State Transitions

Suppose the following description of state transitions

From state 0 - if you see a 0, go to state 0; if you see a 1, go to state 1
From state 1 - if you see a 0, go back to state 0; if you see a 1, go to state 2
If you get to state 2, then stay in state 2 no matter what you see.


The above set of transitions can be described as

0 0 -> 0
0 1 -> 1
1 0 -> 0
1 1 -> 2
2 0 -> 2
2 1 -> 2

We can then represent this as a matrix where the element at [0;0] has a value of 0, at [0;1] has a value of 1 and so forth

m:(0 1
   0 2
   2 2)

We can now apply this state transition matrix over a sequence of inputs. m\ will produce the sequence of transitions and m/ will result in just the final state

  i:0 0 0 1 0 1 1 0 0 0 1 0 1 0
  m\i
0 0 0 1 0 1 2 2 2 2 2 2 2 2
  m/i
2

m': will calculate the state transitions for each pair of values in the input, i[1]->i[0] i[2]->i[1] i[3]->i[2] i[4]->i[3] and so forth. This operation is useful when we have overlapping pairs of inputs and states, or we can use it with just 1 input and 1 starting state

  i
0 0 0 1 0 1 1 0 0 0 1 0 1 0
  m':i
0 0 0 1 0 2 1 0 0 0 1 0 1
  m':0 2
,2
  m':1 2
,2