Understanding how recursive functions work

Question

1.The function is called recursively until a condition is met. That condition is a > b. When this condition is met, return 0. At first glance, I would expect the return value to be 0 which is obviously incorrect.

Here is what the computer computing sumInts(2,5) would think if it were able to:

I want to compute sumInts(2, 5)
for this, I need to compute sumInts(3, 5)
and add 2 to the result.
  I want to compute sumInts(3, 5)
  for this, I need to compute sumInts(4, 5)
  and add 3 to the result.
    I want to compute sumInts(4, 5)
    for this, I need to compute sumInts(5, 5)
    and add 4 to the result.
      I want to compute sumInts(5, 5)
      for this, I need to compute sumInts(6, 5)
      and add 5 to the result.
        I want to compute sumInts(6, 5)
        since 6 > 5, this is zero.
      The computation yielded 0, therefore I shall return 5 = 5 + 0.
    The computation yielded 5, therefore I shall return 9 = 4 + 5.
  The computation yielded 9, therefore I shall return 12 = 3 + 9.
The computation yielded 12, therefore I shall return 14 = 2 + 12.

As you see, some call to the function sumInts actually returns 0 however this not the final value because the computer still has to add 5 to that 0, then 4 to the result, then 3, then 2, as described by the four last sentences of the thoughts of our computer. Note that in the recursion, the computer does not only have to compute the recursive call, it also has to remember what to do with the value returned by the recursive call. There is a special area of computer's memory called the stack where this kind of information is saved, this space is limited and functions that are too recursive can exhaust the stack: this is the stack overflow giving its name to our most loved website.

Your statement seems to make the implicit assumption that the computer forgets what it were at when doing a recursive call, but it does not, this is why your conclusion does not match your observation.

2.Printing out the value of 'a' on each iteration yields a value which I would expect: 2, 3, 4, 5 (at which point 5+1 > b which meets the first condition: a > b) but I still don't see how the value of 14 is achieved.

This is because the return value is not an a itself but the sum of the value of a and the value returned by the recursive call.

I think the confusion is stemming from thinking of it as "the same function" being called many times. If you think of it as "many copies of the same function being called", then it may be clearer:

Only one copy of the function ever returns 0, and it's not the first one (it's the last one). So the result of calling the first one is not 0.

For the second bit of confusion, I think it will be easier to spell out the recursion in English. Read this line:

return a + sumInts(a + 1, b: b)

as "return the value of 'a' plus (the return value of another copy of the function, which is the copy's value of 'a' plus (the return value of another copy of the function, which is the second copy's value of 'a' plus (...", with each copy of the function spawning a new copy of itself with a increased by 1, until the a > b condition is met.

By the time you reach the the a > b condition being true, you have a (potentially arbitrarily) long stack of copies of the function all in the middle of being run, all waiting on the result of the next copy to find out what they should add to 'a'.

(edit: also, something to be aware of is that the stack of copies of the function I mention is a real thing that takes up real memory, and will crash your program if it gets too large. The compiler can optimize it out in some cases, but exhausting stack space is a significant and unfortunate limitation of recursive functions in many languages)

To understand recursion you must think of the problem in a different way. Instead of a large logical sequence of steps that makes sense as a whole you instead take a large problem and break up into smaller problems and solve those, once you have an answer for the sub problems you combine the results of the sub problems to make the solution to the bigger problem. Think of you and your friends needing to count the number of marbles in a huge bucket. You do each take a smaller bucket and go count those individually and when you are done you add the totals together.. Well now if each of you find some friend and split the buckets further, then you just need to wait for these other friends to figure out their totals, bring it back to each of you, you add it up. And so on. The special case is when you only get 1 marble to count then you just return it back and say 1. let the other people above you do the adding you are done.

You must remember every time the function calls itself recursively it creates a new context with a subset of the problem, once that part is resolved it gets returned so that the previous iteration can complete.

Let me show you the steps:

sumInts(a: 2, b: 5) will return: 2 + sumInts(a: 3, b: 5)
sumInts(a: 3, b: 5) will return: 3 + sumInts(a: 4, b: 5)
sumInts(a: 4, b: 5) will return: 4 + sumInts(a: 5, b: 5)
sumInts(a: 5, b: 5) will return: 5 + sumInts(a: 6, b: 5)
sumInts(a: 6, b: 5) will return: 0

once sumInts(a: 6, b: 5) has executed, the results can be computed so going back up the chain with the results you get:

 sumInts(a: 6, b: 5) = 0
 sumInts(a: 5, b: 5) = 5 + 0 = 5
 sumInts(a: 4, b: 5) = 4 + 5 = 9
 sumInts(a: 3, b: 5) = 3 + 9 = 12
 sumInts(a: 2, b: 5) = 2 + 12 = 14.

Another way to represent the structure of the recursion:

 sumInts(a: 2, b: 5) = 2 + sumInts(a: 3, b: 5)
 sumInts(a: 2, b: 5) = 2 + 3 + sumInts(a: 4, b: 5)  
 sumInts(a: 2, b: 5) = 2 + 3 + 4 + sumInts(a: 5, b: 5)  
 sumInts(a: 2, b: 5) = 2 + 3 + 4 + 5 + sumInts(a: 6, b: 5)
 sumInts(a: 2, b: 5) = 2 + 3 + 4 + 5 + 0
 sumInts(a: 2, b: 5) = 14 

Recursion is a tricky topic to understand and I don't think I can fully do it justice here. Instead, I'll try to focus on the particular piece of code you have here and try to describe both the intuition for why the solution works and the mechanics of how the code computes its result.

The code you've given here solves the following problem: you want to know the sum of all the integers from a to b, inclusive. For your example, you want the sum of the numbers from 2 to 5, inclusive, which is

2 + 3 + 4 + 5

When trying to solve a problem recursively, one of the first steps should be to figure out how to break the problem down into a smaller problem with the same structure. So suppose that you wanted to sum up the numbers from 2 to 5, inclusive. One way to simplify this is to notice that the above sum can be rewritten as

2 + (3 + 4 + 5)

Here, (3 + 4 + 5) happens to be the sum of all the integers between 3 and 5, inclusive. In other words, if you want to know the sum of all the integers between 2 and 5, start by computing the sum of all the integers between 3 and 5, then add 2.

So how do you compute the sum of all the integers between 3 and 5, inclusive? Well, that sum is

3 + 4 + 5

which can be thought of instead as

3 + (4 + 5)

Here, (4 + 5) is the sum of all the integers between 4 and 5, inclusive. So, if you wanted to compute the sum of all the numbers between 3 and 5, inclusive, you'd compute the sum of all the integers between 4 and 5, then add 3.

There's a pattern here! If you want to compute the sum of the integers between a and b, inclusive, you can do the following. First, compute the sum of the integers between a + 1 and b, inclusive. Next, add a to that total. You'll notice that "compute the sum of the integers between a + 1 and b, inclusive" happens to be pretty much the same sort of problem we're already trying to solve, but with slightly different parameters. Rather than computing from a to b, inclusive, we're computing from a + 1 to b, inclusive. That's the recursive step - to solve the bigger problem ("sum from a to b, inclusive"), we reduce the problem to a smaller version of itself ("sum from a + 1 to b, inclusive.").

If you take a look at the code you have above, you'll notice that there's this step in it:

return a + sumInts(a + 1, b: b)

This code is simply a translation of the above logic - if you want to sum from a to b, inclusive, start by summing a + 1 to b, inclusive (that's the recursive call to sumInts), then add a.

Of course, by itself this approach won't actually work. For example, how would you compute the sum of all the integers between 5 and 5 inclusive? Well, using our current logic, you'd compute the sum of all the integers between 6 and 5, inclusive, then add 5. So how do you compute the sum of all the integers between 6 and 5, inclusive? Well, using our current logic, you'd compute the sum of all the integers between 7 and 5, inclusive, then add 6. You'll notice a problem here - this just keeps on going and going!

In recursive problem solving, there needs to be some way to stop simplifying the problem and instead just go solve it directly. Typically, you'd find a simple case where the answer can be determined immediately, then structure your solution to solve simple cases directly when they arise. This is typically called a base case or a recursive basis.

So what's the base case in this particular problem? When you're summing up integers from a to b, inclusive, if a happens to be bigger than b, then the answer is 0 - there aren't any numbers in the range! Therefore, we'll structure our solution as follows:

1. If a > b, then the answer is 0.
2. Otherwise (a ≤ b), get the answer as follows:
   1. Compute the sum of the integers between a + 1 and b.
   2. Add a to get the answer.

Now, compare this pseudocode to your actual code:

func sumInts(a: Int, b: Int) -> Int {
    if (a > b) {
        return 0
    } else {
        return a + sumInts(a + 1, b: b)
    }
}

Notice that there's almost exactly a one-to-one map between the solution outlined in pseudocode and this actual code. The first step is the base case - in the event that you ask for the sum of an empty range of numbers, you get 0. Otherwise, compute the sum between a + 1 and b, then go add a.

So far, I've given just a high-level idea behind the code. But you had two other, very good questions. First, why doesn't this always return 0, given that the function says to return 0 if a > b? Second, where does the 14 actually come from? Let's look at these in turn.

Let's try a very, very simple case. What happens if you call sumInts(6, 5)? In this case, tracing through the code, you see that the function just returns 0. That's the right thing to do, to - there aren't any numbers in the range. Now, try something harder. What happens when you call sumInts(5, 5)? Well, here's what happens:

1. You call sumInts(5, 5). We fall into the else branch, which return the value of `a + sumInts(6, 5).
2. In order for sumInts(5, 5) to determine what sumInts(6, 5) is, we need to pause what we're doing and make a call to sumInts(6, 5).
3. sumInts(6, 5) gets called. It enters the if branch and returns 0. However, this instance of sumInts was called by sumInts(5, 5), so the return value is communicated back to sumInts(5, 5), not to the top-level caller.
4. sumInts(5, 5) now can compute 5 + sumInts(6, 5) to get back 5. It then returns it to the top-level caller.

Notice how the value 5 was formed here. We started off with one active call to sumInts. That fired off another recursive call, and the value returned by that call communicated the information back to sumInts(5, 5). The call to sumInts(5, 5) then in turn did some computation and returned a value back to the caller.

If you try this with sumInts(4, 5), here's what will happen:

• sumInts(4, 5) tries to return 4 + sumInts(5, 5). To do that, it calls sumInts(5, 5).
  • sumInts(5, 5) tries to return 5 + sumInts(6, 5). To do that, it calls sumInts(6, 5).
  • sumInts(6, 5) returns 0 back to sumInts(5, 5).</li> <li>sumInts(5, 5)now has a value forsumInts(6, 5), namely 0. It then returns5 + 0 = 5`.
• sumInts(4, 5) now has a value for sumInts(5, 5), namely 5. It then returns 4 + 5 = 9.

In other words, the value that's returned is formed by summing up values one at a time, each time taking one value returned by a particular recursive call to sumInts and adding on the current value of a. When the recursion bottoms out, the deepest call returns 0. However, that value doesn't immediately exit the recursive call chain; instead, it just hands the value back to the recursive call one layer above it. In that way, each recursive call just adds in one more number and returns it higher up in the chain, culminating with the overall summation. As an exercise, try tracing this out for sumInts(2, 5), which is what you wanted to begin with.

Hope this helps!

You've got some good answers here so far, but I'll add one more that takes a different tack.

First off, I have written many articles on simple recursive algorithms that you might find interesting; see

http://ericlippert.com/tag/recursion/

http://blogs.msdn.com/b/ericlippert/archive/tags/recursion/

Those are in newest-on-top order, so start from the bottom.

Second, so far all of the answers have described recursive semantics by considering function activation. That each, each call makes a new activation, and the recursive call executes in the context of this activation. That is a good way to think of it, but there is another, equivalent way: smart text seach-and-replace.

Let me rewrite your function into a slightly more compact form; don't think of this as being in any particular language.

s = (a, b) => a > b ? 0 : a + s(a + 1, b)

I hope that makes sense. If you're not familiar with the conditional operator, it is of the form condition ? consequence : alternative and its meaning will become clear.

Now we wish to evaluate s(2,5) We do so by doing a textual replacing of the call with the function body, then replace a with 2 and b with 5:

s(2, 5) 
---> 2 > 5 ? 0 : 2 + s(2 + 1, 5)

Now evaluate the conditional. We textually replace 2 > 5 with false.

---> false ? 0 : 2 + s(2 + 1, 5)

Now textually replace all false conditionals with the alternative and all true conditionals with the consequence. We have only false conditionals, so we textually replace that expression with the alternative:

---> 2 + s(2 + 1, 5)

Now, to save me having to type all those + signs, textually replace constant arithmetic with its value. (This is a bit of a cheat, but I don't want to have to keep track of all the parentheses!)

---> 2 + s(3, 5)

Now search-and-replace, this time with the body for the call, 3 for a and 5 for b. We'll put the replacement for the call in parentheses:

---> 2 + (3 > 5 ? 0 : 3 + s(3 + 1, 5))

And now we just keep on doing those same textual substitution steps:

---> 2 + (false ? 0 : 3 + s(3 + 1, 5))  
---> 2 + (3 + s(3 + 1, 5))                
---> 2 + (3 + s(4, 5))                     
---> 2 + (3 + (4 > 5 ? 0 : 4 + s(4 + 1, 5)))
---> 2 + (3 + (false ? 0 : 4 + s(4 + 1, 5)))
---> 2 + (3 + (4 + s(4 + 1, 5)))
---> 2 + (3 + (4 + s(5, 5)))
---> 2 + (3 + (4 + (5 > 5 ? 0 : 5 + s(5 + 1, 5))))
---> 2 + (3 + (4 + (false ? 0 : 5 + s(5 + 1, 5))))
---> 2 + (3 + (4 + (5 + s(5 + 1, 5))))
---> 2 + (3 + (4 + (5 + s(6, 5))))
---> 2 + (3 + (4 + (5 + (6 > 5 ? 0 : s(6 + 1, 5)))))
---> 2 + (3 + (4 + (5 + (true ? 0 : s(6 + 1, 5)))))
---> 2 + (3 + (4 + (5 + 0)))
---> 2 + (3 + (4 + 5))
---> 2 + (3 + 9)
---> 2 + 12
---> 14

All we did here was just straightforward textual substitution. Really I shouldn't have substituted "3" for "2+1" and so on until I had to, but pedagogically it would have gotten hard to read.

Function activation is nothing more than replacing the function call with the body of the call, and replacing the formal parameters with their corresponding arguments. You have to be careful about introducing parentheses intelligently, but aside from that, it's just text replacement.

Of course, most languages do not actually implement activation as text replacement, but logically that's what it is.

So what then is an unbounded recursion? A recursion where the textual substitution doesn't stop! Notice how eventually we got to a step where there was no more s to replace, and we could then just apply the rules for arithmetic.

The way that I usually figure out how a recursive function works is by looking at the base case and working backwards. Here's that technique applied to this function.

First the base case:

sumInts(6, 5) = 0

Then the call just above that in the call stack:

sumInts(5, 5) == 5 + sumInts(6, 5)
sumInts(5, 5) == 5 + 0
sumInts(5, 5) == 5

Then the call just above that in the call stack:

sumInts(4, 5) == 4 + sumInts(5, 5)
sumInts(4, 5) == 4 + 5
sumInts(4, 5) == 9

And so on:

sumInts(3, 5) == 3 + sumInts(4, 5)
sumInts(3, 5) == 3 + 9
sumInts(3, 5) == 12

And so on:

sumInts(2, 5) == 2 + sumInts(3, 5)
sumInts(4, 5) == 2 + 12
sumInts(4, 5) == 14

Notice that we've arrived at our original call to the function sumInts(2, 5) == 14

The order in which these calls are executed:

sumInts(2, 5)
sumInts(3, 5)
sumInts(4, 5)
sumInts(5, 5)
sumInts(6, 5)

The order in which these calls return:

sumInts(6, 5)
sumInts(5, 5)
sumInts(4, 5)
sumInts(3, 5)
sumInts(2, 5)

Note that we came to a conclusion about how the function operates by tracing the calls in the order that they return.

Recursion. In Computer Science recursion is covered in depth under the topic of Finite Automata.

In its simplest form it is a self reference. For example, saying that "my car is a car" is a recursive statement. The problem is that the statement is an infinite recursion in that it will never end. The definition in the statement of a "car" is that it is a "car" so it may be substituted. However, there is no end because in the case of substitution, it still becomes "my car is a car".

This could be different if the statement were "my car is a bentley. my car is blue." In which case the substitution in the second situation for car could be "bentley" resulting in "my bentley is blue". These types of substitutions are mathematically explained in Computer Science through Context-Free Grammars.

The actual substitution is a production rule. Given that the statement is represented by S and that car is a variable which can be a "bentley" this statement can be recursively reconstructed.

S -> "my"S | " "S | CS | "is"S | "blue"S | ε
C -> "bentley"

This can be constructed in multiple ways, as each | means there is a choice. S can be replaced by any one of those choices, and S always starts empty. The ε means to terminate the production. Just as S can be replaced, so can other variables (there is only one and it is C which would represent "bentley").

So starting with S being empty, and replacing it with the first choice "my"S S becomes

"my"S

S can still be substituted as it represents a variable. We could choose "my" again, or ε to end it, but lets continue making our original statement. We choose the space which means S is replaced with " "S

"my "S

Next lets choose C

"my "CS

And C only has one choice for replacement

"my bentley"S

And the space again for S

"my bentley "S

And so on "my bentley is"S, "my bentley is "S, "my bentley is blue"S, "my bentley is blue" (replacing S for ε ends the production) and we have recursively built our statement "my bentley is blue".

Think of recursion as these productions and replacements. Each step in the process replaces its predecessor in order to produce the end result. In the exact example of the recursive sum from 2 to 5, you end up with the production

S -> 2 + A
A -> 3 + B
B -> 4 + C
C -> 5 + D
D -> 0

This becomes

2 + A
2 + 3 + B
2 + 3 + 4 + C
2 + 3 + 4 + 5 + D
2 + 3 + 4 + 5 + 0
14