Wednesday, May 16, 2018

Diamond Bracket Forms and How to Count to Two; 3 of 10


From "Cybernetics & Human Knowing", Vol. 24 (2017), No. 3-4, pp. 161-188
2. Re-Entrance and Fixedpoints

Consider the Liar Paradox as a Brownian form:
                   ______    
                        |
      L      =       L  |
                        |
This form contains itself. That can be represented via re-entrance, thus:
                      ___________
                                |
         L        =       |     |
                          |_____|

Let re-entrance permit any mark within a Brownian form to extend a tendril to a distant space, where its endpoint shall be deemed enclosed. This form sends a tendril into itself. Other re-entrant expressions include:
                     ___________
                      _______  |
                            |  |
                         |  |  |
                         |_____|
                     _____________
                       _____     |
                           |  |  |
                        |  |  |  |
                        |_____|__|

                  _____________________
                        ____________  |
                    |   ________   |  |
                    |          |   |  |
                    |      |   |   |  |
                    |      |_______|  |
                    |_________________|

                __________________________
                  __________  _________  |
                           |          |  |
                     |  |  |    |  |  |  |
                     |  |  |____|  |  |  |
                     |  |__________|__|  |
                     |_____________|_____|

Self-reference can be expressed as a re-entrant brownian form, as a switching circuit, as a vector of forms, as an indexed list, and as a diamond-logic fixedpoint. For example: 
     ___________
            _______ |
                  | |                          Brownian form
               |  | |
               |____|

     @
      __________________________    
/                          \        Switching Circuit
     \________|\________|\______/     (triangles = “not” gates)
         |/        |/          (wires and gates doubled)

    @
 ___             ___
|           ____   |
|   A   =    B |   |
|           ____   |                          Brownian Form Vector
|   B   =    A |   |
|__              __|

     @

   A  =  [ [ A ]B ]A                                          Indexed List

     @

     (A , B) = (not B , not A)             Diamond-logic Fixedpoint

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