The goal of this post is to show how to use graphic lambda calculus for understanding the SKI combinators. For the graphs associated to the SKI combinators see this post.

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**UPDATE:** See also the post “Combinators and stickers“.

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Zippers have been introduced here. In particular, the first three zippers are depicted in the following figure.

The combinator has the expression and it satisfies the relation , where means any combination of beta reduction and alpha renaming (in this case is just one beta reduction: ).

In the next figure it is shown that the combinator (figured in green) is just a half of the zipper_1, with an arrow added (figured in blue).

When you open the zipper you get , as it should.

The combinator satisfies . In the next figure the combinator (in green) appears as half of the zipper_2, with one arrow and one termination gate added (in blue).

When you open the zipper you obtain a pair made by and which gets the termination gate on top of it. GLOBAL PRUNING sends B to the trash bin.

Finally, the combinator satisfies . The combinator (in green) appears to be made by half of the zipper_3, with some arrows added and also with a “diamond” added (all in blue). Look well at the “diamond”, it is very much alike the emergent sum gate from this post.

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