A361211 Busy Beaver for the Binary Lambda Calculus (BLC) language BBλ2: the maximum output size of self-delimiting BLC programs of size n, or 0 if no program of size n exists.

0, 0, 0, 0, 0, 4, 0, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 30, 42, 52, 44, 64, 223, 160

Offset: 1

Author

John Tromp, Apr 09 2023

Keywords

• nonn
• more

Comments

Self-delimiting BLC programs are inputs p to the BLC universal machine U (defined in first link) that make U read all bits of p and none beyond. Formally, the only prefix p' of p for which U(p':Omega) has a normal form is p itself. The output of p is that normal form.

This Busy Beaver is directly related to Kolmogorov Complexity: a(n) = max {size(x)| KP(x) = n }, where KP is the prefix Kolmogorov complexity (defined in first link).

Because programs for U are at most a constant number of bits longer than programs for any prefix-free programming language, this busy beaver is optimal: for any other busy beaver BB based on self-delimiting programs, there is a constant c such that a(c+n) >= BB(n).

In particular, a(2+n) >= A333479(n), since for every closed term T, U(00 code(T) : Omega) = (lambda _. T) Omega = T. All entries above except for n=30 are of this form.

We can show that for some k, a(ceiling((113/114)*n) + k) > A333479(n), i.e., universality eventually pays off for BLC. See program link for the supporting computation. - Bertram Felgenhauer, Apr 10 2023

Examples

			The smallest closed lambda term is lambda x.x but its application to the unsolvable Omega lacks a normal form. The next smallest is lambda x.lambda y.y with encoding 000010 of size 6, which applied to Omega yields the normal form lambda x.x, giving a(6)=4.
a(30) = 64 because, with T=lambda x.lambda y.lambda z.x(z x), (lambda x.x x) T applied to Omega yields maximum size normal form lambda x.lambda y.lambda z.x T(z (x T)).
		

Links

• John Tromp, Binary Lambda Calculus
• John Tromp, Haskell program to compute initial terms
• John Tromp, program demonstrating BLC compressibility

Crossrefs

Cf. A333479.