zlacker

>>tromp+(OP)
It's interesting, trying to figure out the conceptual basis for the greater power-per-bit of BLC compared to classical TMs. I wonder if it simply has to do with how β-reduction can pack so many more copy-and-pastes into a byte than TM states can, especially relative to the normal form that we use to define BB_λ. At small sizes, classical TMs are forced to do weird Collatz-like tricks, whereas BLC gets nearly-instant exponentiation. Do you have any thoughts on this?

(Also, I've been thinking of how that BB_λ conjecture might be proven. One strategy I'm thinking of for sufficiently large n would be to create a compression scheme for TMs which omits many copies of machines and trivial machines that cannot contribute to BB_TM, to get past the naive 2n(2+log₂(n+1))-bit bound. Then, we create a BLC program with a decompressor and a TM interpreter, to which a compressed TM can be appended. But the compression would have to be good enough to get around the overhead of bit sequences not being natively expressible in BLM.)

>>Legion+2c1
One thing that comes to mind is function application, which begets recursion. This is the last quarter of the example program's encoding, while the example TM has enough space for 6 states and no concept of a "goto" or "jmp" that would simulate recursion.

>>tadfis+4s1
On the contrary, 36 of its 60 bits are dedicated to goto labels. Although you're right that TMs cannot "call" a state then "return" to the caller, as would be expected from a reusable function. Regardless, for busy beaver purposes, most of TMs' state tends to be encoded in the tape rather than the state graph, so I suspect it's the differences in the means of data manipulation that create such a gap. (E.g., with substitution, you can easily move a term across an arbitrary amount of garbage.)