learning a grid

Now, an idea motivated by the concept of grid cells in rat brains.
So, we define a big grid using this python:

c = new_context("grid play")

def ket_elt(j,i):
  return ket("grid: " + str(j) + " " + str(i))

# Makes use of the fact that context.learn() ignores rules that are the empty ket |>.
def ket_elt_bd(j,i,I,J):
# finite universe model:
  if i <= 0 or j <= 0 or i > I or j > J:
    return ket("",0)
# torus model:
#  i = (i - 1)%I + 1
#  j = (j - 1)%J + 1
  return ket("grid: " + str(j) + " " + str(i))


def create_grid(c,I,J):
  c.learn("dim-1","grid",str(I))
  c.learn("dim-2","grid",str(J))

  for j in range(1,J+1):
    for i in range(1,I+1):
      elt = ket_elt(j,i)
      c.add_learn("elements","grid",elt)
      c.learn("N",elt,ket_elt_bd(j-1,i,I,J))
      c.learn("NE",elt,ket_elt_bd(j-1,i+1,I,J))
      c.learn("E",elt,ket_elt_bd(j,i+1,I,J))
      c.learn("SE",elt,ket_elt_bd(j+1,i+1,I,J))
      c.learn("S",elt,ket_elt_bd(j+1,i,I,J))
      c.learn("SW",elt,ket_elt_bd(j+1,i-1,I,J))
      c.learn("W",elt,ket_elt_bd(j,i-1,I,J))
      c.learn("NW",elt,ket_elt_bd(j-1,i-1,I,J))

And once we run this, we have example grid locations such as:

supported-ops |grid: 4 39> => |op: N> + |op: NE> + |op: E> + |op: SE> + |op: S> + |op: SW> + |op: W> + |op: NW>
N |grid: 4 39> => |grid: 3 39>
NE |grid: 4 39> => |grid: 3 40>
E |grid: 4 39> => |grid: 4 40>
SE |grid: 4 39> => |grid: 5 40>
S |grid: 4 39> => |grid: 5 39>
SW |grid: 4 39> => |grid: 5 38>
W |grid: 4 39> => |grid: 4 38>
NW |grid: 4 39> => |grid: 3 38>

supported-ops |grid: 4 40> => |op: N> + |op: NE> + |op: E> + |op: SE> + |op: S> + |op: SW> + |op: W> + |op: NW>
N |grid: 4 40> => |grid: 3 40>
NE |grid: 4 40> => |grid: 3 41>
E |grid: 4 40> => |grid: 4 41>
SE |grid: 4 40> => |grid: 5 41>
S |grid: 4 40> => |grid: 5 40>
SW |grid: 4 40> => |grid: 5 39>
W |grid: 4 40> => |grid: 4 39>
NW |grid: 4 40> => |grid: 3 39>

So, what is the point of this? Well, once you have a grid defined (or more generally, any topology of interest. eg for a mobile robot in a hospital you can map out the locations of corridors and walls), you can then associate objects with locations.
eg:
value |grid: 4 25> => |39>
building |grid: 30 22> => |building: hotel>
smell |grid: 5 7> => |smell: dead fish>
place |grid: 13 17> => |my place of work>
place |grid: 57 97> => |my home>

eg, "what is two steps north of the hotel?"
N N inverse-building |building: hotel>

eg, "what is seven steps SE of the dead fish smell?"
SE^7 inverse-smell |smell: dead fish>

And you can use it to update your knowledge of position (say when dead-reckoning):
|current location> => inverse-building |building: hotel>

I guess that is about it. This is a very powerful construct, and I don't mean just a grid topology, I mean for the case of more general topologies (eg a calendar!).

Update: Indeed, we can define a near current location too.
Something like:

-- and for other topologies, we define different near operators.
sa: near |grid: *> #=> |_self> + N|_self> + NE|_self> + E|_self> + SE|_self> + S|_self> + SW|_self> + W|_self> + NW|_self>
sa: building |grid: 4 40> => |building: cafe>
sa: create inverse
sa: current |location> => inverse-building |building: cafe>
sa: near-current |location> => near inverse-building |building: cafe>

-- now ask:
sa: current |location>
|grid: 4 40>

sa: near-current |location>
|grid: 4 40> + |grid: 3 40> + |grid: 3 41> + |grid: 4 41> + |grid: 5 41> + |grid: 5 40> + |grid: 5 39> + |grid: 4 39> + |grid: 3 39>

-- alternatively, we can save a step:
sa: near current |location>
|grid: 4 40> + |grid: 3 40> + |grid: 3 41> + |grid: 4 41> + |grid: 5 41> + |grid: 5 40> + |grid: 5 39> + |grid: 4 39> + |grid: 3 39>

-- "what is 13 steps NW of your current location?"
NW^13 current |location>

Neat!

Update: And we can easily enough define alias's for north and so on:

north |*> #=> N |_self>
north-east |*> #=> NE |_self>
east |*> #=> E |_self>
south-east |*> #=> SE |_self>
south |*> #=> S |_self>
south-west |*> #=> SW|_self>
west |*> #=> W |_self>
north-west |*> #=> NW|_self>

Update: an explanation about what I mean by a calendar topology. So, just like grid elements can be linked by north, south, east, west, and so on. A calendar has next-hour, previous-hour, tomorrow, yesterday, next-week, next-thursday, last-friday, etc. Then we build up a generic calendar, with time/date slots all linked by those operators, and then later you associate appointments, and so on with each calendar slot.