[NEM]: The ''fold'' function reduces a list (or other structure) to a single value by "folding" a binary operator between successive elements of the list. To illustrate, if you have a list containing numbers such as: 1 2 3 4 5 Then to sum that list you can fold a binary + operator between the elements, resulting in something like: 1 + 2 + 3 + 4 + 5 (assuming infix notation). This is essentially what the "fold" function does. It takes a binary function, together with an initial value (to tack on to the end of the list) and a list: proc fold {func init list} { if {[llength $list] == 0} { return $init } else { apply $func [lindex $list 0] \ [fold $func $init [lrange $list 1 end]] } } proc apply {func args} { uplevel #0 $func $args } So, we can do our sum over a list by: proc + {a b} { expr {$a + $b} } fold + 0 {1 2 3 4 5} ;# produces 15 as the answer You may have noticed that the version of fold above is right-associative. In other words, the result it produces is equivalent to doing: 1 + (2 + (3 + (4 + (5 + 0)))) With some operators, it might make more sense to have the reduction be done in a left-associative manner. We can define a foldl which does this easily enough. I'll leave that (and converting the definition given above to a more efficient iterative version) as an exercise. You might wonder if we can perform the reverse operation, i.e., go from a value and a function that takes a single argument and produces a pair of results and from that generate a sequence/list. Well, you can indeed, and this is roughly equivalent to [iterators]. There is a demonstration of this ''unfold'' operation on that page. ---- [LV] If the above discussion doesn't seem to match the context of fold that you were seeking, another use of the word ''fold'', in the context of text editing, is the idea of [text widget elision], or representing one or more lines of text with a place holder. Some text editors call this ''text folding''. [escargo] - This use of ''fold'' reminded me of something, and I couldn't remember what. Then I recalled: It reminds me of the [APL] '''reduce''' operator '''/''', where +/A would add up all of the values of the vector A. [NEM] - Yup, lots of languages have a similar function, and "reduce" is a common name for it, e.g. that is what it is called in Python (although, not for much longer [http://www.artima.com/weblogs/viewpost.jsp?thread=98196]). Fold is a very powerful and general function [http://www.cs.nott.ac.uk/~gmh/bib.html#fold] which captures a particular pattern of recursion (or iteration) over a container. Used with other higher-order functions like [map], [filter], and [zip] you can start doing some really powerful programming with a few statements (e.g., this [http://c2.com/cgi/wiki?TransfoldPattern] discussion of the Transfold Pattern). ---- [RS] 2005-09-21: for non-empty lists and [expr] operators, the following does a very simple fold (but see [Elegance vs. performance]): expr [join $list $op] The neutral element, if ''list'' is empty, would be 0 for "+", 1 for "*"... and the others? [escargo] - By "neutral element" do you mean "identity value"? (That is, the assumed element would be whatever is the identify value for the particular operation?) [NEM]: Yes, that's right. What I labelled "init" in the above definition is often more correctly labelled "id". Here's a handful of operations defined using folds: proc foldl {f id list} { if {[llength $list] == 0} { return $id } else { foldl $f [apply $f $id [lindex $list 0]] \ [lrange $list 1 end] } } # Expose some of expr as procs: foreach op {+ - * / && ||} { proc $op {a b} "expr \$a $op \$b" } proc count {x n} { incr n } proc swap {head tail} { linsert $tail end $head } proc def {name = args} { interp alias {} $name {} {expand}$args } def sum = foldl + 0 def product = foldl * 1 def diff = fold - 0 def div = fold / 1.0 def and = fold && 1 ;# 1 == true def or = fold || 0 ;# 0 == false def length = fold count 0 def reverse = fold swap [list] And so on. You can also define [map] and [filter] in terms of fold, but it's easier with [lambda]. As a more powerful example of higher-order functions in action, here is the innerProd mentioned on the C2 page given above (look for Transfold Pattern) using [zip]With: # First, define function composition: proc compose {f g args} { apply $f [uplevel #0 $g $args] } # Reproduce zipWith here for convenience proc zipWith {f cola colb} { set ret [list] foreach a $cola b $colb { lappend ret [apply $f $a $b] } return $ret } # Now, the inner-product (i.e. (a0*b0)+(a1+b1)+...): def innerProd = compose sum {zipWith *} innerProd {2 3 4} {5 6 7} ;# -> 56 Lovely! You might still be wondering what the advantage of this over a traditional loop is. Well, basically, these higher-order functions capture a more specific usage pattern. This means that you can prove more interesting properties about them, but also you can optimise the heck out of them (not done here). For instance, people writing efficient containers can provide their own versions of fold etc which are especially optimised for traversing that particular container in an efficient manner. With more generic looping constructs you are not just specifying what you want to achieve, but also to some extent how to achieve it, which leaves very little room for optimisations. It is my belief that high-level scripting languages should be moving towards these higher-level container-oriented constructs where possible, as they offer conciseness and the possibility for speed. As shown above you can also define lots of other constructs very simply in terms of fold and chums, without adding much extra code at all. This means two things: firstly, if you optimise fold, then you get speed ups in all the other operations (same applies to e.g. [foreach], but there are less opportunities); and, secondly, you can make fold a generic operation over different containers so that all a new container author has to do is provide a couple of generic higher-order functions and gets a whole bunch of functionality for free (e.g., sum, product etc over the new container). I think these are real wins of this type of programming, and indeed wins over list comprehensions which are often seen as a better replacement -- well they are, if all you use are lists. ---- [RS] on 2005-09-22 played with the above ''fold'' and rewrote it with [func] sugar like this (also, an [{expand}]-less def for people not yet on 8.5): proc func {name argl body} {proc $name $argl [list expr $body]} func null list {[llength $list] == 0} proc head list {lindex $list 0} proc tail list {lrange $list 1 end} func fold {f id list} { [null $list]? $id : [apply $f [head $list] [fold $f $id [tail $list]]] } proc apply {func args} { uplevel #0 $func $args } proc swap {head tail} { linsert $tail end $head } proc def {name = args} { eval [list interp alias {} $name {}] $args } ---- [[ [Category Glossary] | [Category Concept] | [Functional Programming] ]]