User blog:Wythagoras/Googological problems II.

1. Define TREE(a,b) as the following:

''Suppose we have a sequence of a-labeled trees T1, T2 ... with the following properties: '' TREE(a,b) is the maximal length of the sequence is a function of a and b now.
 * Each tree Ti has at most i+b vertices. 
 * No tree is homeomorphically embeddable into any tree following it in the sequence. 

Show that TREE(3,0) = TREE(2,1)+1.

2. Define the weak array notation as following:

\(\circ\) is an array of ones.

\(\bullet\) is any array.

\(\{a,b,\circ\} = a^b\)

\(\{a,1,\bullet\} = a\)

\(\{a,b,\circ,c,d,\bullet\} = \{\{a,b-1,\circ,c,d,\bullet\},b,\circ,c,d-1,\bullet\}\)

Find the exact value of the function \(\{a,b(1)2\} = \underbrace{\{a,a...a,a\}}_b\) in terms of addition, multiplication, exponentiation and the factorial extension \(a!b\) by Aalbert Torsius.

3. Prove \(a!2 < (\frac{2a+1}{3})^{\frac{a(a+1)}{2}}\) (\(a!2\) is the factorial extension by Aalbert Torsius)

4. a) Find an ITTM with 3 states, 3 tapes that clocks \(\omega^\omega\)

b) Find an ITTM with 4 states, 3 tapes that clocks \(\omega^{\omega2}\)