OFFSET
1,4
COMMENTS
Sizes of terms are defined as in Binary Lambda Calculus (see Lambda encoding link) by size(lambda M)=2+size(M), size(M N)=2+size(M)+size(N), and size(V)=1+i for a variable V bound by the i-th enclosing lambda.
a(34), a(35) and a(36) correspond to Church numerals 2^2^2^2, 3^3^3, and 2^2^2^3, where numeral n has size 5*n+6.
a(38) > 10^19729, corresponding to Church numeral 2^2^2^2^2.
Only a finite number of entries can be known, as the function is uncomputable.
Quoting from Chaitin's paper below: "to information theorists it is clear that the correct measure is bits, not states [...] to deal with Sigma(number of bits) one would need a model of a binary computer as simple and compelling as the Turing machine model, and no obvious natural choice is at hand."
a(49) > Graham's number, as shown in program melo.lam in the links. - John Tromp, Dec 04 2023
a(111) > f_{ε_0+1}(4), as shown in program E00.lam in the links. - John Tromp, Aug 24 2024
a(404) > f_{PTO(Z_2)+1}(4), as shown in 1st Stackexchange link. John Tromp, Dec 17 2024
a(1850) > Loader's number, as shown in 2nd Stackexchange link. John Tromp, Dec 17 2024
A universal form of this Busy Beaver, using the binary input feature of Binary Lambda Calculus, is given in sequence A361211. - John Tromp, May 24 2023
REFERENCES
Gregory Chaitin, Computing the Busy Beaver Function, in T. M. Cover and B. Gopinath, Open Problems in Communication and Computation, Springer, 1987, pages 108-112.
John Tromp, Binary Lambda Calculus and Combinatory Logic, in Randomness And Complexity, from Leibniz To Chaitin, ed. Cristian S. Calude, World Scientific Publishing Company, October 2008, pages 237-260.
LINKS
AIT repo on Github, melo.lam.
Code Golf Stack Exchange, Comment on BMS[26] entry
Code Golf Stack Exchange, 1850 bit lambda term exceeding Loader's number
MathOverflow, What's the smallest lambda-calculus term not known to have a normal form?, 2020.
John Tromp, The largest number representable in 64 bits, John's Blog.
John Tromp, Lambda encoding
John Tromp, program output analysis
EXAMPLE
The smallest closed lambda term is lambda x.x with encoding 0010 of size 4, giving a(4)=4, as it is in normal form. There is no closed term of size 5, so a(5)=0. a(21)=22 because of term lambda x. (lambda y. y y) (x (lambda y. x)).
CROSSREFS
KEYWORD
nonn
AUTHOR
John Tromp, Mar 23 2020
EXTENSIONS
a(33)-a(34) from John Tromp, Apr 10 2020
a(35) from John Tromp, Apr 18 2020
a(36) from John Tromp, Apr 19 2020
STATUS
approved