500 lines
11 KiB
Markdown
500 lines
11 KiB
Markdown
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# Treating Trees II
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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 followed by a sequence of zero or more child trees. (The asterisk is meant to indicate the [Kleene star](https://en.wikipedia.org/wiki/Kleene_star).)
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tree = [] | [node tree*]
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## `treestep`
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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]`.
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tree [B] [N] [C] treestep
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If the current tree node is empty then just execute `B`:
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[] [B] [N] [C] treestep
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---------------------------
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[] B
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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`.
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[node tree*] [B] [N] [C] treestep
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--------------------------------------- w/ K == [B] [N] [C] treestep
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node N [tree*] [K] map C
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(Later on we'll experiment with making `map` part of `C` so you can use other combinators.)
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## Derive the recursive function.
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We can begin to derive it by finding the `ifte` stage that `genrec` will produce.
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K == [not] [B] [R0] [R1] genrec
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== [not] [B] [R0 [K] R1] ifte
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So we just have to derive `J`:
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J == R0 [K] R1
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The behavior of `J` is to accept a (non-empty) tree node and arrive at the desired outcome.
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[node tree*] J
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------------------------------
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node N [tree*] [K] map C
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So `J` will have some form like:
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J == ... [N] ... [K] ... [C] ...
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Let's dive in. First, unquote the node and `dip` `N`.
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[node tree*] uncons [N] dip
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node [tree*] [N] dip
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node N [tree*]
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Next, `map` `K` over the child trees and combine with `C`.
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node N [tree*] [K] map C
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node N [tree*] [K] map C
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node N [K.tree*] C
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So:
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J == uncons [N] dip [K] map C
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Plug it in and convert to `genrec`:
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K == [not] [B] [J ] ifte
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== [not] [B] [uncons [N] dip [K] map C] ifte
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== [not] [B] [uncons [N] dip] [map C] genrec
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## Extract the givens to parameterize the program.
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Working backwards:
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[not] [B] [uncons [N] dip] [map C] genrec
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[B] [not] swap [uncons [N] dip] [map C] genrec
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[B] [uncons [N] dip] [[not] swap] dip [map C] genrec
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^^^^^^^^^^^^^^^^
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[B] [[N] dip] [uncons] swoncat [[not] swap] dip [map C] genrec
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[B] [N] [dip] cons [uncons] swoncat [[not] swap] dip [map C] genrec
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^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Extract a couple of auxiliary definitions:
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TS.0 == [[not] swap] dip
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TS.1 == [dip] cons [uncons] swoncat
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[B] [N] TS.1 TS.0 [map C] genrec
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[B] [N] [map C] [TS.1 TS.0] dip genrec
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[B] [N] [C] [map] swoncat [TS.1 TS.0] dip genrec
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The givens are all to the left so we have our definition.
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### (alternate) Extract the givens to parameterize the program.
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Working backwards:
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[not] [B] [uncons [N] dip] [map C] genrec
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[not] [B] [N] [dip] cons [uncons] swoncat [map C] genrec
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[B] [N] [not] roll> [dip] cons [uncons] swoncat [map C] genrec
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^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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## Define `treestep`
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```python
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from notebook_preamble import D, J, V, define, DefinitionWrapper
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```
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```python
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DefinitionWrapper.add_definitions('''
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_treestep_0 == [[not] swap] dip
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_treestep_1 == [dip] cons [uncons] swoncat
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treegrind == [_treestep_1 _treestep_0] dip genrec
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treestep == [map] swoncat treegrind
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''', D)
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```
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## Examples
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Consider trees, the nodes of which are integers. We can find the sum of all nodes in a tree with this function:
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sumtree == [pop 0] [] [sum +] treestep
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```python
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define('sumtree == [pop 0] [] [sum +] treestep')
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```
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Running this function on an empty tree value gives zero:
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[] [pop 0] [] [sum +] treestep
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------------------------------------
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0
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```python
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J('[] sumtree') # Empty tree.
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```
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0
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Running it on a non-empty node:
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[n tree*] [pop 0] [] [sum +] treestep
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n [tree*] [[pop 0] [] [sum +] treestep] map sum +
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n [ ... ] sum +
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n m +
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n+m
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```python
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J('[23] sumtree') # No child trees.
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```
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23
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```python
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J('[23 []] sumtree') # Child tree, empty.
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```
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23
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```python
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J('[23 [2 [4]] [3]] sumtree') # Non-empty child trees.
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```
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32
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```python
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J('[23 [2 [8] [9]] [3] [4 []]] sumtree') # Etc...
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```
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49
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```python
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J('[23 [2 [8] [9]] [3] [4 []]] [pop 0] [] [cons sum] treestep') # Alternate "spelling".
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```
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49
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```python
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J('[23 [2 [8] [9]] [3] [4 []]] [] [pop 23] [cons] treestep') # Replace each node.
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```
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[23 [23 [23] [23]] [23] [23 []]]
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```python
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J('[23 [2 [8] [9]] [3] [4 []]] [] [pop 1] [cons] treestep')
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```
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[1 [1 [1] [1]] [1] [1 []]]
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```python
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J('[23 [2 [8] [9]] [3] [4 []]] [] [pop 1] [cons] treestep sumtree')
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```
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6
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```python
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J('[23 [2 [8] [9]] [3] [4 []]] [pop 0] [pop 1] [sum +] treestep') # Combine replace and sum into one function.
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```
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6
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```python
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J('[4 [3 [] [7]]] [pop 0] [pop 1] [sum +] treestep') # Combine replace and sum into one function.
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```
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3
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## Redefining the Ordered Binary Tree in terms of `treestep`.
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Tree = [] | [[key value] left right]
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What kind of functions can we write for this with our `treestep`?
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The pattern for processing a non-empty node is:
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node N [tree*] [K] map C
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Plugging in our BTree structure:
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[key value] N [left right] [K] map C
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### Traversal
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[key value] first [left right] [K] map i
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key [value] [left right] [K] map i
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key [left right] [K] map i
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key [lkey rkey ] i
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key lkey rkey
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This doesn't quite work:
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```python
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J('[[3 0] [[2 0] [][]] [[9 0] [[5 0] [[4 0] [][]] [[8 0] [[6 0] [] [[7 0] [][]]][]]][]]] ["B"] [first] [i] treestep')
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```
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3 'B' 'B'
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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.
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[key value] N [left right] [K] map C
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[key value] first [left right] [K] map flatten cons
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key [left right] [K] map flatten cons
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key [[lk] [rk] ] flatten cons
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key [ lk rk ] cons
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[key lk rk ]
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So:
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[] [first] [flatten cons] treestep
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```python
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J('[[3 0] [[2 0] [] []] [[9 0] [[5 0] [[4 0] [] []] [[8 0] [[6 0] [] [[7 0] [] []]] []]] []]] [] [first] [flatten cons] treestep')
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```
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[3 2 9 5 4 8 6 7]
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There we go.
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### In-order traversal
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From here:
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key [[lk] [rk]] C
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key [[lk] [rk]] i
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key [lk] [rk] roll<
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[lk] [rk] key swons concat
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[lk] [key rk] concat
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[lk key rk]
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So:
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[] [i roll< swons concat] [first] treestep
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```python
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J('[[3 0] [[2 0] [] []] [[9 0] [[5 0] [[4 0] [] []] [[8 0] [[6 0] [] [[7 0] [] []]] []]] []]] [] [uncons pop] [i roll< swons concat] treestep')
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```
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[2 3 4 5 6 7 8 9]
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## With `treegrind`?
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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:
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node N [tree*] [K] C
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Plugging in our BTree structure:
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[key value] N [left right] [K] C
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```python
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J('[["key" "value"] ["left"] ["right"] ] ["B"] ["N"] ["C"] treegrind')
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```
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['key' 'value'] 'N' [['left'] ['right']] [[not] ['B'] [uncons ['N'] dip] ['C'] genrec] 'C'
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## `treegrind` with `step`
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Iteration through the nodes
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```python
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J('[[3 0] [[2 0] [] []] [[9 0] [[5 0] [[4 0] [] []] [[8 0] [[6 0] [] [[7 0] [] []]] []]] []]] [pop] ["N"] [step] treegrind')
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```
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[3 0] 'N' [2 0] 'N' [9 0] 'N' [5 0] 'N' [4 0] 'N' [8 0] 'N' [6 0] 'N' [7 0] 'N'
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Sum the nodes' keys.
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```python
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J('0 [[3 0] [[2 0] [] []] [[9 0] [[5 0] [[4 0] [] []] [[8 0] [[6 0] [] [[7 0] [] []]] []]] []]] [pop] [first +] [step] treegrind')
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```
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44
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Rebuild the tree using `map` (imitating `treestep`.)
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```python
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J('[[3 0] [[2 0] [] []] [[9 0] [[5 0] [[4 0] [] []] [[8 0] [[6 0] [] [[7 0] [] []]] []]] []]] [] [[100 +] infra] [map cons] treegrind')
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```
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[[103 0] [[102 0] [] []] [[109 0] [[105 0] [[104 0] [] []] [[108 0] [[106 0] [] [[107 0] [] []]] []]] []]]
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## Do we have the flexibility to reimplement `Tree-get`?
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I think we do:
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[B] [N] [C] treegrind
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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:
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[B] [query_key] [C] treegrind
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This means we just have to define `C` from:
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[key value] query_key [left right] [K] C
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Let's try `cmp`:
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C == P [T>] [E] [T<] cmp
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[key value] query_key [left right] [K] P [T>] [E] [T<] cmp
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### The predicate `P`
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Seems pretty easy (we must preserve the value in case the keys are equal):
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[key value] query_key [left right] [K] P
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[key value] query_key [left right] [K] roll<
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[key value] [left right] [K] query_key [roll< uncons swap] dip
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[key value] [left right] [K] roll< uncons swap query_key
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[left right] [K] [key value] uncons swap query_key
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[left right] [K] key [value] swap query_key
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[left right] [K] [value] key query_key
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P == roll< [roll< uncons swap] dip
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(Possibly with a swap at the end? Or just swap `T<` and `T>`.)
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So now:
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[left right] [K] [value] key query_key [T>] [E] [T<] cmp
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Becomes one of these three:
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[left right] [K] [value] T>
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[left right] [K] [value] E
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[left right] [K] [value] T<
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### `E`
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Easy.
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E == roll> popop first
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### `T<` and `T>`
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T< == pop [first] dip i
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T> == pop [second] dip i
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## Putting it together
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T> == pop [first] dip i
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T< == pop [second] dip i
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E == roll> popop first
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P == roll< [roll< uncons swap] dip
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Tree-get == [P [T>] [E] [T<] cmp] treegrind
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To me, that seems simpler than the `genrec` version.
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```python
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DefinitionWrapper.add_definitions('''
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T> == pop [first] dip i
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T< == pop [second] dip i
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E == roll> popop first
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P == roll< [roll< uncons swap] dip
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Tree-get == [P [T>] [E] [T<] cmp] treegrind
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''', D)
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```
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```python
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from joy.library import FunctionWrapper
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from joy.utils.stack import pushback
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@FunctionWrapper
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def cmp_(stack, expression, dictionary):
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'''
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cmp takes two values and three quoted programs on the stack and runs
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one of the three depending on the results of comparing the two values:
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a b [G] [E] [L] cmp
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------------------------- a > b
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G
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a b [G] [E] [L] cmp
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------------------------- a = b
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E
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a b [G] [E] [L] cmp
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------------------------- a < b
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L
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'''
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L, (E, (G, (b, (a, stack)))) = stack
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expression = pushback(G if a > b else L if a < b else E, expression)
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return stack, expression, dictionary
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D['cmp'] = cmp_
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```
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```python
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J('''\
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[[3 13] [[2 12] [] []] [[9 19] [[5 15] [[4 14] [] []] [[8 18] [[6 16] [] [[7 17] [] []]] []]] []]]
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[] [5] Tree-get
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''')
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```
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15
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```python
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J('''\
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[[3 13] [[2 12] [] []] [[9 19] [[5 15] [[4 14] [] []] [[8 18] [[6 16] [] [[7 17] [] []]] []]] []]]
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[pop "nope"] [25] Tree-get
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''')
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```
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'nope'
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