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I have a function $f(x, a)$ which is invoked over all the elements of a sequence feeding the result to the next call, with $x$ being the next element in the list and $a$ the accumulated result. What is the correct (and popular) notation to write this properly?

Lets say $F(p, a)$ is the recursive function which takes the sequence, $p$, and runs $f(x, a)$ on each element of $p$.

I was thinking of defining it something like this:

$F(p, a) = \begin{cases} a & \text{if } p = \langle \rangle\\ f(F(\langle x_1, ..., x_{n-1} \rangle, a), x_n) & \text{if } p = \langle x_1, ..., x_n \rangle \end{cases} $

But for some reason it seems incorrect for the case when $p$ has just one element. Is there a better way to define it? Maybe some way of denoting the head and tail of the list?

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You have the concept of folding. Your definition seem to be right for me. It's the right fold. Note, that there is also left folding:

$$ F(p,a) = \begin{cases} a & ; p=[] \\ F([x_1,\ldots,x_{n-1}], f(x_n, a)) & ; p = [x_1, \ldots, x_n]; n \ge 1 \end{cases} $$

EDIT: You can also define it with the colon operator (which is the concatenation of an element to a list):

$$ F(p,a) = \begin{cases} a & ; p=[] \\ F(xs, f(x, a)) & ; p = x : xs \end{cases} $$

That's the way it is usually done in haskell...

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  • $\begingroup$ Thanks. My concern is that if $n = 1$ then $x_1, ..., x_{n-1}$ becomes invalid, because $n-1$ becomes $0$, so $x_1, ..., x_0$ is incorrect, when it should be the empty sequence. I once saw somewhere a different syntax, but I cannot recall where, but it was something like $F(p, h:t) = F(t, f(h, a))$ but I am not sure if it was with the colon and whether it is correct. $\endgroup$ – jbx May 10 '15 at 22:05
  • $\begingroup$ Regarding the right fold (vs left fold), the result would be different though right? I want to first apply $f()$ to $x_1$, then $x_2$ etc. while with the left fold it applies $f()$ to $x_n$ first. $\endgroup$ – jbx May 10 '15 at 22:08
  • $\begingroup$ @jbx: depending on $f$ from the right and left fold you might get different results... the colon is the concatenation of an element to a list... $\endgroup$ – Stephan Kulla May 11 '15 at 8:36
  • $\begingroup$ Thanks. Yes I was looking for something like that, but I wasn't sure if the colon was accepted mathematical notation for the head / tail of a list. (In Haskell its written :, in Scala its written ::). To me it seems a bit more correct, because $xs$ can be the empty list. $\endgroup$ – jbx May 11 '15 at 14:59

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