User:Jon Awbrey/MNO

=Minimal negation operator=

JA: This is a critter that goes back to Leibniz, I think. In some dialects of graph theory it's called the link operator. It's a discrete analogue of the point-deleted neighborhood in calculus. I used to call it the boundary operator, because that's what it looks like in venn diagrams, and I think that there's an official name for it in co/homology theory, but I've forgotten what little I knew of that.

JA: If you are thinking of a n-cube (incubus) as your favorite graph, then each point of the cube can be represented as a conjunction of n posited or negated variables. So the all 1's node is the product of all positives x_1 x_2 … x_[n-1] x_n and the all 0's node is the product of all negatives (x_1)(x_2) … (x_[n-1])(x_n), here using parentheses for negation a la Peirce. These are called singular propositions (or singular boolean functions) because the fibre of truth under such a function is a single point of the cube. To be continued … Jon Awbrey 17:08, 17 August 2006 (UTC)

JA: Let's say that p is a product of literals, that is, a conjunction of posited or negated basis elements. Then p = e_1 e_2 … e_[n-1] e_n, where e_j = x_j or e_j = (x_j) = &not;x_j, for each j = 1 to n. Then the fibre of 1 under p, that is, (p^(-1))(1) is a single node of the n-cube.

JA: Given p as above, and representing the boundary operator or minimal negation operator (mno) of rank n by a bracket of the form "(, …, )", the proposition (e_1, e_2, …, e_[n-1], e_n) is true on the nodes adjacent to the node where p is true, and false everywhere else on the cube. Jon Awbrey 19:00, 17 August 2006 (UTC)

JA: For example, let's take the case where n = 3. Then the minimal negation opus $$\nu (p, q, r)\!$$, which we will eventually get so lax as to write "(p, q, r)" when there is minimal risk of misinterpretation, has the following venn diagram:

o-o |                                                | |                                                 | |                 o-o                 | |               /               \                | |               /                 \               | |              /                   \              | |             /                     \             | |            o                       o            | |           |           P           |            | |           |                       |            | |            |                       |            | |        o---o-o   o-o---o        | |      /     \`````````\ /`````````/     \       | |      /       \`````````o`````````/       \      | |    /         \```````/ \```````/         \     | |    /           \`````/   \`````/           \    | |   o             o---o-o---o             o   | |  |                 |`````|                 |   | |   |                 |`````|                 |   | |   |        Q        |`````|        R        |   | |  o                 o`````o                 o   | |   \                 \```/                 /    | |     \                 \`/                 /     | |      \                 o                 /      | |      \               / \               /       | |        o-o   o-o        | |                                                | |                                                 | o-o Figure 1. (p, q, r)

JA: Back in a flash ... Jon Awbrey 13:18, 18 August 2006 (UTC)

JA: (Some flashes are slower than others.) For a contrasting example, consider the boolean function expressed by the form $$((p),(q),(r))\!$$, whose venn diagram is as follows:

o-o |                                                | |                                                 | |                 o-o                 | |               /```````````````\                | |               /`````````````````\               | |              /```````````````````\              | |             /`````````````````````\             | |            o```````````````````````o            | |           |`````````` P ``````````|            | |           |```````````````````````|            | |            |```````````````````````|            | |        o---o-o```o-o---o        | |      /`````\         \`/         /`````\       | |      /```````\         o         /```````\      | |    /`````````\       / \       /`````````\     | |    /```````````\     /   \     /```````````\    | |   o`````````````o---o-o---o`````````````o   | |  |`````````````````|     |`````````````````|   | |   |`````````````````|     |`````````````````|   | |   |``````` Q ```````|     |``````` R ```````|   | |  o`````````````````o     o`````````````````o   | |   \`````````````````\   /`````````````````/    | |     \`````````````````\ /`````````````````/     | |      \`````````````````o`````````````````/      | |      \```````````````/ \```````````````/       | |        o-o   o-o        | |                                                | |                                                 | o-o Figure 2. ((p),(q),(r))

JA: TGIF! Jon Awbrey 21:32, 18 August 2006 (UTC)

Truth Tables

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Version 1
Table 1. A Family of Propositional Forms On Three Variables o-oo-o---o | L_1    | L_2        | L_3             | L_4               | |        |            |                 |                   | | Decimal | Binary     | Vector          | Cactus            | o-oo-o---o |        |          p : 1 1 1 1 0 0 0 0 |                   | |        |          q : 1 1 0 0 1 1 0 0 |                   | |        |          r : 1 0 1 0 1 0 1 0 |                   | o-oo-o---o |        |            |                 |                   | | q_22    | q_00010110 | 0 0 0 1 0 1 1 0 |  ((p), (q), (r))  | |        |            |                 |                   | | q_41    | q_00101001 | 0 0 1 0 1 0 0 1 |  ((p), (q),  r )  | |        |            |                 |                   | | q_73    | q_01001001 | 0 1 0 0 1 0 0 1 |  ((p),  q, (r))  | |        |            |                 |                   | | q_134   | q_10000110 | 1 0 0 0 0 1 1 0 |  ((p),  q,  r )  | |        |            |                 |                   | | q_97    | q_01100001 | 0 1 1 0 0 0 0 1 |  ( p, (q), (r))  | |        |            |                 |                   | | q_146   | q_10010010 | 1 0 0 1 0 0 1 0 |  ( p, (q),  r )  | |        |            |                 |                   | | q_148   | q_10010100 | 1 0 0 1 0 1 0 0 |  ( p,  q , (r))  | |        |            |                 |                   | | q_104   | q_01101000 | 0 1 1 0 1 0 0 0 |  ( p,  q ,  r )  | |        |            |                 |                   | o-oo-o---o |        |            |                 |                   | | q_233   | q_11101001 | 1 1 1 0 1 0 0 1 | (((p), (q), (r))) | |        |            |                 |                   | | q_214   | q_11010110 | 1 1 0 1 0 1 1 0 | (((p), (q),  r )) | |        |            |                 |                   | | q_182   | q_10110110 | 1 0 1 1 0 1 1 0 | (((p),  q, (r))) | |        |            |                 |                   | | q_121   | q_01111001 | 0 1 1 1 1 0 0 1 | (((p),  q,  r )) | |        |            |                 |                   | | q_158   | q_10011110 | 1 0 0 1 1 1 1 0 | (( p, (q), (r))) | |        |            |                 |                   | | q_109   | q_01101101 | 0 1 1 0 1 1 0 1 | (( p, (q),  r )) | |        |            |                 |                   | | q_107   | q_01101011 | 0 1 1 0 1 0 1 1 | (( p,  q , (r))) | |        |            |                 |                   | | q_151   | q_10010111 | 1 0 0 1 0 1 1 1 | (( p,  q ,  r )) | |        |            |                 |                   | o-oo-o---o