TITLE: An intuitive reason why the "Rule 30" CA is random/pseudorandom?
QUESTION [13 upvotes]: I'm a little bit hesitant to ask this here, so please notice the tag.  My hope is that someone will have a more satisfying answer than what I've heard before...
A long time ago I read (perhaps 'browsed' is a better word) Wolfram's "A New Kind of Science".  There are many many references to the "Rule 30" CA - http://mathworld.wolfram.com/Rule30.html.  However, no intuitive reasoning for it's random/pseudorandom behavior is provided prior to a digression to the importance of the discovery.  I was recently reminded of this when I heard that Mathematica (a program I use quite frequently) uses certain outputs from this CA as its random number generator.  
So my question is - beyond 'numerical phenomenology' is there an intuitive understanding why Rule 30 should behave in this random/pseudorandom manner?

REPLY [2 votes]: As I understand it Mathematica in fact uses rule 30 to generate pseudorandom numbers by just reading down a single column: so what it's exploiting is not just that rule 30 behaves pseudorandomly but that columns look more or less like strings of independent uniform random bits.  If we label the four cells on which the rule is defined

a b c
  d

it seems a lot easier to achieve this if (b,d) takes on all its possible values equally often, as rule 30 does.  
In fact, the stronger property is true that rule 30 can be run leftwards.  a is a (total) function of (b,c,d), therefore any two columns of cells can be completed uniquely to a halfplane extending leftwards satisfying rule 30.  This is suggestive at the very least -- if more things could appear to the left of some columns than others, the columns probably wouldn't be nice and uniform and all that.