# 功能編程和狀態算法 Here's a simple answer relying on `mapConcat`.

`````` mapConcat :: (a -> [b]) -> [a] -> [b]
-- mapConcat is in the std libs, mapConcat = concat . map
type Path = []

isReachable :: a -> Auto a -> a -> [Path a]
isReachable to auto from | to == from = [[]]
isReachable to auto from | otherwise =
map (from:) . mapConcat (isReachable to auto) \$ neighbors auto from
``````

Where `neighbors` returns the states immediately connected to a state. This returns a series of paths.

Functional programming doesn't get rid of state. It only makes it explicit! While it's true that functions like map will often "unravel" a "shared" data structure, if all you want to do is write a reachability algorithm then it's just a matter of keeping track of what nodes you already visited:

``````import qualified Data.Set as S
data Node = Node Int [Node] deriving (Show)

-- Receives a root node, returns a list of the node keyss visited in a depth-first search
dfs :: Node -> [Int]
dfs x = fst (dfs' (x, S.empty))

-- This worker function keeps track of a set of already-visited nodes to ignore.
dfs' :: (Node, S.Set Int) -> ([Int], S.Set Int)
dfs' ([email protected](Node k ns), s )
| k  `S.member` s = ([], s)
| otherwise =
let (childtrees, s') = loopChildren ns (S.insert k s) in
(k:(concat childtrees), s')

--This function could probably be implemented as just a fold but Im lazy today...
loopChildren :: [Node] -> S.Set Int -> ([[Int]], S.Set Int)
loopChildren []  s = ([], s)
loopChildren (n:ns) s =
let (xs, s') = dfs' (n, s) in
let (xss, s'') = loopChildren ns s' in
(xs:xss, s'')

na = Node 1 [nb, nc, nd]
nb = Node 2 [ne]
nc = Node 3 [ne, nf]
nd = Node 4 [nf]
ne = Node 5 [ng]
nf = Node 6 []
ng = Node 7 []

main = print \$ dfs na -- [1,2,5,7,3,6,4]
``````

Now, I must confess that keeping track of all this state by hand is pretty annoying and error prone (it's easy to use s' instead of s'', it's easy to pass the same s' to more than one computation...). This is where monads come in: they don't add anything you couldn't already do before but they let you pass the state variable around implicitly and the interface guarantees that it happens in a single-threaded manner.

Edit: I'll attempt to give a more reasoning of what I did now: first of all, instead of just testing for reachability I coded a depth-first search. The implementation is going to look pretty much the same but the debugging looks a bit better.

In a stateful language, the DFS would look kind of like this:

``````visited = set()  #mutable state
visitlist = []   #mutable state
def dfs(node):
if isMember(node, visited):
//do nothing
else:
visited[node.key] = true
visitlist.append(node.key)
for child in node.children:
dfs(child)
``````

Now we need to find a way to get rid of the mutable state. First of all we get rid of the "visitlist" variable by making dfs return that instead of void:

``````visited = set()  #mutable state
def dfs(node):
if isMember(node, visited):
return []
else:
visited[node.key] = true
return [node.key] + concat(map(dfs, node.children))
``````

And now comes the tricky part: getting rid of the "visited" variable. The basic trick is to use a convention where we pass the state as an additional parameter to functions that need it and have those functions return the new version of the state as an extra return value if they want to modify it.

``````let increment_state s = s+1 in
let extract_state s = (s, 0) in

let s0 = 0 in
let s1 = increment_state s0 in
let s2 = increment_state s1 in
let (x, s3) = extract_state s2 in
-- and so on...
``````

To apply this pattern to the dfs, we need to change it to receive the "visited" set as an extra parameter and to return the updated version of "visited" as an extra return value. Additionally, we need to rewrite the code so that we are always passing forward the "most recent" version of the "visited" array:

``````def dfs(node, visited1):
if isMember(node, visited1):
return ([], visited1) #return the old state because we dont want to  change it
else:
curr_visited = insert(node.key, visited1) #immutable update, with a new variable for the new value
childtrees = []
for child in node.children:
(ct, curr_visited) = dfs(child, curr_visited)
child_trees.append(ct)
return ([node.key] + concat(childTrees), curr_visited)
``````

The Haskell version does pretty much what I did here, except that it goes all the way and uses an inner recursive function instead of mutable "curr_visited" and "childtrees" variables.

As for monads, what they basically accomplish is implicitly passing the "curr_visited" around, instead of forcing you to do it by hand. Not only does this remove clutter from the code, but it also prevents you from making mistakes, such as forking state (passing the same "visited" set to two subsequent calls instead of chaining the state).