Evaluate GT / LT / GE / LE

Context

We already went through the EQ / NEQ evaluation, let's now see how to check, given 2 numbers A and B, if A > B or if B ≤ A
Initial state

• Memory: A B 0 0
• Cursor: first cell
• Input: any

Process

• Set else bit on fourth cell
• While A is not null
• Decrease A by 1
• If B is not null
• Decrease B by 1
• Reset else bit
• If B is null (i.e. if else bit is not null)
• A is definitely greater than or equal to B: no need to continue the loop, we can reset A
• Set else bit to 1 and go back to A
• Do some result formatting (A is always null, B is null if and only if B ≤ A, so this would be our result)

Code - try it

set else bit
>>+<<
[
  decrease A by 1
  -
  if B is not null: decrease B and reset else bit
  >[->]>
  if B is null (else bit not null)
  [
    reset A
    <<[-]
    clear else
    >>-
    go to same position
    >>
  ]
  set else bit and back to A
  <<+<<
]

format / copy result
if B is not null then set result to true
>[[-]<+>]
reset else bit
>-<<

Code - minified

>>+<<[->[->]>[<<[-]>>->>]<<+<<]>[[-]<+>]>-<< 

Final state

• Memory: result 0 0 0 with result = 1 if and only if A < B
• Cursor: on result
• Input: unchanged
• Ouput: unchanged

Note

This can be easily modified to compute GE / LT: simply set result to true before final check on B, that will do a reset instead of set. As B is not null if and only if B > A, then

• set result to true
• and reset if B > A

will give true for A ≥ B

Evaluate EQ / NEQ

Context

To know whether 2 values A and B are equal or not, we need to compute A - B: if the result is 0, then A=B, otherwise A<>B.
So, we can say that "A<>B" = (A-B).
Optionally, we can use some extra code to have evaluate A=B

Initial state

• Memory: A, B
• Cursor: first cell
• Input: any

Process

• While second cell is not null
• Decrease first and second cells by 1
• Loop invariant: first cell - second cell = A - B
• When second cell equals 0, first cell equals A - B
• Move to first cell

Code 

>[-<->]<

Final state

• Memory: A <> B, 0
• Cursor: first cell
• Input: any
• Output: unchanged

Note: to compute A=B, use second cell as 'else' bit, if A <> B then reset result and else bit; and finally move else bit to result position

>[-<->]+<[>-<[-]]>[-<+>]<