Open Problems

(1)
What is the probability of the following ?
   A finite set of proper combinators 
   satisfying Craig's conditions + ?
   is a basis.

Where 
proper combinator = closed normal form
                    with no embedded lambda's
Craig's conditions = compositive
                     permutative
                     calcellative &
                     duplicative effect
                     (Curry & Feys Vol. 1)
+ = at least one more condition is necessary;
    that at least one combinator has order > 2.
    But add more if needed to get a nice result.

(2) 
For normalizable combinatory terms (conditional
prob.)
true or false? 
     A random reduction sequence is normalizing.

(3) 
For non-normalizable lambda terms (conditonal 
prob.)
true or false?
      A random reduction sequence is confinal.

(4) Is there a first order theory of lambda/
    combinatory terms with a decidable theory
    of sentences which hold with probability 1
    (like the theory for graphs)?