Treating Trees II: treestep
¶
Let’s consider a tree structure, similar to one described “Why functional programming matters” by John Hughes, that consists of a node value followed by zero or more child trees. (The asterisk is meant to indicate the Kleene star.)
tree = [] | [node tree*]
In the spirit of step
we are going to define a combinator
treestep
which expects a tree and three additional items: a
base-case function [B]
, and two quoted programs [N]
and [C]
.
tree [B] [N] [C] treestep
If the current tree node is empty then just execute B
:
[] [B] [N] [C] treestep
---------------------------
[] B
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*] [B] [N] [C] treestep
--------------------------------------- w/ K == [B] [N] [C] treestep
node N [tree*] [K] map C
(Later on we’ll experiment with making map
part of C
so you can
use other combinators.)
Derive the recursive function.¶
We can begin to derive it by finding the ifte
stage that genrec
will produce.
K == [not] [B] [R0] [R1] genrec
== [not] [B] [R0 [K] R1] ifte
So we just have to derive J
:
J == R0 [K] R1
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*] uncons [N] dip
node [tree*] [N] dip
node N [tree*]
Next, map
K
over the 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 == uncons [N] dip [K] map C
Plug it in and convert to genrec
:
K == [not] [B] [J ] ifte
== [not] [B] [uncons [N] dip [K] map C] ifte
== [not] [B] [uncons [N] dip] [map C] genrec
Extract the givens to parameterize the program.¶
Working backwards:
[not] [B] [uncons [N] dip] [map C] genrec
[B] [not] swap [uncons [N] dip] [map C] genrec
[B] [uncons [N] dip] [[not] swap] dip [map C] genrec
^^^^^^^^^^^^^^^^
[B] [[N] dip] [uncons] swoncat [[not] swap] dip [map C] genrec
[B] [N] [dip] cons [uncons] swoncat [[not] swap] dip [map C] genrec
^^^^^^^^^^^^^^^^^^^^^^^^^^^
Extract a couple of auxiliary definitions:
TS.0 == [[not] swap] dip
TS.1 == [dip] cons [uncons] swoncat
[B] [N] TS.1 TS.0 [map C] genrec
[B] [N] [map C] [TS.1 TS.0] dip genrec
[B] [N] [C] [map] swoncat [TS.1 TS.0] dip genrec
The givens are all to the left so we have our definition.
(alternate) Extract the givens to parameterize the program.¶
Working backwards:
[not] [B] [uncons [N] dip] [map C] genrec
[not] [B] [N] [dip] cons [uncons] swoncat [map C] genrec
[B] [N] [not] roll> [dip] cons [uncons] swoncat [map C] genrec
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Define treestep
¶
from notebook_preamble import D, J, V, define, DefinitionWrapper
DefinitionWrapper.add_definitions('''
_treestep_0 == [[not] swap] dip
_treestep_1 == [dip] cons [uncons] swoncat
treegrind == [_treestep_1 _treestep_0] dip genrec
treestep == [map] swoncat treegrind
''', D)
Examples¶
Consider trees, the nodes of which are integers. We can find the sum of all nodes in a tree with this function:
sumtree == [pop 0] [] [sum +] treestep
define('sumtree == [pop 0] [] [sum +] treestep')
Running this function on an empty tree value gives zero:
[] [pop 0] [] [sum +] treestep
------------------------------------
0
J('[] sumtree') # Empty tree.
0
Running it on a non-empty node:
[n tree*] [pop 0] [] [sum +] treestep
n [tree*] [[pop 0] [] [sum +] treestep] map sum +
n [ ... ] sum +
n m +
n+m
J('[23] sumtree') # No child trees.
23
J('[23 []] sumtree') # Child tree, empty.
23
J('[23 [2 [4]] [3]] sumtree') # Non-empty child trees.
32
J('[23 [2 [8] [9]] [3] [4 []]] sumtree') # Etc...
49
J('[23 [2 [8] [9]] [3] [4 []]] [pop 0] [] [cons sum] treestep') # Alternate "spelling".
49
J('[23 [2 [8] [9]] [3] [4 []]] [] [pop 23] [cons] treestep') # Replace each node.
[23 [23 [23] [23]] [23] [23 []]]
J('[23 [2 [8] [9]] [3] [4 []]] [] [pop 1] [cons] treestep')
[1 [1 [1] [1]] [1] [1 []]]
J('[23 [2 [8] [9]] [3] [4 []]] [] [pop 1] [cons] treestep sumtree')
6
J('[23 [2 [8] [9]] [3] [4 []]] [pop 0] [pop 1] [sum +] treestep') # Combine replace and sum into one function.
6
J('[4 [3 [] [7]]] [pop 0] [pop 1] [sum +] treestep') # Combine replace and sum into one function.
3
Redefining the Ordered Binary Tree in terms of treestep
.¶
Tree = [] | [[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
Traversal¶
[key value] first [left right] [K] map i
key [value] [left right] [K] map i
key [left right] [K] map i
key [lkey rkey ] i
key lkey rkey
This doesn’t quite work:
J('[[3 0] [[2 0] [][]] [[9 0] [[5 0] [[4 0] [][]] [[8 0] [[6 0] [] [[7 0] [][]]][]]][]]] ["B"] [first] [i] treestep')
3 'B' 'B'
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:
[] [first] [flatten cons] treestep
J('[[3 0] [[2 0] [] []] [[9 0] [[5 0] [[4 0] [] []] [[8 0] [[6 0] [] [[7 0] [] []]] []]] []]] [] [first] [flatten cons] treestep')
[3 2 9 5 4 8 6 7]
There we go.
In-order traversal¶
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
J('[[3 0] [[2 0] [] []] [[9 0] [[5 0] [[4 0] [] []] [[8 0] [[6 0] [] [[7 0] [] []]] []]] []]] [] [uncons pop] [i roll< swons concat] treestep')
[2 3 4 5 6 7 8 9]
With treegrind
?¶
The treegrind
function doesn’t include the map
combinator, so
the [C]
function must arrange to use some combinator on the quoted
recursive copy [K]
. With this function, the pattern for processing a
non-empty node is:
node N [tree*] [K] C
Plugging in our BTree structure:
[key value] N [left right] [K] C
J('[["key" "value"] ["left"] ["right"] ] ["B"] ["N"] ["C"] treegrind')
['key' 'value'] 'N' [['left'] ['right']] [[not] ['B'] [uncons ['N'] dip] ['C'] genrec] 'C'
treegrind
with step
¶
Iteration through the nodes
J('[[3 0] [[2 0] [] []] [[9 0] [[5 0] [[4 0] [] []] [[8 0] [[6 0] [] [[7 0] [] []]] []]] []]] [pop] ["N"] [step] treegrind')
[3 0] 'N' [2 0] 'N' [9 0] 'N' [5 0] 'N' [4 0] 'N' [8 0] 'N' [6 0] 'N' [7 0] 'N'
Sum the nodes’ keys.
J('0 [[3 0] [[2 0] [] []] [[9 0] [[5 0] [[4 0] [] []] [[8 0] [[6 0] [] [[7 0] [] []]] []]] []]] [pop] [first +] [step] treegrind')
44
Rebuild the tree using map
(imitating treestep
.)
J('[[3 0] [[2 0] [] []] [[9 0] [[5 0] [[4 0] [] []] [[8 0] [[6 0] [] [[7 0] [] []]] []]] []]] [] [[100 +] infra] [map cons] treegrind')
[[103 0] [[102 0] [] []] [[109 0] [[105 0] [[104 0] [] []] [[108 0] [[106 0] [] [[107 0] [] []]] []]] []]]
Do we have the flexibility to reimplement Tree-get
?¶
I think we do:
[B] [N] [C] treegrind
We’ll start by saying that the base-case (the key is not in the tree) is user defined, and the per-node function is just the query key literal:
[B] [query_key] [C] treegrind
This means we just have to define C
from:
[key value] query_key [left right] [K] C
Let’s try cmp
:
C == P [T>] [E] [T<] cmp
[key value] query_key [left right] [K] P [T>] [E] [T<] cmp
The predicate P
¶
Seems pretty easy (we must preserve the value in case the keys are equal):
[key value] query_key [left right] [K] P
[key value] query_key [left right] [K] roll<
[key value] [left right] [K] query_key [roll< uncons swap] dip
[key value] [left right] [K] roll< uncons swap query_key
[left right] [K] [key value] uncons swap query_key
[left right] [K] key [value] swap query_key
[left right] [K] [value] key query_key
P == roll< [roll< uncons swap] dip
(Possibly with a swap at the end? Or just swap T<
and T>
.)
So now:
[left right] [K] [value] key query_key [T>] [E] [T<] cmp
Becomes one of these three:
[left right] [K] [value] T>
[left right] [K] [value] E
[left right] [K] [value] T<
T<
and T>
¶
T< == pop [first] dip i
T> == pop [second] dip i
Putting it together¶
T> == pop [first] dip i
T< == pop [second] dip i
E == roll> popop first
P == roll< [roll< uncons swap] dip
Tree-get == [P [T>] [E] [T<] cmp] treegrind
To me, that seems simpler than the genrec
version.
DefinitionWrapper.add_definitions('''
T> == pop [first] dip i
T< == pop [second] dip i
E == roll> popop first
P == roll< [roll< uncons swap] dip
Tree-get == [P [T>] [E] [T<] cmp] treegrind
''', D)
J('''\
[[3 13] [[2 12] [] []] [[9 19] [[5 15] [[4 14] [] []] [[8 18] [[6 16] [] [[7 17] [] []]] []]] []]]
[] [5] Tree-get
''')
15
J('''\
[[3 13] [[2 12] [] []] [[9 19] [[5 15] [[4 14] [] []] [[8 18] [[6 16] [] [[7 17] [] []]] []]] []]]
[pop "nope"] [25] Tree-get
''')
'nope'