that includes a discrete lambda calculus prior (not for induction).