Back to device_0
-- Guide to the Ridge Route
DEVICE_1
Device_0 is re-organized around the principal diagonal
Device_0 has a constraint that, for every ensemble, the sets of contexts and details must be disjoint. (No context can be a detail, and no detail can be a context.)
The constraint forbids a hierarchical structure as in the adjacent image. Example: book contains parts; parts contain chapters; chapters contain sections.
Re-organization overcomes the constraint.
Data Field: Re-defining the ensemble
An Ensemble is:
- a set of locations, {l}
- a set of ordered pairs of locations.
- the first location in the ordered pair is a context; the second, a detail
- Each location is involved in at least one ordered pair.
- Each ordered pair is a structural event, the appearance of a detail in a context
Hardware Realization of Device_1
1. The resistance matrix is organized into a diagonal form and
each location is identified with a position along the diagonal:
- The matrix is of size n x n, where n is the total number of data locations
-- any row or column from device_0 qualifies as a data location
- Each resistance is connected in series with a diode. The diode allows
current to flow only in the direction that, in device_0, would point from the context to the
detail (from a row-wire to a column-wire)
- Each data location connects through one row and one column
-- the row contains stuctural events where the data location acts as a context
-- the column contains structural events where it acts as a detail
- The order of locations along the diagonal can be changed without affecting the data field.
A simple change is accomplished by switching two locations. When two locations are switched, the two rows are switched and the two columns are switched.
Any re-arrangement
can be accomplished by a series of simple changes.
The possible re-arrangements are the same as (isomorphic to) the mathematical object known as the permutation group
2. Each location has terminal hardware on the diagonal:
- separate input and output hardware simultaneously operable
- input can be activated or null ("inactive") according to command
- a one-to-one transfer command that can activatate input hardware
for output on_locations previously triggered
- controlled through common lines that reset switches for on_locations,
set input level and execute commands
Interpretation
There are:
- data locations -- in connection with a particular command, a data location takes on the role of context or detail (only one at a time)
- on_locations
There are three forms of notation for structural events:
- Event notation -- the ordered pair from device_0: e.g, (context - detail)
- Matrix notation -- each location identified with a diagonal element (marked with a dot)
- Dot-arrow notation -- each location marked with a dot
Structural events can be assembled into structures. A structure familiar to programmers, the linked list,
is notated in:
- matrix notation (see adjacent image)
- dot-arrow notation (see adjacent image)
- event notation for the adjacent image (arranged in visually convenient groups):
- (list - item_1), (list - item_2), (list - item_3), (list - item_4)
- (item_1 - item_2), (item_2 - item_3), (item_3 - item_4)
- (item_2 - item_1), (item_3 - item_2), (item_4 - item_3)
The notations and the device:
- If you have a structure in one notation, you can construct the other two notations
- This is a useful property. The following rules make it work:
- you can change the order of events in event notation and nothing significant changes
- you can change the appearance of the dot-arrow network by continuous deformation and nothing significant changes
- you can change the order of entries along the diagonal of matrix notation and nothing significant changes
- All the notations can be used to describe the operation of the device.
- Equivalently:
- commands are invariant under diagonal re-arrangements (isomorphic to the permutation group) and/or deformation of networks
- the operation of the device does not depend on the notation used to interpret it
Problems interpreted through notation developed in Stage 3 of Water Trail -- 
- Thesaurus (an exercise for the visitor)
- Hypertext organization of this website
- Clusters of citations, where one document in a set of documents cites or names other documents: e.g., finding "hotspots" in the law by looking for dense clusters of citations
- Search methods involving co-ordination of hierarchical outlines, classification schemes,
citations and keywords
Forward to Device_2.
All materials copyright by Robert Kovsky, 1997.