question

[Alexander Shen]

I was asked (being at CiE) the following question (by a nice guy whom I
met on some AI meeting, but cannot remember the name now - I hope to see
him in September again).

Let a(x) be the maximal continuous semimeasure, and let m(x) be the
maximal discrete semimeasure. Then

a(xy) \ge m(x) a(y|x)

where string x is treated as a condition (or oracle) in the right hand
side. Indeed, imagine a machine that produces x and then produces y or
its continuations using x as an oracle.

In other terms,

m(x) \le min_y (a(xy)/(a(y|x))

Now the question: is this inequality an equality (up to a bounded
factors)? Probably not, it sounds to good to be true, and there should
be some game argument to show this (but I haven't found anything like
that yet - and may be I miss something important)