I record here some differences in reduction between chemlambda and dirIC, then I’ll talk very briefly about chemSKI for SKI or SK calculus.
Chemlambda vs dirIC
Chemlambda and directed Interaction Combinators (dirIC) are not equal.
Play with div from https://github.com/tromp/AIT/blob/master/numerals/div.lam
Use https://mbuliga.github.io/quinegraphs/lambda2mol.html
As chemlambda or dirIC want only one term to be inputted, we have to build it progressively, so here is the process:
with all parantheses…
1tuple = (\x.(\f.(f x)))
id = (\x.x)
F = (\m.(\f.(\t.((((m (\x.(\f.(f x)))) (\c. f (c t))) (\x.x))))))
K = (\x.(\y.x))
div = (\n.(\m.(\f.(\x.(((n (\x.(\f.(f x)))) ((\x.(\y.x)) x)) ((n (((\m.(\f.(\t.((((m (\x.(\f.(f x)))) (\c. f (c t))) (\x.x)))))) m) f)) x))))))
we prepare the term
term = (((div number1) number2) a) b
replaced div
term = ((((\n.(\m.(\f.(\x.(((n (\x.(\f.(f x)))) ((\x.(\y.x)) x)) ((n (((\m.(\f.(\t.((((m (\x.(\f.(f x)))) (\c. f (c t))) (\x.x)))))) m) f)) x)))))) number1) number2) a) b
for
number1 = (\f.(\x.(f (f (f (f (f (f x)))))))) (ie Church 6)
number2 = (\f.(\x.(f (f (f x))))) (ie Church 3)
we get
term = ((((\n.(\m.(\f.(\x.(((n (\x.(\f.(f x)))) ((\x.(\y.x)) x)) ((n (((\m.(\f.(\t.((((m (\x.(\f.(f x)))) (\c. f (c t))) (\x.x)))))) m) f)) x)))))) (\f.(\x.(f (f (f (f (f (f x))))))))) (\f.(\x.(f (f (f x)))))) a) b
which works in chemlambda but not in dirIC.
For the term
term1 = (div number1) number2
we replace div, number1 and number2
term1 = ((\n.(\m.(\f.(\x.(((n (\x.(\f.(f x)))) ((\x.(\y.x)) x)) ((n (((\m.(\f.(\t.((((m (\x.(\f.(f x)))) (\c. f (c t))) (\x.x)))))) m) f)) x)))))) (\f.(\x.(f (f (f (f (f (f x))))))))) (\f.(\x.(f (f (f x)))))
works in chemlambda, not in dirIC.
For
number1 = (\f.(\x.(f (f (f (f (f (f (f x))))))))) (ie Church 7)
number2 = (\f.(\x.(f (f (f x))))) (ie Church 3)
we get
term1 = ((\n.(\m.(\f.(\x.(((n (\x.(\f.(f x)))) ((\x.(\y.x)) x)) ((n (((\m.(\f.(\t.((((m (\x.(\f.(f x)))) (\c. f (c t))) (\x.x)))))) m) f)) x)))))) (\f.(\x.(f (f (f (f (f (f (f x)))))))))) (\f.(\x.(f (f (f x)))))
works in chemlambda 7 div 3 = 2
_________________
ChemSKI for SKI or SK calculus
I does not matter which one is used, in the sense that if you don’t input any I combinator, then you don’t get any. There is a lot of waste, but now this is managed by waste rewrites.
Maybe relevant in the larger discussion chemSKI not Kind.
Examples:
Use https://mbuliga.github.io/chemski/chemski-with-tokens.html#PlayPlace
Down the page tromp.github.io is a minimal SK version of the Y combinator.
S S K (S (K (S S (S(S S K)))) K)
Applied to I should give a quine
((((S S) K) ((S (K ((S S) (S ((S S) K))))) K)) I)
Applied to S K K instead should work as well
((((S S) K) ((S (K ((S S) (S ((S S) K))))) K)) (S K K))
It does. ChemSKI rules rule!
chorasimilarity 