# Combinators and zippers

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

The combinator $I$ has the expression $I = \lambda x . x$ and it satisfies the relation $I A \rightarrow A$, where $\rightarrow$ means any combination of beta reduction and alpha renaming (in this case is just one beta reduction: $I A = \left( \lambda x . x \right) A \rightarrow A$).

In the next figure it is shown that the combinator $I$  (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 $A$, as it should.

The combinator $K = \lambda xy.x$ satisfies $K A B = (KA)B \rightarrow A$. In the next figure the combinator $K$ (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 $A$   and $B$ which gets the termination gate on top of it. GLOBAL PRUNING sends B to the trash bin.

Finally, the combinator $S = \lambda xyz. ((xz)(yz))$ satisfies $SABC = ((SA)B)C \rightarrow (AC)(BC)$. The combinator $S$ (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.

# The value of the article given by the journal? It’s like that tomcat joke

At least in mathematics, but I suppose in almost any domain, people use to justify the huge inertia which stops us moving to a better publishing system by appeal to the following argument:

• publishing in a well recognized journal gives prestige to the article,
• most of well recognized journals are based on the traditional publication system
• therefore, if you want your article to be considered as valuable then you better submit it to a traditional, well seen journal.

There is a big hole in this argument, namely that the value of an article is primarily coming from the value of the journal where the article appeared. While it is true that good journals offer a better platform of dissemination for the research works, this is also on the border of recognizing that vanity is the leading force for doing good research (which might be true, after all we are humans and vanity plays an important role in our lives, but “leading force”? this would be just sad).

The value of a good journal is more of a statistical nature, it comes from having several very good articles, written by some very good researchers,   all this spread in a sea of all the other not fantastically important articles which appeared in the journal.

This shaky but so powerful argument pro traditional publishing  reminded me a joke which I want to share. Maybe you know it, but have you thought about it from the viewpoint of publishing habits?

Here is the joke, I kind of formulated it such as to not contain offensive words:

There are an old tomcat and a very young one, inside the house, during a rainy night. The old tomcat finds an open window and wants to get out. The young one asks him:

– Where are you going?

– Well, I’m going out to get some cats, junior, would you like to come with me?

They get out and up on the roofs. It rains, but the old experienced tomcat knows that it is only a matter of time until some cat appears. The young one is not very happy, is wet and bored, so he says:

-You know, I’m going to get some cat for five minutes more, then I’m going back home.

So, you see,  just from publishing in the same place as the old tomcat, you won’t get some cat time of your own.

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UPDATE: On a more serious note read “Impacting our young” by Eve Marder, Helmut Kettenmann and Sten Grillner, (PNAS vol 107, no. 50).  (I learned about this article by following a link from the article “How to bury your academic writing“, discovered by the intermediary of a post by  +Michael Rowe, thanks!)

# Ancient CS: Arbor Porphyriana

I could not resist to the title, so I started to read Umberto Eco, Dall’albero al labirinto. Studi storici sul segno e l’interpretazione (Bompiani 2007). (The brute English translation of the title is “From the tree to the labyrinth. Historical Studies on  sign and  interpretation “.) Just opening the book, I learned about Arbor Porphyriana.

I shall cite from the very rough description given in the wiki page (before, let me laugh a bit by reading from the talk page of the mentioned wiki page: “This article has been rated as Low-importance on the project’s importance scale.” Ha-ha, an encyclopedia to state that arbor porphyriana is a low importance subject, that’s weird. Is like the egg stating that chickens are not important on it’s scale.)

The Porphyrian tree, Tree of Porphyry or Arbor Porphyriana is a classic classification of a “Scale of being”,[1] invented by one of the earliest Greek logicians Porphyry.[2] It is also known as scala praedicamentalis.

The Greek Neoplatonist Porphyry introduced the Porphyrian tree in his introduction to Aristotle‘s Categories. Porphyry presented the basis of Aristotle’s thought as a tree-like scheme of dichotomous divisions, which indicates that a species is defined by genus-differentia and that the process continues until the lowest species is reached.

This work was translated into Latin by Boethius and became the standard philosophical textbook in the Middle Ages.[3] Until the late 19th century, it was still being taught to students of logic.

So, that’s a huge subject. A short google search for “arbor porphyriana ontology semantic web” gives on the first place these very interesting slides by Harald Sack: “Semantic Web Technologies“.

# On graphic lambda calculus and the dual of the graphic beta move

Much of the research reported in the article “On graphic lambda calculus and the dual of the graphic beta move”  arXiv:1303.0778  appeared first time in posts from this blog, in places indicated by links given in the article  (here is the link to the preview, before the appearance in the arxiv).

The abstract is:

This is a short description of graphic lambda calculus, with special emphasis on a duality suggested by the two different appearances of knot diagrams, in  lambda calculus  and emergent algebra sectors of the graphic lambda calculus respectively. This duality leads to the introduction of the dual of the graphic beta move. While the graphic beta move corresponds to beta reduction in untyped lambda calculus, the dual graphic beta move appears in relation to emergent algebras.

See the page Graphic lambda calculus for details.

I played also with the bibliography, in two ways: I tried to cite only articles from arxiv and I give each time in the text the link to the respective article, also I preferred to indicate web pages as sources of bibliographic information whenever possible. This way, the bibliography is reduced to the bare minimum, it is there mostly by convention. Finally, there are no theorems or definitions in the text (I mean, there are, as well as proofs, only that they are not “encrypted”  into the respective formats), I preferred instead to use a more relaxed writing, more alike  wiki pages, according to views expressed in “Comments in epijournals: we may learn from Wikipedia”  and “Wiki journals over arxiv“.

I first wanted to make a tutorial article on graphic lambda calculus, but for the moment I don’t see the point, there already is such a tutorial here; much of it is in several arxiv articles. But probably this article will have updates, if it will be submitted to a “regular” publisher. For the moment it is a bit of an experiment (but mind you, it is a rigorous mathematical article).

# Wiki journals over arxiv

Just dreaming. The technical part first. Then comes the social part, which is trickier.

• The author A of an arxiv article submits the latex version to an editor E of the wiki-journal.
• The editor transforms the latex file into the wiki format of the journal. There seem to be tools for this, a quick google search gives this latex2wiki.
• The editor E creates a wiki page for the article. We can use MediaWiki, we can go to the WikiWikiWeb, details to be discussed. At this moment the wiki page can be deleted only if both A and E agree.
• This wiki page is modified by anyone in the PEER COMMUNITY of the wiki-journal. A link to the original version arxiv article is given, this can be modified only by the author A.

Now, the social part:  only  suggestions.

• Any author A becomes member of a PEER COMMUNITY, there is some mathoverflow type reputation and badges system.
• PEER COMMUNITIES and wiki-journals are different parts of the system, one PEER COMMUNITY may act on several wiki-journals, one wiki-journal may contact several PEER COMMUNITIES, but only one per article.
• anybody can be member of several PEER COMMUNITIES
• to make a very rough comparison, wiki journals are like companies and PEER COMMUNITIES are like syndicates

The most important point: we can start it now, the soft (open source) exists, anybody can try to do it. There is no need to wait for anybody’s approval, no need to wait several months to  see what exactly are   epijournals  (however see epimath), anybody can just try and contribute, instead of us (mathematicians) being one of the least reactive communities when it comes to the future of publication.

What do you think?

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UPDATE: Something close to this idea already exists, see knowledgeblog.org.

UPDATE 2: This kind of proposal has already been made, see these two articles:

… however, both papers look like minor adaptations of the new system of the world, made in order to fit into the old one. This may be good for starters, or it may be not good enough. We still long for a really great idea, for the moment.

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Background: