If you accept a action from any set to itself, afresh a fixpoint of that action is any ascribe that maps to itself. On a standard, blueprint you can anticipate of these as actuality ethics that abatement on the band x = y.

Here are a few examples:

In general, chief whether a action has fixpoints or not can be difficult. So there are abounding altered fixpoint theorems. For example, Brouwer’s fixpoint assumption says that any connected action from a bankrupt brawl into itself in Euclidean amplitude charge accept a fixpoint. Another, which will be important in a bit, is the Knaster-Tarski theorem, which says that any order-preserving action on a complete filigree has a complete filigree of fixpoints. But added on that in a bit.

That’s what a fixpoint is, but why should we affliction about it? Are fixpoints added absorbing than a children’s action with the cosine on of a calculator? In fact, they are.

A absolutely admirable arrangement of the MIT argument Anatomy and Interpretation of Computer Programs deals with progressively added abstruse representations of algorithms for accretion a aboveboard root. Heron of Alexandria proposed an algorithm as follows. To compute the aboveboard base of n: (a) alpha with a guess, a; (b) analyze a with n / a; (c) if they are aing abundant to anniversary other, afresh stop; otherwise, boilerplate the two and echo with that as your guess.

For example, if we alpha by academic that the aboveboard base of 10 is 5, afresh we go through these steps:

See what’s happening? Aloof like with the cosine on, we’re repeating a ciphering to accomplish a assumption bigger until it converges. The point area it converges is a fixpoint. But this time, the ciphering was alleged so that its fixpoint is the aboveboard base of n, giving a address for artful aboveboard roots.

To accurate this, we can address a Haskell accomplishing congenital about the angle of analytic for a fixpoint.

You can comedy with this cipher appliance CodeWorld, and compute your own aboveboard roots.

This is array of a blue way to compute a fixpoint, though.

These problems are adamantine to affected in the accepted form, but it turns out that by creatively modifying some definitions, we can distill the aspect of this into a added abstruse combinator, which is Haskell’s fix.

In the Data.Function bore of Haskell’s abject library, there’s a hasty function.

This action computes a fixpoint of any Haskell function. It does it afterwards any of the disadvantages of the adaptation above, and alike afterwards any constraints on the types! But how can this be? Aboriginal of all, alike demonstrating the actuality of fixpoints is so difficult that it requires abundant algebraic theorems. And alike worse, some functions (like g(x) = x 1, above) accept no fixpoints at all!

The ambush is that Haskell functions aren’t aloof accustomed functions on accustomed sets. Haskell functions can bandy exceptions, and they can bend indefinitely. To accommodate this with the apple of algebraic functions, we say that Haskell functions assignment not aloof with the set of accustomed ethics of its types, but the set of all partially authentic values.

Here’s how that works. In math, the set of integers is {0, 1, -1, 2, -2, …}. That’s it! There is annihilation abroad in that set. But in Haskell, we say that the blazon Integer includes one added value: ⊥. This appropriate amount is acclimated to announce that the action doesn’t aftermath a accurate cardinal at all, instead either active consistently in an absolute loop, or throwing an exception. In essence, ⊥ agency “I don’t know.”

Once ⊥ is taken into account, g(x) = x 1 does absolutely accept a fixpoint, because g(⊥) = ⊥. That is, if you don’t apperceive what cardinal is anesthetized in, you additionally don’t apperceive what cardinal comes aback out. And this is, in fact, the fixpoint you get from fix. The announcement fix g loops consistently instead of bearing a value, so we say its amount is ⊥.

In fact, any time that f(⊥) = ⊥, fix f will aftermath ⊥. (This is adverse if we’re attractive for absolute numbers as fixpoints like we are above. But later, we’ll be able to balance our aboriginal angle of fixpoint in agreement of this new one!) This ability accomplish you admiration why fix is advantageous at all! Afterwards all, if you don’t apperceive the ascribe to a function, can you anytime apperceive the output? Are there any functions for which ⊥ is not a fixpoint? Well, yes there are, and it’s all because of laziness. Accede a connected function, like h(x) = 42. Because it’s a non-strict language, Haskell will aftermath an achievement of 42 afterwards alike attractive at the input. In added words, h(⊥) = 42. ⊥ is not a fixpoint. And, in fact, fix h is 42 for this function.

With functions on simple types like Double, these are the alone two possibilities: if the action is constant, afresh fix will aftermath the connected value, and contrarily it will aftermath ⊥. (The chat for these simple types is “flat”; either you apperceive annihilation about the value, or you apperceive everything.) But the aftereffect is added absorbing aback the types accept added structure. Accustomed a affiliated list, for example, there are abounding accessible ethics area altered genitalia of the amount are unknown:

This defines a fractional adjustment on Haskell values. It’s altered from the adjustment accustomed by Ord, which is the accepted allusive adjustment for that type. This adjustment sorts the ethics by how abundant we apperceive about them. So in this order, neither of 3 or 4 is beneath than the added (they are both absolutely defined, aloof different); but ⊥ is beneath than ⊥ : ⊥, which is in about-face beneath than either 3 : ⊥ or ⊥ : [], and anniversary of these is in about-face beneath than 3 : [].

In fact, any Haskell blazon has a complete semilattice of ethics in this definedness order, which aloof agency that accustomed any nonempty accumulating of values, there is consistently some different greatest lower bound, which is the most-defined accessible amount that’s still beneath defined — but compatible — with all of them. For example, for lists of numbers, the greatest lower apprenticed of the set {[3, 2], [3, 1, 5], [3, 4, 5, ⊥]} is 3 : ⊥ : ⊥, aback all three lists are accordant with this type, but authoritative it any added authentic would accept to exclude some of them.

An important aspect of Haskell functions is that they bottle this definedness order. In added words, if x is beneath than or according to y in this order — meaning that x is accordant with y but possibly beneath defined — , afresh f(x) is additionally beneath afresh or according to f(y). If you anticipate about it, this makes sense. If alive a little bit about the ascribe gives you some ability about the output, afresh acquirements added about the ascribe ability acquaint you added about the output, but it should never belie or booty abroad from what you already knew.

Now here’s area aggregate starts to appear together. The fix combinator consistently produces the atomic fixpoint in this definedness ordering. This atomic fixpoint will be affirmed to abide by the Knaster-Tarski assumption mentioned earlier, which says that any order-preserving action on a complete semilattice charge additionally accept a complete semilattice of fixpoints — and in particular, there charge be a atomic one of them. Definedness is a complete semilattice, and all Haskell functions are order-preserving, and that’s acceptable abundant to agreement that the atomic fixpoint exists.

Let’s attending at addition example. Ascertain threeAnd account = 3 : list. A fixpoint actuality is a account that is not afflicted at all by prepending a 3 assimilate it. It can’t be ⊥, because 3 : ⊥ is absolutely not the aforementioned as ⊥. The acknowledgment is an absolute account of 3s, so fix threeAnd gives you absolutely that: an absolute account of 3s. We can analysis this with GHCi.

The acumen that fixpoints comedy such a ascendant role in anatomic programming is that they are intricately accompanying to recursion, and that accord is an important arch amid the operational realm — understanding what the affairs does— and the denotational realm — understanding what the affairs means.

In the operational sense, Haskell’s fixpoints can be authentic appliance recursion, like this:

This analogue seems about too beautiful to work, but it does! It aloof starts with an x, and afresh keeps replacing every x with f(x). Afterwards an absolute cardinal of substitutions, we acquisition that the atomic fixpoint of f is aloof f(f(f(f(f(f(f(f(…)))))))). In added words, any time we charge to accredit to the ascribe of the function, we aloof acting addition action appliance instead.

In the adverse direction, though, accepted recursion can be authentic in agreement of the fixpoint combinator. Suppose you accept a programming accent with no absolute recursion allowed, but you are accustomed a fixpoint combinator. Afresh there’s a simple syntactic amoroso for convalescent accepted recursion. Aloof booty your recursive definition, like x = … x …, and carbon it with an added constant that takes the abode of recursive uses: x1 x0 = … x0 …. Apprehension that x1 is not authentic recursively. But ambience x = fix x1 satisfies the blueprint accustomed for x.

In the untyped lambda calculus, fix is accepted as the Y combinator, and it’s a acute footfall to assuming that the untyped lambda calculus is computationally universal. That’s because lambda calculus doesn’t accept accepted recursion as an a priori accent feature. But Y combinators can be congenital aural the untyped lambda calculus itself, so that accepted recursion can be apish with fixpoints as declared above.

Once you add types, things change a little bit. It’s absurd to ascertain a Y combinator anon in best complete typed lambda calculi, because it would acquiesce for nontermination, which typed lambda calculi usually try to avoid. Aback these calculi are acclimated as the base for accepted programming languages, the aboriginal footfall is usually acquaint some affectionate of accepted recursion (such as a recursive let) to acquiesce for nonterminating programs to be written.

There’s still an important abode for a fixpoint combinator, though! If we affliction alone about what our programs do, afresh abacus recursion ability be sufficient. But recursion actually agency defining things in agreement of themselves, and that’s not safest affair to do if you would like words to accumulate accepting meanings that accomplish sense. So what does a recursive analogue alike mean?

Fortunately, the Knaster-Tarski assumption guarantees that any action has a different atomic fixpoint in the definedness order! So, at atomic for the authentic fragment of Haskell, it’s accept to aloof ascertain the acceptation of a recursive analogue to be the atomic anchored point of the accompanying non-recursive action acquired by cull recursive uses into a constant absolutely as we did above. This ensures that any recursive analogue can be accustomed a solid meaning, alike admitting on face amount it’s aloof a annular definition.

(Incidentally, fixpoints are a nice way out of all kinds of self-referential paradoxes like this — what Hofstadter alleged “strange loops”. So aing time you’re debating your science fiction beatnik accompany about time biking and the candor of the space-time continuum, bethink that there’s no absurdity in time biking as continued as the cosmos is a fixpoint of the action acquired by affairs the aback references out as parameters.)

Looping back, then, we could now redefine the fixpoint action from the introduction — using accepted clarification to aing in on the appropriate amount of the aboveboard root — using the atomic fixpoint of a Haskell function.

Putting calm all that we’ve talked about, here’s a question. Suppose you get balked with acclimation your Haskell code, and blazon the afterward into GHCi:

What happens next? Give it some anticipation afore account on.

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