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Treating Trees
==============
Although any expression in Joy can be considered to describe a
`tree <https://en.wikipedia.org/wiki/Tree_structure>`__ with the quotes
as compound nodes and the non-quote values as leaf nodes, in this page I
want to talk about `ordered binary
trees <https://en.wikipedia.org/wiki/Binary_search_tree>`__ and how to
make and use them.
The basic structure, in a `crude type
notation <https://en.wikipedia.org/wiki/Algebraic_data_type>`__, is:
::
BTree :: [] | [key value BTree BTree]
That says that a BTree is either the empty quote ``[]`` or a quote with
four items: a key, a value, and two BTrees representing the left and
right branches of the tree.
A Function to Traverse this Structure
-------------------------------------
Let's take a crack at writing a function that can recursively iterate or
traverse these trees.
Base case ``[]``
^^^^^^^^^^^^^^^^
The stopping predicate just has to detect the empty list:
::
BTree-iter == [not] [E] [R0] [R1] genrec
And since there's nothing at this node, we just ``pop`` it:
::
BTree-iter == [not] [pop] [R0] [R1] genrec
Node case ``[key value left right]``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Now we need to figure out ``R0`` and ``R1``:
::
BTree-iter == [not] [pop] [R0] [R1] genrec
== [not] [pop] [R0 [BTree-iter] R1] ifte
Let's look at it *in situ*:
::
[key value left right] R0 [BTree-iter] R1
Processing the current node.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^
``R0`` is almost certainly going to use ``dup`` to make a copy of the
node and then ``dip`` on some function to process the copy with it:
::
[key value left right] [F] dupdip [BTree-iter] R1
[key value left right] F [key value left right] [BTree-iter] R1
For example, if we're getting all the keys ``F`` would be ``first``:
::
R0 == [first] dupdip
[key value left right] [first] dupdip [BTree-iter] R1
[key value left right] first [key value left right] [BTree-iter] R1
key [key value left right] [BTree-iter] R1
Recur
^^^^^
Now ``R1`` needs to apply ``[BTree-iter]`` to ``left`` and ``right``. If
we drop the key and value from the node using ``rest`` twice we are left
with an interesting situation:
::
key [key value left right] [BTree-iter] R1
key [key value left right] [BTree-iter] [rest rest] dip
key [key value left right] rest rest [BTree-iter]
key [left right] [BTree-iter]
Hmm, will ``step`` do?
::
key [left right] [BTree-iter] step
key left BTree-iter [right] [BTree-iter] step
key left-keys [right] [BTree-iter] step
key left-keys right BTree-iter
key left-keys right-keys
Wow. So:
::
R1 == [rest rest] dip step
Putting it together
^^^^^^^^^^^^^^^^^^^
We have:
::
BTree-iter == [not] [pop] [[F] dupdip] [[rest rest] dip step] genrec
When I was reading this over I realized ``rest rest`` could go in
``R0``:
::
BTree-iter == [not] [pop] [[F] dupdip rest rest] [step] genrec
(And ``[step] genrec`` is such a cool and suggestive combinator!)
Parameterizing the ``F`` per-node processing function.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
::
[F] BTree-iter == [not] [pop] [[F] dupdip rest rest] [step] genrec
Working backward:
::
[not] [pop] [[F] dupdip rest rest] [step] genrec
[not] [pop] [F] [dupdip rest rest] cons [step] genrec
[F] [not] [pop] roll< [dupdip rest rest] cons [step] genrec
Ergo:
::
BTree-iter == [not] [pop] roll< [dupdip rest rest] cons [step] genrec
.. code:: ipython2
from notebook_preamble import J, V, define
.. code:: ipython2
define('BTree-iter == [not] [pop] roll< [dupdip rest rest] cons [step] genrec')
.. code:: ipython2
J('[] [23] BTree-iter') # It doesn't matter what F is as it won't be used.
.. parsed-literal::
.. code:: ipython2
J('["tommy" 23 [] []] [first] BTree-iter')
.. parsed-literal::
'tommy'
.. code:: ipython2
J('["tommy" 23 ["richard" 48 [] []] ["jenny" 18 [] []]] [first] BTree-iter')
.. parsed-literal::
'tommy' 'richard' 'jenny'
.. code:: ipython2
J('["tommy" 23 ["richard" 48 [] []] ["jenny" 18 [] []]] [second] BTree-iter')
.. parsed-literal::
23 48 18
Adding Nodes to the BTree
=========================
Let's consider adding nodes to a BTree structure.
::
BTree value key BTree-add == BTree
Adding to an empty node.
^^^^^^^^^^^^^^^^^^^^^^^^
If the current node is ``[]`` then you just return
``[key value [] []]``:
::
BTree-add == [popop not] [[pop] dipd BTree-new] [R0] [R1] genrec
Where ``BTree-new`` is:
::
value key BTree-new == [key value [] []]
value key swap [[] []] cons cons
key value [[] []] cons cons
key [value [] []] cons
[key value [] []]
BTree-new == swap [[] []] cons cons
.. code:: ipython2
define('BTree-new == swap [[] []] cons cons')
.. code:: ipython2
V('"v" "k" BTree-new')
.. parsed-literal::
. 'v' 'k' BTree-new
'v' . 'k' BTree-new
'v' 'k' . BTree-new
'v' 'k' . swap [[] []] cons cons
'k' 'v' . [[] []] cons cons
'k' 'v' [[] []] . cons cons
'k' ['v' [] []] . cons
['k' 'v' [] []] .
(As an implementation detail, the ``[[] []]`` literal used in the
definition of ``BTree-new`` will be reused to supply the *constant* tail
for *all* new nodes produced by it. This is one of those cases where you
get amortized storage "for free" by using `persistent
datastructures <https://en.wikipedia.org/wiki/Persistent_data_structure>`__.
Because the tail, which is ``((), ((), ()))`` in Python, is immutable
and embedded in the definition body for ``BTree-new``, all new nodes can
reuse it as their own tail without fear that some other code somewhere
will change it.)
If the current node isn't empty.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
We now have to derive ``R0`` and ``R1``, consider:
::
[key_n value_n left right] value key R0 [BTree-add] R1
In this case, there are three possibilites: the key can be greater or
less than or equal to the node's key. In two of those cases we will need
to apply a copy of ``BTree-add``, so ``R0`` is pretty much out of the
picture.
::
[R0] == []
A predicate to compare keys.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The first thing we need to do is compare the the key we're adding to see
if it is greater than the node key and ``branch`` accordingly, although
in this case it's easier to write a destructive predicate and then use
``ifte`` to apply it ``nullary``:
::
[key_n value_n left right] value key [BTree-add] R1
[key_n value_n left right] value key [BTree-add] [P >] [T] [E] ifte
[key_n value_n left right] value key [BTree-add] P >
[key_n value_n left right] value key [BTree-add] pop roll> pop first >
[key_n value_n left right] value key roll> pop first >
key [key_n value_n left right] value roll> pop first >
key key_n >
Boolean
P > == pop roll> pop first >
P < == pop roll> pop first <
P == pop roll> pop first
.. code:: ipython2
define('P == pop roll> pop first')
.. code:: ipython2
V('["k" "v" [] []] "vv" "kk" [0] P >')
.. parsed-literal::
. ['k' 'v' [] []] 'vv' 'kk' [0] P >
['k' 'v' [] []] . 'vv' 'kk' [0] P >
['k' 'v' [] []] 'vv' . 'kk' [0] P >
['k' 'v' [] []] 'vv' 'kk' . [0] P >
['k' 'v' [] []] 'vv' 'kk' [0] . P >
['k' 'v' [] []] 'vv' 'kk' [0] . pop roll> pop first >
['k' 'v' [] []] 'vv' 'kk' . roll> pop first >
'kk' ['k' 'v' [] []] 'vv' . pop first >
'kk' ['k' 'v' [] []] . first >
'kk' 'k' . >
True .
If the key we're adding is greater than the node's key.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Here the parantheses are meant to signify that the right-hand side (RHS)
is not literal, the code in the parentheses is meant to have been
evaluated:
::
[key_n value_n left right] value key [BTree-add] T == [key_n value_n left (BTree-add key value right)]
Use ``infra`` on ``K``.
^^^^^^^^^^^^^^^^^^^^^^^
So how do we do this? We know we're going to want to use ``infra`` on
some function ``K`` that has the key and value to work with, as well as
the quoted copy of ``BTree-add`` to apply somehow:
::
right left value_n key_n value key [BTree-add] K
...
right value key BTree-add left value_n key_n
Pretty easy:
::
right left value_n key_n value key [BTree-add] cons cons dipdd
right left value_n key_n [value key BTree-add] dipdd
right value key BTree-add left value_n key_n
So:
::
K == cons cons dipdd
And:
::
[key_n value_n left right] [value key [BTree-add] K] infra
Derive ``T``.
^^^^^^^^^^^^^
So now we're at getting from this to this:
::
[key_n value_n left right] value key [BTree-add] T
...
[key_n value_n left right] [value key [BTree-add] K] infra
And so ``T`` is just:
::
value key [BTree-add] T == [value key [BTree-add] K] infra
T == [ K] cons cons cons infra
.. code:: ipython2
define('K == cons cons dipdd')
define('T == [K] cons cons cons infra')
.. code:: ipython2
V('"r" "l" "v" "k" "vv" "kk" [0] K')
.. parsed-literal::
. 'r' 'l' 'v' 'k' 'vv' 'kk' [0] K
'r' . 'l' 'v' 'k' 'vv' 'kk' [0] K
'r' 'l' . 'v' 'k' 'vv' 'kk' [0] K
'r' 'l' 'v' . 'k' 'vv' 'kk' [0] K
'r' 'l' 'v' 'k' . 'vv' 'kk' [0] K
'r' 'l' 'v' 'k' 'vv' . 'kk' [0] K
'r' 'l' 'v' 'k' 'vv' 'kk' . [0] K
'r' 'l' 'v' 'k' 'vv' 'kk' [0] . K
'r' 'l' 'v' 'k' 'vv' 'kk' [0] . cons cons dipdd
'r' 'l' 'v' 'k' 'vv' ['kk' 0] . cons dipdd
'r' 'l' 'v' 'k' ['vv' 'kk' 0] . dipdd
'r' . 'vv' 'kk' 0 'l' 'v' 'k'
'r' 'vv' . 'kk' 0 'l' 'v' 'k'
'r' 'vv' 'kk' . 0 'l' 'v' 'k'
'r' 'vv' 'kk' 0 . 'l' 'v' 'k'
'r' 'vv' 'kk' 0 'l' . 'v' 'k'
'r' 'vv' 'kk' 0 'l' 'v' . 'k'
'r' 'vv' 'kk' 0 'l' 'v' 'k' .
.. code:: ipython2
V('["k" "v" "l" "r"] "vv" "kk" [0] T')
.. parsed-literal::
. ['k' 'v' 'l' 'r'] 'vv' 'kk' [0] T
['k' 'v' 'l' 'r'] . 'vv' 'kk' [0] T
['k' 'v' 'l' 'r'] 'vv' . 'kk' [0] T
['k' 'v' 'l' 'r'] 'vv' 'kk' . [0] T
['k' 'v' 'l' 'r'] 'vv' 'kk' [0] . T
['k' 'v' 'l' 'r'] 'vv' 'kk' [0] . [K] cons cons cons infra
['k' 'v' 'l' 'r'] 'vv' 'kk' [0] [K] . cons cons cons infra
['k' 'v' 'l' 'r'] 'vv' 'kk' [[0] K] . cons cons infra
['k' 'v' 'l' 'r'] 'vv' ['kk' [0] K] . cons infra
['k' 'v' 'l' 'r'] ['vv' 'kk' [0] K] . infra
'r' 'l' 'v' 'k' . 'vv' 'kk' [0] K [] swaack
'r' 'l' 'v' 'k' 'vv' . 'kk' [0] K [] swaack
'r' 'l' 'v' 'k' 'vv' 'kk' . [0] K [] swaack
'r' 'l' 'v' 'k' 'vv' 'kk' [0] . K [] swaack
'r' 'l' 'v' 'k' 'vv' 'kk' [0] . cons cons dipdd [] swaack
'r' 'l' 'v' 'k' 'vv' ['kk' 0] . cons dipdd [] swaack
'r' 'l' 'v' 'k' ['vv' 'kk' 0] . dipdd [] swaack
'r' . 'vv' 'kk' 0 'l' 'v' 'k' [] swaack
'r' 'vv' . 'kk' 0 'l' 'v' 'k' [] swaack
'r' 'vv' 'kk' . 0 'l' 'v' 'k' [] swaack
'r' 'vv' 'kk' 0 . 'l' 'v' 'k' [] swaack
'r' 'vv' 'kk' 0 'l' . 'v' 'k' [] swaack
'r' 'vv' 'kk' 0 'l' 'v' . 'k' [] swaack
'r' 'vv' 'kk' 0 'l' 'v' 'k' . [] swaack
'r' 'vv' 'kk' 0 'l' 'v' 'k' [] . swaack
['k' 'v' 'l' 0 'kk' 'vv' 'r'] .
If the key we're adding is less than the node's key.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
This is very very similar to the above:
::
[key_n value_n left right] value key [BTree-add] E
[key_n value_n left right] value key [BTree-add] [P <] [Te] [Ee] ifte
In this case ``Te`` works that same as ``T`` but on the left child tree
instead of the right, so the only difference is that it must use
``dipd`` instead of ``dipdd``:
::
Te == [cons cons dipd] cons cons cons infra
This suggests an alternate factorization:
::
ccons == cons cons
T == [ccons dipdd] ccons cons infra
Te == [ccons dipd] ccons cons infra
But whatever.
.. code:: ipython2
define('Te == [cons cons dipd] cons cons cons infra')
.. code:: ipython2
V('["k" "v" "l" "r"] "vv" "kk" [0] Te')
.. parsed-literal::
. ['k' 'v' 'l' 'r'] 'vv' 'kk' [0] Te
['k' 'v' 'l' 'r'] . 'vv' 'kk' [0] Te
['k' 'v' 'l' 'r'] 'vv' . 'kk' [0] Te
['k' 'v' 'l' 'r'] 'vv' 'kk' . [0] Te
['k' 'v' 'l' 'r'] 'vv' 'kk' [0] . Te
['k' 'v' 'l' 'r'] 'vv' 'kk' [0] . [cons cons dipd] cons cons cons infra
['k' 'v' 'l' 'r'] 'vv' 'kk' [0] [cons cons dipd] . cons cons cons infra
['k' 'v' 'l' 'r'] 'vv' 'kk' [[0] cons cons dipd] . cons cons infra
['k' 'v' 'l' 'r'] 'vv' ['kk' [0] cons cons dipd] . cons infra
['k' 'v' 'l' 'r'] ['vv' 'kk' [0] cons cons dipd] . infra
'r' 'l' 'v' 'k' . 'vv' 'kk' [0] cons cons dipd [] swaack
'r' 'l' 'v' 'k' 'vv' . 'kk' [0] cons cons dipd [] swaack
'r' 'l' 'v' 'k' 'vv' 'kk' . [0] cons cons dipd [] swaack
'r' 'l' 'v' 'k' 'vv' 'kk' [0] . cons cons dipd [] swaack
'r' 'l' 'v' 'k' 'vv' ['kk' 0] . cons dipd [] swaack
'r' 'l' 'v' 'k' ['vv' 'kk' 0] . dipd [] swaack
'r' 'l' . 'vv' 'kk' 0 'v' 'k' [] swaack
'r' 'l' 'vv' . 'kk' 0 'v' 'k' [] swaack
'r' 'l' 'vv' 'kk' . 0 'v' 'k' [] swaack
'r' 'l' 'vv' 'kk' 0 . 'v' 'k' [] swaack
'r' 'l' 'vv' 'kk' 0 'v' . 'k' [] swaack
'r' 'l' 'vv' 'kk' 0 'v' 'k' . [] swaack
'r' 'l' 'vv' 'kk' 0 'v' 'k' [] . swaack
['k' 'v' 0 'kk' 'vv' 'l' 'r'] .
Else the keys must be equal.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^
This means we must find:
::
[key_n value_n left right] value key [BTree-add] Ee
...
[key value left right]
This is another easy one:
::
Ee == pop swap roll< rest rest cons cons
[key_n value_n left right] value key [BTree-add] pop swap roll< rest rest cons cons
[key_n value_n left right] value key swap roll< rest rest cons cons
[key_n value_n left right] key value roll< rest rest cons cons
key value [key_n value_n left right] rest rest cons cons
key value [ left right] cons cons
[key value left right]
.. code:: ipython2
define('Ee == pop swap roll< rest rest cons cons')
.. code:: ipython2
V('["k" "v" "l" "r"] "vv" "k" [0] Ee')
.. parsed-literal::
. ['k' 'v' 'l' 'r'] 'vv' 'k' [0] Ee
['k' 'v' 'l' 'r'] . 'vv' 'k' [0] Ee
['k' 'v' 'l' 'r'] 'vv' . 'k' [0] Ee
['k' 'v' 'l' 'r'] 'vv' 'k' . [0] Ee
['k' 'v' 'l' 'r'] 'vv' 'k' [0] . Ee
['k' 'v' 'l' 'r'] 'vv' 'k' [0] . pop swap roll< rest rest cons cons
['k' 'v' 'l' 'r'] 'vv' 'k' . swap roll< rest rest cons cons
['k' 'v' 'l' 'r'] 'k' 'vv' . roll< rest rest cons cons
'k' 'vv' ['k' 'v' 'l' 'r'] . rest rest cons cons
'k' 'vv' ['v' 'l' 'r'] . rest cons cons
'k' 'vv' ['l' 'r'] . cons cons
'k' ['vv' 'l' 'r'] . cons
['k' 'vv' 'l' 'r'] .
.. code:: ipython2
define('E == [P <] [Te] [Ee] ifte')
Now we can define ``BTree-add``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
::
BTree-add == [popop not] [[pop] dipd BTree-new] [] [[P >] [T] [E] ifte] genrec
Putting it all together:
::
BTree-new == swap [[] []] cons cons
P == pop roll> pop first
T == [cons cons dipdd] cons cons cons infra
Te == [cons cons dipd] cons cons cons infra
Ee == pop swap roll< rest rest cons cons
E == [P <] [Te] [Ee] ifte
BTree-add == [popop not] [[pop] dipd BTree-new] [] [[P >] [T] [E] ifte] genrec
.. code:: ipython2
define('BTree-add == [popop not] [[pop] dipd BTree-new] [] [[P >] [T] [E] ifte] genrec')
.. code:: ipython2
J('[] 23 "b" BTree-add') # Initial
.. parsed-literal::
['b' 23 [] []]
.. code:: ipython2
J('["b" 23 [] []] 88 "c" BTree-add') # Less than
.. parsed-literal::
['b' 23 [] ['c' 88 [] []]]
.. code:: ipython2
J('["b" 23 [] []] 88 "a" BTree-add') # Greater than
.. parsed-literal::
['b' 23 ['a' 88 [] []] []]
.. code:: ipython2
J('["b" 23 [] []] 88 "b" BTree-add') # Equal to
.. parsed-literal::
['b' 88 [] []]
.. code:: ipython2
J('[] 23 "a" BTree-add 88 "b" BTree-add 44 "c" BTree-add') # Series.
.. parsed-literal::
['a' 23 [] ['b' 88 [] ['c' 44 [] []]]]
We can use this to make a set-like datastructure by just setting values
to e.g. 0 and ignoring them. It's set-like in that duplicate items added
to it will only occur once within it, and we can query it in
`:math:`O(\log_2 N)` <https://en.wikipedia.org/wiki/Binary_search_tree#cite_note-2>`__
time.
.. code:: ipython2
J('[] [3 9 5 2 8 6 7 8 4] [0 swap BTree-add] step')
.. parsed-literal::
[3 0 [2 0 [] []] [9 0 [5 0 [4 0 [] []] [8 0 [6 0 [] [7 0 [] []]] []]] []]]
.. code:: ipython2
define('to_set == [] swap [0 swap BTree-add] step')
.. code:: ipython2
J('[3 9 5 2 8 6 7 8 4] to_set')
.. parsed-literal::
[3 0 [2 0 [] []] [9 0 [5 0 [4 0 [] []] [8 0 [6 0 [] [7 0 [] []]] []]] []]]
And with that we can write a little program to remove duplicate items
from a list.
.. code:: ipython2
define('unique == [to_set [first] BTree-iter] cons run')
.. code:: ipython2
J('[3 9 3 5 2 9 8 8 8 6 2 7 8 4 3] unique') # Filter duplicate items.
.. parsed-literal::
[7 6 8 4 5 9 2 3]
``cmp`` combinator
==================
Instead of all this mucking about with nested ``ifte`` let's just go
whole hog and define ``cmp`` which takes two values and three quoted
programs on the stack and runs one of the three depending on the results
of comparing the two values:
::
a b [G] [E] [L] cmp
------------------------- a > b
G
a b [G] [E] [L] cmp
------------------------- a = b
E
a b [G] [E] [L] cmp
------------------------- a < b
L
We need a new non-destructive predicate ``P``:
::
[key_n value_n left right] value key [BTree-add] P
[key_n value_n left right] value key [BTree-add] over [Q] nullary
[key_n value_n left right] value key [BTree-add] key [Q] nullary
[key_n value_n left right] value key [BTree-add] key Q
[key_n value_n left right] value key [BTree-add] key popop popop first
[key_n value_n left right] value key popop first
[key_n value_n left right] first
key_n
[key_n value_n left right] value key [BTree-add] key [Q] nullary
[key_n value_n left right] value key [BTree-add] key key_n
P == over [popop popop first] nullary
Here are the definitions again, pruned and renamed in some cases:
::
BTree-new == swap [[] []] cons cons
P == over [popop popop first] nullary
T> == [cons cons dipdd] cons cons cons infra
T< == [cons cons dipd] cons cons cons infra
E == pop swap roll< rest rest cons cons
Using ``cmp`` to simplify `our code above at
``R1`` <#If-the-current-node-isn't-empty.>`__:
::
[key_n value_n left right] value key [BTree-add] R1
[key_n value_n left right] value key [BTree-add] P [T>] [E] [T<] cmp
The line above becomes one of the three lines below:
::
[key_n value_n left right] value key [BTree-add] T>
[key_n value_n left right] value key [BTree-add] E
[key_n value_n left right] value key [BTree-add] T<
The definition is a little longer but, I think, more elegant and easier
to understand:
::
BTree-add == [popop not] [[pop] dipd BTree-new] [] [P [T>] [E] [T<] cmp] genrec
.. code:: ipython2
from joy.library import FunctionWrapper
from joy.utils.stack import pushback
from notebook_preamble import D
@FunctionWrapper
def cmp_(stack, expression, dictionary):
L, (E, (G, (b, (a, stack)))) = stack
expression = pushback(G if a > b else L if a < b else E, expression)
return stack, expression, dictionary
D['cmp'] = cmp_
.. code:: ipython2
J("1 0 ['G'] ['E'] ['L'] cmp")
.. parsed-literal::
'G'
.. code:: ipython2
J("1 1 ['G'] ['E'] ['L'] cmp")
.. parsed-literal::
'E'
.. code:: ipython2
J("0 1 ['G'] ['E'] ['L'] cmp")
.. parsed-literal::
'L'
.. code:: ipython2
from joy.library import DefinitionWrapper
DefinitionWrapper.add_definitions('''
P == over [popop popop first] nullary
T> == [cons cons dipdd] cons cons cons infra
T< == [cons cons dipd] cons cons cons infra
E == pop swap roll< rest rest cons cons
BTree-add == [popop not] [[pop] dipd BTree-new] [] [P [T>] [E] [T<] cmp] genrec
''', D)
.. code:: ipython2
J('[] 23 "b" BTree-add') # Initial
.. parsed-literal::
['b' 23 [] []]
.. code:: ipython2
J('["b" 23 [] []] 88 "c" BTree-add') # Less than
.. parsed-literal::
['b' 23 [] ['c' 88 [] []]]
.. code:: ipython2
J('["b" 23 [] []] 88 "a" BTree-add') # Greater than
.. parsed-literal::
['b' 23 ['a' 88 [] []] []]
.. code:: ipython2
J('["b" 23 [] []] 88 "b" BTree-add') # Equal to
.. parsed-literal::
['b' 88 [] []]
.. code:: ipython2
J('[] 23 "a" BTree-add 88 "b" BTree-add 44 "c" BTree-add') # Series.
.. parsed-literal::
['a' 23 [] ['b' 88 [] ['c' 44 [] []]]]
Factoring and naming
====================
It may seem silly, but a big part of programming in Forth (and therefore
in Joy) is the idea of small, highly-factored definitions. If you choose
names carefully the resulting definitions can take on a semantic role.
::
get-node-key == popop popop first
remove-key-and-value-from-node == rest rest
pack-key-and-value == cons cons
prep-new-key-and-value == pop swap roll<
pack-and-apply == [pack-key-and-value] swoncat cons pack-key-and-value infra
BTree-new == swap [[] []] pack-key-and-value
P == over [get-node-key] nullary
T> == [dipdd] pack-and-apply
T< == [dipd] pack-and-apply
E == prep-new-key-and-value remove-key-and-value-from-node pack-key-and-value
A Version of ``BTree-iter`` that does In-Order Traversal
========================================================
If you look back to the `non-empty case of the ``BTree-iter``
function <#Node-case-%5Bkey-value-left-right%5D>`__ we can design a
varient that first processes the left child, then the current node, then
the right child. This will allow us to traverse the tree in sort order.
::
BTree-iter-order == [not] [pop] [R0 [BTree-iter] R1] ifte
To define ``R0`` and ``R1`` it helps to look at them as they will appear
when they run:
::
[key value left right] R0 [BTree-iter-order] R1
Process the left child.
^^^^^^^^^^^^^^^^^^^^^^^
Staring at this for a bit suggests ``dup third`` to start:
::
[key value left right] R0 [BTree-iter-order] R1
[key value left right] dup third [BTree-iter-order] R1
[key value left right] left [BTree-iter-order] R1
Now maybe:
::
[key value left right] left [BTree-iter-order] [cons dip] dupdip
[key value left right] left [BTree-iter-order] cons dip [BTree-iter-order]
[key value left right] [left BTree-iter-order] dip [BTree-iter-order]
left BTree-iter-order [key value left right] [BTree-iter-order]
Process the current node.
^^^^^^^^^^^^^^^^^^^^^^^^^
So far, so good. Now we need to process the current node's values:
::
left BTree-iter-order [key value left right] [BTree-iter-order] [[F] dupdip] dip
left BTree-iter-order [key value left right] [F] dupdip [BTree-iter-order]
left BTree-iter-order [key value left right] F [key value left right] [BTree-iter-order]
If ``F`` needs items from the stack below the left stuff it should have
``cons``'d them before beginning maybe? For functions like ``first`` it
works fine as-is.
::
left BTree-iter-order [key value left right] first [key value left right] [BTree-iter-order]
left BTree-iter-order key [key value left right] [BTree-iter-order]
Process the right child.
^^^^^^^^^^^^^^^^^^^^^^^^
First ditch the rest of the node and get the right child:
::
left BTree-iter-order key [key value left right] [BTree-iter-order] [rest rest rest first] dip
left BTree-iter-order key right [BTree-iter-order]
Then, of course, we just need ``i`` to run ``BTree-iter-order`` on the
right side:
::
left BTree-iter-order key right [BTree-iter-order] i
left BTree-iter-order key right BTree-iter-order
Defining ``BTree-iter-order``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The result is a little awkward:
::
R1 == [cons dip] dupdip [[F] dupdip] dip [rest rest rest first] dip i
Let's do a little semantic factoring:
::
fourth == rest rest rest first
proc_left == [cons dip] dupdip
proc_current == [[F] dupdip] dip
proc_right == [fourth] dip i
BTree-iter-order == [not] [pop] [dup third] [proc_left proc_current proc_right] genrec
Now we can sort sequences.
.. code:: ipython2
define('BTree-iter-order == [not] [pop] [dup third] [[cons dip] dupdip [[first] dupdip] dip [rest rest rest first] dip i] genrec')
.. code:: ipython2
J('[3 9 5 2 8 6 7 8 4] to_set BTree-iter-order')
.. parsed-literal::
2 3 4 5 6 7 8 9
Getting values by key
=====================
Let's derive a function that accepts a tree and a key and returns the
value associated with that key.
::
tree key BTree-get
------------------------
value
The base case ``[]``
^^^^^^^^^^^^^^^^^^^^
As before, the stopping predicate just has to detect the empty list:
::
BTree-get == [pop not] [E] [R0] [R1] genrec
But what do we do if the key isn't in the tree? In Python we might raise
a ``KeyError`` but I'd like to avoid exceptions in Joy if possible, and
here I think it's possible. (Division by zero is an example of where I
think it's probably better to let Python crash Joy. Sometimes the
machinery fails and you have to "stop the line", methinks.)
Let's pass the buck to the caller by making the base case a given, you
have to decide for yourself what ``[E]`` should be.
::
tree key [E] BTree-get
---------------------------- key in tree
value
tree key [E] BTree-get
---------------------------- key not in tree
tree key E
Now we define:
::
BTree-get == [pop not] swap [R0] [R1] genrec
Note that this ``BTree-get`` creates a slightly different function than
itself and *that function* does the actual recursion. This kind of
higher-level programming is unusual in most languages but natural in
Joy.
::
tree key [E] [pop not] swap [R0] [R1] genrec
tree key [pop not] [E] [R0] [R1] genrec
The anonymous specialized recursive function that will do the real work.
::
[pop not] [E] [R0] [R1] genrec
Node case ``[key value left right]``
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Now we need to figure out ``R0`` and ``R1``:
::
[key value left right] key R0 [BTree-get] R1
We want to compare the search key with the key in the node, and if they
are the same return the value and if they differ then recurse on one of
the child nodes. So it's very similar to the above funtion, with
``[R0] == []`` and ``R1 == P [T>] [E] [T<] cmp``:
::
[key value left right] key [BTree-get] P [T>] [E] [T<] cmp
So:
::
get-node-key == pop popop first
P == over [get-node-key] nullary
The only difference is that ``get-node-key`` does one less ``pop``
because there's no value to discard. Now we have to derive the branches:
::
[key_n value_n left right] key [BTree-get] T>
[key_n value_n left right] key [BTree-get] E
[key_n value_n left right] key [BTree-get] T<
The cases of ``T>`` and ``T<`` are similar to above but instead of using
``infra`` we have to discard the rest of the structure:
::
[key_n value_n left right] key [BTree-get] T> == right key BTree-get
[key_n value_n left right] key [BTree-get] T< == left key BTree-get
So:
::
T> == [fourth] dipd i
T< == [third] dipd i
E.g.:
::
[key_n value_n left right] key [BTree-get] [fourth] dipd i
[key_n value_n left right] fourth key [BTree-get] i
right key [BTree-get] i
right key BTree-get
And:
::
[key_n value_n left right] key [BTree-get] E == value_n
E == popop second
So:
::
fourth == rest rest rest first
get-node-key == pop popop first
P == over [get-node-key] nullary
T> == [fourth] dipd i
T< == [third] dipd i
E == popop second
BTree-get == [pop not] swap [] [P [T>] [E] [T<] cmp] genrec
.. code:: ipython2
# I don't want to deal with name conflicts with the above so I'm inlining everything here.
# The original Joy system has "hide" which is a meta-command which allows you to use named
# definitions that are only in scope for a given definition. I don't want to implement
# that (yet) so...
define('''
BTree-get == [pop not] swap [] [
over [pop popop first] nullary
[[rest rest rest first] dipd i]
[popop second]
[[third] dipd i]
cmp
] genrec
''')
.. code:: ipython2
J('[] "gary" [popop "err"] BTree-get')
.. parsed-literal::
'err'
.. code:: ipython2
J('["gary" 23 [] []] "gary" [popop "err"] BTree-get')
.. parsed-literal::
23
.. code:: ipython2
J('''
[] [[0 'a'] [1 'b'] [2 'c']] [i BTree-add] step
'c' [popop 'not found'] BTree-get
''')
.. parsed-literal::
2
TODO: BTree-delete
==================
Then, once we have add, get, and delete we can see about abstracting
them.
::
tree key [E] BTree-delete
---------------------------- key in tree
tree
tree key [E] BTree-delete
---------------------------- key not in tree
tree key E
So:
::
BTree-delete == [pop not] [] [R0] [R1] genrec
And:
::
[n_key n_value left right] key R0 [BTree-get] R1
[n_key n_value left right] key [dup first] dip [BTree-get] R1
[n_key n_value left right] n_key key [BTree-get] R1
[n_key n_value left right] n_key key [BTree-get] roll> [T>] [E] [T<] cmp
[n_key n_value left right] [BTree-get] n_key key [T>] [E] [T<] cmp
BTree-delete == [pop not] swap [[dup first] dip] [roll> [T>] [E] [T<] cmp] genrec
::
[n_key n_value left right] [BTree-get] T>
[n_key n_value left right] [BTree-get] E
[n_key n_value left right] [BTree-get] T<
::
[n_key n_value left right] [BTree-get]
[n_key n_value left right] [BTree-get] E
[n_key n_value left right] [BTree-get] T<
Tree with node and list of trees.
=================================
Let's consider a tree structure, similar to one described `"Why
functional programming matters" by John
Hughes <https://www.cs.kent.ac.uk/people/staff/dat/miranda/whyfp90.pdf>`__,
that consists of a node value and a sequence of zero or more child
trees. (The asterisk is meant to indicate the `Kleene
star <https://en.wikipedia.org/wiki/Kleene_star>`__.)
::
tree = [] | [node [tree*]]
``treestep``
~~~~~~~~~~~~
In the spirit of ``step`` we are going to define a combinator
``treestep`` which expects a tree and three additional items: a
base-case value ``z``, and two quoted programs ``[C]`` and ``[N]``.
::
tree z [C] [N] treestep
If the current tree node is empty then just leave ``z`` on the stack in
lieu:
::
[] z [C] [N] treestep
---------------------------
z
Otherwise, evaluate ``N`` on the node value, ``map`` the whole function
(abbreviated here as ``k``) over the child trees recursively, and then
combine the result with ``C``.
::
[node [tree*]] z [C] [N] treestep
--------------------------------------- w/ K == z [C] [N] treestep
node N [tree*] [K] map C
Derive the recursive form.
~~~~~~~~~~~~~~~~~~~~~~~~~~
Since this is a recursive function, we can begin to derive it by finding
the ``ifte`` stage that ``genrec`` will produce. The predicate and
base-case functions are trivial, so we just have to derive ``J``.
::
K == [not] [pop z] [J] ifte
The behavior of ``J`` is to accept a (non-empty) tree node and arrive at
the desired outcome.
::
[node [tree*]] J
------------------------------
node N [tree*] [K] map C
So ``J`` will have some form like:
::
J == .. [N] .. [K] .. [C] ..
Let's dive in. First, unquote the node and ``dip`` ``N``.
::
[node [tree*]] i [N] dip
node [tree*] [N] dip
node N [tree*]
Next, ``map`` ``K`` over teh child trees and combine with ``C``.
::
node N [tree*] [K] map C
node N [tree*] [K] map C
node N [K.tree*] C
So:
::
J == i [N] dip [K] map C
Plug it in and convert to ``genrec``:
::
K == [not] [pop z] [i [N] dip [K] map C] ifte
K == [not] [pop z] [i [N] dip] [map C] genrec
Extract the givens to parameterize the program.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
::
[not] [pop z] [i [N] dip] [map C] genrec
[not] [pop z] [i [N] dip] [map C] genrec
[not] [z] [pop] swoncat [i [N] dip] [map C] genrec
[not] z unit [pop] swoncat [i [N] dip] [map C] genrec
z [not] swap unit [pop] swoncat [i [N] dip] [map C] genrec
\ .........TS0............./
\/
z TS0 [i [N] dip] [map C] genrec
z [i [N] dip] [TS0] dip [map C] genrec
z [[N] dip] [i] swoncat [TS0] dip [map C] genrec
z [N] [dip] cons [i] swoncat [TS0] dip [map C] genrec
\ ......TS1........./
\/
z [N] TS1 [TS0] dip [map C] genrec
z [N] [map C] [TS1 [TS0] dip] dip genrec
z [N] [C] [map] swoncat [TS1 [TS0] dip] dip genrec
z [C] [N] swap [map] swoncat [TS1 [TS0] dip] dip genrec
The givens are all to the left so we have our definition.
Define ``treestep``
~~~~~~~~~~~~~~~~~~~
::
TS0 == [not] swap unit [pop] swoncat
TS1 == [dip] cons [i] swoncat
treestep == swap [map] swoncat [TS1 [TS0] dip] dip genrec
.. code:: ipython2
DefinitionWrapper.add_definitions('''
TS0 == [not] swap unit [pop] swoncat
TS1 == [dip] cons [i] swoncat
treestep == swap [map] swoncat [TS1 [TS0] dip] dip genrec
''', D)
::
[] 0 [C] [N] treestep
---------------------------
0
[n [tree*]] 0 [sum +] [] treestep
--------------------------------------------------
n [tree*] [0 [sum +] [] treestep] map sum +
.. code:: ipython2
J('[] 0 [sum +] [] treestep')
.. parsed-literal::
0
.. code:: ipython2
J('[23 []] 0 [sum +] [] treestep')
.. parsed-literal::
23
.. code:: ipython2
J('[23 [[2 []] [3 []]]] 0 [sum +] [] treestep')
.. parsed-literal::
28
A slight modification.
----------------------
Let's simplify the tree datastructure definition slightly by just
letting the children be the ``rest`` of the tree:
::
tree = [] | [node tree*]
The ``J`` function changes slightly.
::
[node tree*] J
------------------------------
node N [tree*] [K] map C
[node tree*] uncons [N] dip [K] map C
node [tree*] [N] dip [K] map C
node N [tree*] [K] map C
node N [tree*] [K] map C
node N [K.tree*] C
J == uncons [N] dip [K] map C
K == [not] [pop z] [uncons [N] dip] [map C] genrec
.. code:: ipython2
define('TS1 == [dip] cons [uncons] swoncat') # We only need to redefine one word.
.. code:: ipython2
J('[23 [2] [3]] 0 [sum +] [] treestep')
.. parsed-literal::
28
.. code:: ipython2
J('[23 [2 [8] [9]] [3] [4 []]] 0 [sum +] [] treestep')
.. parsed-literal::
49
I think these trees seem a little easier to read.
Redefining our BTree in terms of this form.
-------------------------------------------
::
BTree = [] | [[key value] left right]
What kind of functions can we write for this with our ``treestep``? The
pattern for processing a non-empty node is:
::
node N [tree*] [K] map C
Plugging in our BTree structure:
::
[key value] N [left right] [K] map C
[key value] uncons pop [left right] [K] map i
key [value] pop [left right] [K] map i
key [left right] [K] map i
key [lkey rkey ] i
key lkey rkey
.. code:: ipython2
J('[[3 0] [[2 0] [] []] [[9 0] [[5 0] [[4 0] [] []] [[8 0] [[6 0] [] [[7 0] [] []]] []]] []]] 23 [i] [uncons pop] treestep')
.. parsed-literal::
3 23 23
Doesn't work because ``map`` extracts the ``first`` item of whatever its
mapped function produces. We have to return a list, rather than
depositing our results directly on the stack.
::
[key value] N [left right] [K] map C
[key value] first [left right] [K] map flatten cons
key [left right] [K] map flatten cons
key [[lk] [rk] ] flatten cons
key [ lk rk ] cons
[key lk rk ]
So:
::
[] [flatten cons] [first] treestep
.. code:: ipython2
J('[[3 0] [[2 0] [] []] [[9 0] [[5 0] [[4 0] [] []] [[8 0] [[6 0] [] [[7 0] [] []]] []]] []]] [] [flatten cons] [first] treestep')
.. parsed-literal::
[3 2 9 5 4 8 6 7]
There we go. #### In-order traversal with ``treestep``.
From here:
::
key [[lk] [rk]] C
key [[lk] [rk]] i
key [lk] [rk] roll<
[lk] [rk] key swons concat
[lk] [key rk] concat
[lk key rk]
So:
::
[] [i roll< swons concat] [first] treestep
.. code:: ipython2
J('[[3 0] [[2 0] [] []] [[9 0] [[5 0] [[4 0] [] []] [[8 0] [[6 0] [] [[7 0] [] []]] []]] []]] [] [i roll< swons concat] [uncons pop] treestep')
.. parsed-literal::
[2 3 4 5 6 7 8 9]
Miscellaneous Crap
------------------
Toy with it.
~~~~~~~~~~~~
Let's reexamine:
::
[key value left right] R0 [BTree-iter-order] R1
...
left BTree-iter-order key value F right BTree-iter-order
[key value left right] disenstacken swap
key value left right swap
key value right left
key value right left [BTree-iter-order] [cons dipdd] dupdip
key value right left [BTree-iter-order] cons dipdd [BTree-iter-order]
key value right [left BTree-iter-order] dipdd [BTree-iter-order]
left BTree-iter-order key value right [BTree-iter-order]
left BTree-iter-order key value right [F] dip [BTree-iter-order]
left BTree-iter-order key value F right [BTree-iter-order] i
left BTree-iter-order key value F right BTree-iter-order
So:
::
R0 == disenstacken swap
R1 == [cons dipdd [F] dip] dupdip i
[key value left right] R0 [BTree-iter-order] R1
[key value left right] disenstacken swap [BTree-iter-order] [cons dipdd [F] dip] dupdip i
key value right left [BTree-iter-order] [cons dipdd [F] dip] dupdip i
key value right left [BTree-iter-order] cons dipdd [F] dip [BTree-iter-order] i
key value right [left BTree-iter-order] dipdd [F] dip [BTree-iter-order] i
left BTree-iter-order key value right [F] dip [BTree-iter-order] i
left BTree-iter-order key value F right [BTree-iter-order] i
left BTree-iter-order key value F right BTree-iter-order
BTree-iter-order == [not] [pop] [disenstacken swap] [[cons dipdd [F] dip] dupdip i] genrec
Refactor ``cons cons``
^^^^^^^^^^^^^^^^^^^^^^
::
cons2 == cons cons
Refactoring:
::
BTree-new == swap [[] []] cons2
T == [cons2 dipdd] cons2 cons infra
Te == [cons2 dipd] cons2 cons infra
Ee == pop swap roll< rest rest cons2
It's used a lot because it's tied to the fact that there are two "data
items" in each node. This point to a more general factorization that
would render a combinator that could work for other geometries of trees.
A General Form for Trees
------------------------
A general form for tree data with N children per node:
::
[[data] [child0] ... [childN-1]]
Suggests a general form of recursive iterator, but I have to go walk the
dogs at the mo'.
For a given structure, you would have a structure of operator functions
and sort of merge them and run them, possibly in a different order (pre-
post- in- y'know). The ``Cn`` functions could all be the same and use
the ``step`` trick if the children nodes are all of the right kind. If
they are heterogeneous then we need a way to get the different ``Cn``
into the structure in the right order. If I understand correctly, the
"Bananas..." paper shows how to do this automatically from a type
description. They present, if I have it right, a tiny machine that
accepts `some sort of algebraic data type description and returns a
function that can recusre over
it <https://en.wikipedia.org/wiki/Catamorphism#General_case>`__, I
think.
::
[data.. [c0] [c1] ... [cN]] [F C0 C1 ... CN] infil
--------------------------------------------------------
data F [c0] C0 [c1] C1 ... [cN] CN
Just make ``[F]`` a parameter.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
We can generalize to a sort of pure form:
::
BTree-iter == [not] [pop] [[F]] [R1] genrec
== [not] [pop] [[F] [BTree-iter] R1] ifte
Putting ``[F]`` to the left as a given:
::
[F] unit [not] [pop] roll< [R1] genrec
[[F]] [not] [pop] roll< [R1] genrec
[not] [pop] [[F]] [R1] genrec
Let's us define a parameterized form:
::
BTree-iter == unit [not] [pop] roll< [R1] genrec
So in the general case of non-empty nodes:
::
[key value left right] [F] [BTree-iter] R1
We just define ``R1`` to do whatever it has to to process the node. For
example:
::
[key value left right] [F] [BTree-iter] R1
...
key value F left BTree-iter right BTree-iter
left BTree-iter key value F right BTree-iter
left BTree-iter right BTree-iter key value F
Pre-, ??-, post-order traversals.
::
[key value left right] uncons uncons
key value [left right]
For pre- and post-order we can use the ``step`` trick:
::
[left right] [BTree-iter] step
...
left BTree-iter right BTree-iter
We worked out one scheme for ?in-order? traversal above, but maybe we
can do better?
::
[key value left right] [F] [BTree-iter] [disenstacken] dipd
[key value left right] disenstacken [F] [BTree-iter]
key value left right [F] [BTree-iter]
key value left right [F] [BTree-iter] R1.1
Hmm...
::
key value left right [F] [BTree-iter] tuck
key value left right [BTree-iter] [F] [BTree-iter]
[key value left right] [F] [BTree-iter] [disenstacken [roll>] dip] dipd
[key value left right] disenstacken [roll>] dip [F] [BTree-iter]
key value left right [roll>] dip [F] [BTree-iter]
key value left roll> right [F] [BTree-iter]
left key value right [F] [BTree-iter]
left key value right [F] [BTree-iter] tuck foo
left key value right [BTree-iter] [F] [BTree-iter] foo
...
left BTree-iter key value F right BTree-iter
We could just let ``[R1]`` be a parameter too, for maximum flexibility.
Automatically deriving the recursion combinator for a data type?
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
If I understand it correctly, the "Bananas..." paper talks about a way
to build the processor function automatically from the description of
the type. I think if we came up with an elegant way for the Joy code to
express that, it would be cool. In Joypy the definitions can be circular
because lookup happens at evaluation, not parsing. E.g.:
::
A == ... B ...
B == ... A ...
That's fine. Circular datastructures can't be made though.
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