Definition talk:Minimal Negation Operator

Having spent some time this morning sorting out this page (adding missing definitions and links, tidying the format, etc.) I discover that it's a direct cut-and-paste of the Wikipedia article on the same subject: Minimal negation operator.

Nota Bene. en.wikipedia.org/w/index.php?title=Minimal_negation_operator&limit=500&action=history. Jon Awbrey 03:08, 23 August 2009 (UTC)

Yes, I understand you wrote it, but ProofWiki is trying to do something different from Wikipedia. There's more emphasis on the process of proving than of providing an infodump - no matter, the whole idea is that it evolves into what "works". Good job BTW. --Matt Westwood 16:19, 23 August 2009 (UTC)

Does anyone object if I drastically cut this down, by removing the venn diagrams and moving the truth table discussion into a "proof" page? --Matt Westwood 09:51, 4 July 2009 (UTC)

Go ahead. The text-based venn diagrams were ugly beasts anyway. --Cynic (talk) 14:58, 4 July 2009 (UTC)

Jon - what s/w do you use for generating those neat venn diagrams? Been looking all over for something that does this sort of job. --Matt Westwood 21:46, 22 August 2009 (UTC)

I'm using StarOffice7 — I think the current version is around 9 or so. Jon Awbrey 02:54, 23 August 2009 (UTC)

Hmm ... nightmare. I might have to continue searching ...

JA: What was the problem? I pretty much only use the "Drawing" application, and that works okay for the simple sorts of figures I do. Let me know if I can be of help. Jon Awbrey 11:58, 25 August 2009 (UTC)

I just don't like it very much. Had to wrestle with it at work too often. --Matt Westwood 19:26, 25 August 2009 (UTC)

Working on the exposition …

MW: BTW what's a "boundary" in this context? --Matt Westwood 18:12, 23 August 2009 (UTC)

JA: It's (the proposition that indicates) the set of cells or nodes that are adjacent to a given cell or node, so the proposition $\mathsf{(p, q, r)}$ indicates the cells or nodes adjacent to the the cell or node where $p \land q \land r$ is true.

JA: The name is not cast in stone — "border" or "margin" might work better. There's a species of graph theorists driving under the influence of algebraic topology who call it the "link" of the given point in a graph, but I thought that might be confusing in a Web context. Jon Awbrey 18:28, 23 August 2009 (UTC)

MW: What's "adjacent" in this context? --Matt Westwood 19:30, 23 August 2009 (UTC)

JA: In the $k\!$-cube picture, the points adjacent to a point $x\!$ are the points at a distance of 1 from $x.\!$ Jon Awbrey 20:00, 23 August 2009 (UTC)

MW: Is "boundary", then, a graph-theoretical term in this context? Seems we have some work to do to get that whole area up to scratch. Links are needed, because without this knowledge the page itself could be confusing. At a casual reading, one may not immediately glean that they are taken in the context of graph theory, despite the clue relating to the $k$-cube.

JA: There will be a way of making it clearer, I'm sure.

JA: The first line defines a set of boolean functions, $\nu_0 \in \mathbb{B}, \nu_1 : \mathbb{B}^1 \to \mathbb{B}, \nu_2 : \mathbb{B}^2 \to \mathbb{B}, \ldots, \nu_k : \mathbb{B}^k \to \mathbb{B}, \ldots.$ So the initial context is the subject matter of boolean spaces $\mathbb{B}^k$ and boolean functions $\mathbb{B}^k \to \mathbb{B}.$

JA: The points of $\mathbb{B}^k$ are often pictured as the vertices of a k-dimensional hypercube, so that is where a lot of the geometric and graph-theoretic language comes in. Jon Awbrey 21:50, 23 August 2009 (UTC)

JA: I'm planning to work up some 3-cube figures, maybe this week … Jon Awbrey 21:56, 23 August 2009 (UTC)

Plus vs. Oplus

I'll be avoiding the use of $\oplus$ for XOR, as $\oplus$ always means "direct sum" to me. I always use $+\!$ for the field operation in GF(2), and that interprets as XOR in logic. It is something of a double misnomer to use $+\!$ for $\lor$ and to call the latter "boolean addition" as Boole used $+\!$ for the exclusive disjunction (more or less). Jon Awbrey 12:24, 25 August 2009 (UTC)

I don't think there's any danger of $+$ being used for $\lor$ nowadays, the mathematical community seem to have settled on $\lor$ and it's foolish to use something else and expect to be understood ... OTOH there is no convention for XOR AFAIK, and every book uses something different. Matter of taste, I suppose. I've actually never seen $\oplus$ used as direct sum but then maybe I've been reading the wrong books. My view is: I'm not happy about $+$ being used because it is so easy to confuse it with its conventional meaning. No matter, it's your page. I did the recent change to $\oplus$ so as to avoid two different notations in the same paragraph. --Matt Westwood 19:24, 25 August 2009 (UTC)