Understanding how recursive functions work

Question

As the title explains I have a very fundamental programming question which I have just not been able to grok yet. Filtering out all of the (extremely clever) \"In order to

Answer 1

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
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