1b: Tree Addressing

    Check out the Nock explanation for more information on the tree-addressing system.

    ++cap

    Tree head

    Tests whether the tree address a is in the head or the tail of a noun. Produces the constant atom %2 if it is within the head (subtree +2), or the constant atom %3 if it is within the tail (subtree +3).

    Accepts

    a is an atom.

    Produces

    A constant atom.

    Source

            ++  cap
              ~/  %cap
              |=  a=@
              ^-  ?($2 $3)
              ?-  a
                $2        %2
                $3        %3
                ?(%0 %1)  !!
                *         $(a (div a 2))
              ==
    

    Examples

        > (cap 4)
        %2
    
        > (cap 6)
        %3
    
        > (cap (add 10 9))
        %2
    
        > (cap 1)                                    ::address '1' is in neither the head nor the tail
        ! exit
    
        > (cap 0x40))
        %2
        > `@`0x40
        64
    
        > (cap 'a')
        %3
        > `@`'a'
        97
    

    ++mas

    Address within head/tail

    Computes the tree address of atom a within either the head (+2) or tail (+3) of a noun.

    Accepts

    a is an atom.

    Produces

    An atom.

    Source

          ++  mas
              ~/  %mas
              |=  a=@
              ^-  @
              ?-  a
                1   !!
                2   1
                3   1
                *   (add (mod a 2) (mul $(a (div a 2)) 2))
              ==
    

    Examples

        > (mas 3)
        1
    
        > (mas 4)
        2
    
        > (mas 5)
        3
        > (cap 5)                                    ::`(cap a)` computes whether address `a` is in the head or the tail
        %2
    
        > (mas 7)
        3
        > (cap 7)
        %3
    
        > (mas 11)
        7
    
        > (mas (mas 11))
        3
    
        > (cap (mas 6))
        %3
    
        > (mas 0)
        ! exit                                       ::address `0` is in neither the head nor the tail
    
        > (mas 1)
        ! exit                                       ::address `1` is in neither the head nor the tail
    

    Discussion

                1
              /   \
             /     \
            2       3              <--here are the head (`+2`) and the tail (`+3`)
           / \      /\
          4   5    6  7
         /\   /\  /\  /\
         (continues...)
    

    Running (mas 7) in the Dojo will return 3, because address +3 is what +7 now occupies. The tree below helps illustrate the relationship. With parentheses are a values (if a is in subtree +3), and without parentheses are the values returned with (mas a).

                1(3)                       ::new/(old) addresses
               /    \
              2       3
             (6)     (7)
            / \       /\
           /   \     /  \
          4     5   6    7
         (12) (13) (14) (15)
        / \    / \ / \   / \
           (continues...)
    

    Notice how the old values in the head (subtree +2) were not illustrated in this case, because +7 is within the tail (subtree +3).


    ++peg

    Address within address

    Computes the absolute address of b, a relative address within the subtree a.

    Accepts

    a is an atom.

    b is an atom.

    Produces

    An atom.

    Source

        ++  peg
          ~/  %peg
          |=  [a=@ b=@]
          ?<  =(0 a)
          ^-  @
          ?-  b
            $1  a
            $2  (mul a 2)
            $3  +((mul a 2))
            *   (add (mod b 2) (mul $(b (div b 2)) 2))
          ==
    

    Examples

        > (peg 4 1)
        4
    
        > (peg 1 4)
        4
    
        > (peg 4 2)
        8
    
        > (peg 4 8)
        32
    
        > (peg 4 (4 2))
        32
    
        > (peg 8 45)
        269
    
        > (cap (peg 4 2))                            ::`(cap a)` computes whether address `a` is in the head or the tail
        %2
    
    

    Discussion

    In other words, the subtree at address a is treated as a tree in its own right (starting with root +1, head +2, and tail +3). Relative address b is found with respect to a, and then its absolute address, within the greater tree, is returned.

    Running (peg 3 4) in the Dojo, for example, will return 12. Looking at a tree diagram makes it easy to see why.

                     1
                  /     \
                 /       \
                /         \
               2           3       <- here is the subtree `+3`. The subtree address is `a` in `(peg a b)`
              / \         / \
             /   \       /   \
            4     5     6     7
           / \   / \   / \   / \
          8  9  10 11 12 13 14  15
         /\  /\ /\ /\ /\ /\ /\  /\
             (continues...)
    

    When we consider subtree at address +3 by itself, it has relative addresses that are structured in the same way as its parent tree's absolute addresses. The absolute addresses are given in parentheses in the diagram below. Notice how relative address +4 is at the same position as absolute address +12.

                1(3)                        ::new/(old) addresses
               /    \
              2       3
             (6)     (7)
            / \       /\
           /   \     /  \
          4     5   6    7
        (12) (13) (14)  (15)
        / \    / \ / \   / \
           (continues...)