needing some help (if possible) in how to count the amount of times a recursive function executes itself.
I don't know how to make some sort of counter in OCaml.
Thanks!
Let's consider a very simple recursive function (not Schroder as I don't want to do homework for you) to calculate Fibonacci numbers.
let rec fib n =
match n with
| 0 | 1 -> 1
| _ when n > 0 -> fib (n - 2) + fib (n - 1)
| _ -> raise (Invalid_argument "Negative values not supported")
Now, if we want to know how many times it's been passed in, we can have it take a call number and return a tuple with that call number updated.
To get each updated call count and pass it along, we explicitly call fib in let bindings. Each time c shadows its previous binding, as we don't need that information.
let rec fib n c =
match n with
| 0 | 1 -> (1, c + 1)
| _ when n > 0 ->
let n', c = fib (n - 1) (c + 1) in
let n'', c = fib (n - 2) (c + 1) in
(n' + n'', c)
| _ -> raise (Invalid_argument "Negative values not supported")
And we can shadow that to not have to explicitly pass 0 on the first call.
let fib n = fib n 0
Now:
utop # fib 5;;
- : int * int = (8, 22)
The same pattern can be applied to the Schroder function you're trying to write.
You can create a reference in any higher scope like so
let counter = ref 0 in
let rec f ... =
counter := !counter + 1;
... (* Function body *)
If the higher scope happens to be the module scope (or file top-level scope) you should omit the in
You can return a tuple (x,y) where y you increment by one for each recursive call. It can be useful if your doing for example a Schroder sequence ;)
I am doing some exercises on F#, i have this function that calculate the alternate sum:
let rec altsum = function
| [] -> 0
| [x] -> x
| x0::x1::xs -> x0 - x1 + altsum xs;;
val altsum : int list -> int
The exercise consist in declare the same function with only two clauses...but how to do this?
The answer of mydogisbox is correct and work!
But after some attempts I found a smallest and readable solution of the problem.
let rec altsum2 = function
| [] -> 0
| x0::xs -> x0 - altsum2 xs
Example
altsum2 [1;2;3] essentially do this:
1 - (2 - (3 - 0)
it's is a bit tricky but work!
OFF TOPIC:
Another elegant way to solve the problem, using F# List library is:
let altsum3 list = List.foldBack (fun x acc -> x - acc) list 0;;
After the comment of phoog I started trying to solve the problem with a tail recursive function:
let tail_altsum4 list =
let pl l = List.length l % 2 = 0
let rec rt = function
| ([],acc) -> if pl list then -acc else acc
| (x0::xs,acc) -> rt (xs, x0 - acc)
rt (list,0)
This is also a bit tricky...substraction is not commutative and it's impossible think to revers with List.rev a long list...but I found a workaround! :)
To reduce the number of cases, you need to move your algorithm back closer to the original problem. The problem says to negate alternating values, so that's what your solution should do.
let altsum lst =
let rec altsumRec lst negateNext =
match lst with
| [] -> 0
| head::tail -> (if negateNext then -head else head) + altsumRec tail (not negateNext)
altsumRec lst false
Update: I can't use any List.function stuff.
I'm new to OCaml and I'm learning this course in which I'm supposed to calculate a list of non decreasing values from a list of values.
So for e.g. I have a list [1; 2; 3; 1; 2; 7; 6]
So function mono that takes in a list returns the following:
# mono [1; 2; 3; 1; 2; 7; 6];;
- : int list = [1; 2; 3; 7]
I do the following:
let rec calculateCheck value lst = (
match lst with
[] -> true
| x :: xs -> (
if (value < x) then
false
else
calculateCheck value xs
)
);;
let rec reverse_list lst = (
match lst with
[] -> []
| x :: xs -> (
reverse_list xs # [x]
)
);;
let shouldReverse = ref 1;;
let cancelReverse somelist lst = (
shouldReverse := 0;
reverse_list lst
);;
let rec mono lst = (
let somelist = ref lst in
if (!shouldReverse = 1) then
somelist := cancelReverse somelist lst
else
somelist := lst;
match !somelist with
[] -> []
| x :: xs -> (
if (calculateCheck x xs) then
[x] # mono xs
else
[] # mono xs
);
);;
Problem?
This only works once because of shouldReverse.
I cannot reverse the value; mono list should return non decreasing list.
Question?
Any easy way to do this?
Specifically how to get a subset of the list. For e.g. for [1; 2; 3; 5; 6], I want [1; 2; 3] as an output for 5 so that I can solve this issue recursively. The other thing, is you can have a list as [1; 2; 3; 5; 6; 5]:: so for the second 5, the output should be [1; 2; 3; 5; 6].
Any ideas?
Thanks
A good way to approach this kind of problem is to force yourself to
formulate what you're looking for formally, in a mathematically
correct way. With some training, this will usually get you
a description that is close to the final program you will write.
We are trying to define a function incr li that contains the
a strictly increasing subsequence of li. As Jeffrey Scoffield asked,
you may be looking for the
longest
such subsequence: this is an interesting and non-trivial algorithmic
problem that is well-studied, but given that you're a beginner
I suppose your teacher is asking for something simpler. Here is my
suggestion of a simpler specification: you are looking for all the
elements that are greater than all the elements before them in the
list.
A good way to produce mathematical definitions that are easy to turn
into algorithms is reasoning by induction: define a property on
natural numbers P(n) in terms of the predecessor P(n-1), or define
a property on a given list in terms of this property on a list of one
less element. Consider you want to define incr [x1; x2; x3; x4]. You
may express it either in terms of incr [x1; x2; x3] and x4, or in
terms of x1 and incr [x2; x3; x4].
incr [x1;x2;x3;x4] is incr[x1;x2;x3], plus x4 if it is bigger
than all the elements before it in the list, or, equivalently, the
biggest element of incr[x1;x2;x3]
incr [x1;x2;x3;x4] is incr[x2;x3;x4] where all the elements
smaller than x1 have been removed (they're not bigger than all
elements before them), and x1 added
These two precise definitions can of course be generalized to lists of
any length, and they give two different ways to write incr.
(* `incr1` defines `incr [x1;x2;x3;x4]` from `incr [x1;x2;x3]`,
keeping as intermediate values `subli` that corresponds to
`incr [x1;x2;x3]` in reverse order, and `biggest` the biggest
value encountered so far. *)
let incr1 li =
let rec incr subli biggest = function
| [] -> List.rev subli
| h::t ->
if h > biggest
then incr (h::subli) h t
else incr subli biggest t
in
match li with
| [] -> []
| h::t -> incr [h] h t
(* `incr2` defines `incr [x1;x2;x3;x4]` from `incr [x2;x3;x4]`; it
needs no additional parameter as this is just a recursive call on
the tail of the input list. *)
let rec incr2 = function
| [] -> []
| h::t ->
(* to go from `incr [x2;x3;x4]` to `incr [x1;x2;x3;x4]`, one
must remove all the elements of `incr [x2;x3;x4]` that are
smaller than `x1`, then add `x1` to it *)
let rec remove = function
| [] -> []
| h'::t ->
if h >= h' then remove t
else h'::t
in h :: remove (incr2 t)
As a followup to my earlier question on finding runs of the same character in a string, I would also like to find a functional algorithm to find all substrings of length greater than 2 that are ascending or descending sequences of letters or digits (e,g,: "defgh", "34567", "XYZ", "fedcba", "NMLK", 9876", etc.) in a character string ([Char]). The only sequences that I am considering are substrings of A..Z, a..z, 0..9, and their descending counterparts. The return value should be a list of (zero-based offset, length) pairs. I am translating the "zxcvbn" password strength algorithm from JavaScript (containing imperative code) to Scala. I would like to keep my code as purely functional as possible, for all the usual reasons given for writing in the functional programming style.
My code is written in Scala, but I can probably translate an algorithm in any of Clojure, F#, Haskell, or pseudocode.
Example: For the string qweABCD13987 would return [(3,4),(9,3)].
I have written a rather monsterous function that I will post when I again have access to my work computer, but I am certain that a more elegant solution exists.
Once again, thanks.
I guess a nice solution for this problem is really more complicated than it seems at first.
I'm no Scala Pro, so my solution is surely not optimal and nice, but maybe it gives you some ideas.
The basic idea is to compute the difference between two consecutive characters, afterwards it unfortunately gets a bit messy. Ask me if some of the code is unclear!
object Sequences {
val s = "qweABCD13987"
val pairs = (s zip s.tail) toList // if s might be empty, add a check here
// = List((q,w), (w,e), (e,A), (A,B), (B,C), (C,D), (D,1), (1,3), (3,9), (9,8), (8,7))
// assuming all characters are either letters or digits
val diff = pairs map {case (t1, t2) =>
if (t1.isLetter ^ t2.isLetter) 0 else t1 - t2} // xor could also be replaced by !=
// = List(-6, 18, 36, -1, -1, -1, 19, -2, -6, 1, 1)
/**
*
* #param xs A list indicating the differences between consecutive characters
* #param current triple: (start index of the current sequence;
* number of current elements in the sequence;
* number indicating the direction i.e. -1 = downwards, 1 = upwards, 0 = doesn't matter)
* #return A list of triples similar to the argument
*/
def sequences(xs: Seq[Int], current: (Int, Int, Int) = (0, 1, 0)): List[(Int, Int, Int)] = xs match {
case Nil => current :: Nil
case (1 :: ys) =>
if (current._3 != -1)
sequences(ys, (current._1, current._2 + 1, 1))
else
current :: sequences(ys, (current._1 + current._2 - 1, 2, 1)) // "recompute" the current index
case (-1 :: ys) =>
if (current._3 != 1)
sequences(ys, (current._1, current._2 + 1, -1))
else
current :: sequences(ys, (current._1 + current._2 - 1, 2, -1))
case (_ :: ys) =>
current :: sequences(ys, (current._1 + current._2, 1, 0))
}
sequences(diff) filter (_._2 > 1) map (t => (t._1, t._2))
}
It's always best to split a problem into several smaller subproblems. I wrote a solution in Haskell, which is easier for me. It uses lazy lists, but I suppose you can convert it to Scala either using streams or by making the main function tail recursive and passing the intermediate result as an argument.
-- Mark all subsequences whose adjacent elements satisfy
-- the given predicate. Includes subsequences of length 1.
sequences :: (Eq a) => (a -> a -> Bool) -> [a] -> [(Int,Int)]
sequences p [] = []
sequences p (x:xs) = seq x xs 0 0
where
-- arguments: previous char, current tail sequence,
-- last asc. start offset of a valid subsequence, current offset
seq _ [] lastOffs curOffs = [(lastOffs, curOffs - lastOffs)]
seq x (x':xs) lastOffs curOffs
| p x x' -- predicate matches - we're extending current subsequence
= seq x' xs lastOffs curOffs'
| otherwise -- output the currently marked subsequence and start a new one
= (lastOffs, curOffs - lastOffs) : seq x' xs curOffs curOffs'
where
curOffs' = curOffs + 1
-- Marks ascending subsequences.
asc :: (Enum a, Eq a) => [a] -> [(Int,Int)]
asc = sequences (\x y -> succ x == y)
-- Marks descending subsequences.
desc :: (Enum a, Eq a) => [a] -> [(Int,Int)]
desc = sequences (\x y -> pred x == y)
-- Returns True for subsequences of length at least 2.
validRange :: (Int, Int) -> Bool
validRange (offs, len) = len >= 2
-- Find all both ascending and descending subsequences of the
-- proper length.
combined :: (Enum a, Eq a) => [a] -> [(Int,Int)]
combined xs = filter validRange (asc xs) ++ filter validRange (desc xs)
-- test:
main = print $ combined "qweABCD13987"
Here is my approximation in Clojure:
We can transform the input string so we can apply your previous algorithm to find a solution. The alorithm wont be the most performant but I think you will have a more abstracted and readable code.
The example string can be transformed in the following way:
user => (find-serials "qweABCD13987")
(0 1 2 # # # # 7 8 # # #)
Reusing the previous function "find-runs":
user => (find-runs (find-serials "qweABCD13987"))
([3 4] [9 3])
The final code will look like this:
(defn find-runs [s]
(let [ls (map count (partition-by identity s))]
(filter #(>= (% 1) 3)
(map vector (reductions + 0 ls) ls))))
(def pad "#")
(defn inc-or-dec? [a b]
(= (Math/abs (- (int a) (int b))) 1 ))
(defn serial? [a b c]
(or (inc-or-dec? a b) (inc-or-dec? b c)))
(defn find-serials [s]
(map-indexed (fn [x [a b c]] (if (serial? a b c) pad x))
(partition 3 1 (concat pad s pad))))
find-serials creates a 3 cell sliding window and applies serial? to detect the cells that are the beginning/middle/end of a sequence. The string is conveniently padded so the window is always centered over the original characters.
UPDATE - Solution
Thanks to jacobm for his help, I came up with a solution.
// Folding Recursion
let reverse_list_3 theList =
List.fold_left (fun element recursive_call -> recursive_call::element) [] theList;;
I'm learning about the different ways of recursion in OCaml (for class) and for some exercise, I'm writing a function to reverse a list using different recursion styles.
// Forward Recursion
let rec reverse_list_forward theList =
match theList with [] -> [] | (head::tail) -> (reverse_list_1 tail) # [head];;
// Tail Recursion
let rec reverse_list_tail theList result =
match theList with [] -> result | (head::tail) -> reverse_list_2 tail (head::result);;
Now, I'm trying to write a reverse function using List.fold_left but I'm stuck and can't figure it out. How would I write this reverse function using folding?
Also, if anyone has good references on functional programming, the different types of recursion, higher-order-functions, etc..., links would be greatly appreciated :)
I find it helpful to think of the fold operations as a generalization of what to do with a sequence of operations
a + b + c + d + e
fold_right (+) 0 applies the + operation right-associatively, using 0 as a base case:
(a + (b + (c + (d + (e + 0)))))
fold_left 0 (+) applies it left-associatively:
(((((0 + a) + b) + c) + d) + e)
Now consider what happens if you replace + with :: and 0 with [] in both right- and left-folds.
It may also be useful to think about the way fold_left and fold_right work as "replacing" the :: and [] operators in a list. For instance, the list [1,2,3,4,5] is really just shorthand for 1::(2::(3::(4::(5::[])))). It may be useful to think of fold_right op base as letting you "replace" :: with op and [] with base: for instance
fold_right (+) 0 1::(2::(3::(4::(5::[]))))
becomes
1 + (2 + (3 + (4 + (5 + 0))))
:: became +, [] became 0. From this perspective, it's easy to see that fold_right (::) [] just gives you back your original list. fold_left base op does something a bit weirder: it rewrites all the parentheses around the list to go the other direction, moves [] from the back of the list to the front, and then replaces :: with op and [] with base. So for instance:
fold_left 0 (+) 1::(2::(3::(4::(5::[]))))
becomes
(((((0 + 1) + 2) + 3) + 4) + 5)
With + and 0, fold_left and fold_right produce the same result. But in other cases, that's not so: for instance if instead of + you used - the results would be different: 1 - (2 - (3 - (4 - (5 - 0)))) = 3, but (((((0 - 1) - 2) - 3) - 4) - 5) = -15.
let rev =
List.fold_left ( fun lrev b ->
b::lrev
) [];;
test:
# rev [1;2;3;4];;
- : int list = [4; 3; 2; 1]