Fixed point combinator

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Fixed point combinator

In mathematics and computer science in general, a fixed point of a function is a value that is mapped to itself by the function. In the classical untyped lambda calculusevery function has a fixed point. A particular implementation of fix is Curry's paradoxical combinator Yrepresented by. In functional programmingthe Y combinator can be used to formally define recursive functions in a programming language that does not support recursion. This combinator may be used in implementing Curry's paradox.

The heart of Curry's paradox is that untyped lambda calculus is unsound as a deductive system, and the Y combinator demonstrates that by allowing an anonymous expression to represent zero, or even many values.

This is inconsistent in mathematical logic. Applied to a function with one variable the Y combinator usually does not terminate.

fixed point combinator

More interesting results are obtained by applying the Y combinator to functions of two or more variables. The second variable may be used as a counter, or index.

The resulting function behaves like a while or a for loop in an imperative language. Used in this way the Y combinator implements simple recursion.

In the lambda calculus it is not possible to refer to the definition of a function in a function body. Recursion may only be achieved by passing in a function as a parameter. The Y combinator demonstrates this style of programming. The Y combinator is an implementation of a fixed-point combinator in lambda calculus. Fixed-point combinators may also be easily defined in other functional and imperative languages.

The implementation in lambda calculus is more difficult due to limitations in lambda calculus. Fixed-point combinators may be applied to a range of different functions, but normally will not terminate unless there is an extra parameter.

When the function to be fixed refers to its parameter, another call to the function is invoked, so the calculation never gets started. Instead, the extra parameter is used to trigger the start of the calculation.

The type of the fixed point is the return type of the function being fixed. This may be a real or a function or any other type. In the untyped lambda calculus, the function to apply the fixed-point combinator to may be expressed using an encoding, like Church encoding. In this case particular lambda terms which define functions are considered as values. These different approaches affect how a mathematician and a programmer may regard a fixed-point combinator.

A lambda calculus mathematician may see the Y combinator applied to a function as being an expression satisfying the fixed-point equation, and therefore a solution.

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In contrast, a person only wanting to apply a fixed-point combinator to some general programming task may see it only as a means of implementing recursion. Every expression has one value. This is true in general mathematics and it must be true in lambda calculus. This means that in lambda calculus, applying a fixed-point combinator to a function gives you an expression whose value is the fixed point of the function. However, this is a value in the lambda calculus domainit may not correspond to any value in the domain of the function, so in a practical sense it is not necessarily a fixed point of the function, and only in the lambda calculus domain is it a fixed point of the equation.

Division of Signed numbers may be implemented in the Church encoding, so f may be represented by a lambda term. This equation has no solution in the real numbers.In strict functional programming and the lambda calculusfunctions lambda expressions don't have state and are only allowed to refer to arguments of enclosing functions.

This rules out the usual definition of a recursive function wherein a function is associated with the state of a variable and this variable's state is used in the body of the function.

The Y combinator is itself a stateless function that, when applied to another stateless function, returns a recursive version of the function. The Y combinator is the simplest of the class of such functions, called fixed-point combinators. Define the stateless Y combinator and use it to compute factorials and Fibonacci numbers from other stateless functions or lambda expressions. AppleScript is not particularly "functional" friendly.

It can, however, support the Y combinator. AppleScript does not have anonymous functions, but it does have anonymous objects. The code below implements the latter with the former using a handler i. Unfortunately, an anonymous object can only be created in its own statement 'script' Thus, we have to apply Y to the automatic 'result' variable that holds the value of the previous statement.

The identifier used for Y uses "pipe quoting" to make it obviously distinct from the y used inside the definition. BlitzMax doesn't support anonymous functions or classes, so everything needs to be explicitly named.

Like many other statically typed languages, this involves a recursive type, and like other strict languages, it is the Z-combinator instead. The combinator here is expressed entirely as a lambda expression and is a static property of the generic YCombinator class. Both it and the RecursiveFunc type thus "inherit" the type parameters of the containing class—there effectively exists a separate specialized copy of both for each generic instantiation of YCombinator.

See Higher-order functions C or the documentation for more information. Alternatively, with a non-generic holder class note that Fix is now a method, as properties cannot be generic :. To compare differences in language and runtime instead of in approaches to the task, the following are translations of solutions from other languages.

Two versions of each translation are provided, one seeking to resemble the original as closely as possible, and another that is identical in program control flow but syntactically closer to idiomatic C.

Since C does not have local classes, RecursiveFunc and y1 are declared in a class of their own. Moving the type parameters to the class also prevents type parameter inference. Since Java uses interfaces and C uses delegates, which are the only type that the C compiler will coerce lambda expressions to, this code declares a Functions class for providing a means of converting CLR delegates to objects that implement the Function and RecursiveFunction interfaces.

For demonstrative purposes, to completely avoid using CLR delegates, lambda expressions can be replaced with explicit types that implement the functional interfaces. Closures are thus implemented by replacing all usages of the original local variable with a field of the type that represents the lambda expression; this process, called "hoisting" is actually how variable capturing is implemented by the C compiler for more information, see this Microsoft blog post.

To conclude this chain of decreasing reliance on language features, here is above code translated to C 1. The largest change is the replacement of the generic interfaces with the results of manually substituting their type parameters.

fixed point combinator

A shorter version, taking advantage of generic lambdas. Known to work with GCC 5. Compile with. Another version which is disallowed because a function passes itself, which is also a kind of recursion:. Using recursion, this does not satisfy the task requirements, but is included here for illustrative purposes:. The translation is not literal; e.

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Dhall is not a turing complete language, so there's no way to implement the real Y combinator. This is similar to the Haskell solution below, but uses a strict fixed-point combinator since Elm lacks lazy evaluation.The code below demonstrates all the different ways to express the fully polymorphic fix-point combinator in OCaml: from banal to advanced.

fixed point combinator

Our approach is explicitly based on self-application. Most of the code focuses on the emulation of equi- recursive types via iso-recursive types. We also explore equi-recursive types, explicit ones and those inherent in delimited continuations underlying mutations or exceptions. Along the way we describe two lesser-known expressions of the fix-point combinator: via mutations see FixReferenceA in the code and exceptions. Some of the ML techniques, such as using iso-recursive data types to define fixeasily transfer to Haskell.

The side-effecting approaches e. Surprisingly, ST reference cells let us write the pure fixif we use the lazy ST monad. Type classes and type families offer a Haskell-specific way to express the fix-point combinator. In the literature a polyvariadic fix-point combinator is often presented using " For example. We start by recalling how ordinary general recursion is represented via the ordinary fix-point combinator such as Y.

A recursive function like fact :. We now generalize to mutual recursion, on the familiar example of determining if a natural number is even or odd:. We now derive a mutual fix-point combinator with the better interface and efficiency. We wish the fix-point combinator took a list of functions in the open recursion style, returning the list of functions with their knots tied-up. Ideally, the running example should look like. This combinator has two problems: divergence and inefficiency.

The former is easy to fix with the appropriate eta-expansion. However, not only List. Ideally, the list of clauses should be handled as an abstract collection, using only operations like fold or map rather than random-access List.

It would be easy then to generalize to other collections, such as tuples. We calculate. The result is immediately usable in Haskell. In OCaml, we have to introduce a few eta-expansions to prevent premature looping. After inlining the definition of the ordinary fix-point combinator and simplifying, we obtain the following expression for the polyvariadic fix-point combinator.

It is notably simpler than the polyvariadic combinators by Queinnec and Goldberg.In mathematics and computer science in general, a fixed point of a function is a value that is mapped to itself by the function.

Fixed-point combinator

In the classical untyped lambda calculusevery function has a fixed point. A particular implementation of fix is Curry's paradoxical combinator Yrepresented by. In functional programmingthe Y combinator can be used to formally define recursive functions in a programming language that does not support recursion. This combinator may be used in implementing Curry's paradox.

The heart of Curry's paradox is that untyped lambda calculus is unsound as a deductive system, and the Y combinator demonstrates that by allowing an anonymous expression to represent zero, or even many values.

This is inconsistent in mathematical logic. Applied to a function with one variable the Y combinator usually does not terminate. More interesting results are obtained by applying the Y combinator to functions of two or more variables.

The second variable may be used as a counter, or index. The resulting function behaves like a while or a for loop in an imperative language. Used in this way the Y combinator implements simple recursion. In the lambda calculus it is not possible to refer to the definition of a function in a function body. Recursion may only be achieved by passing in a function as a parameter. The Y combinator demonstrates this style of programming. The Y combinator is an implementation of a fixed-point combinator in lambda calculus.

Fixed-point combinators may also be easily defined in other functional and imperative languages. The implementation in lambda calculus is more difficult due to limitations in lambda calculus.

Fixed-point combinators may be applied to a range of different functions, but normally will not terminate unless there is an extra parameter. When the function to be fixed refers to its parameter, another call to the function is invoked, so the calculation never gets started. Instead, the extra parameter is used to trigger the start of the calculation. The type of the fixed point is the return type of the function being fixed.

This may be a real or a function or any other type. In the untyped lambda calculus, the function to apply the fixed-point combinator to may be expressed using an encoding, like Church encoding. In this case particular lambda terms which define functions are considered as values.

Alternately, a function may be considered as a lambda term defined purely in lambda calculus. These different approaches affect how a mathematician and a programmer may regard a fixed-point combinator. A lambda calculus mathematician may see the Y combinator applied to a function as being an expression satisfying the fixed-point equation, and therefore a solution.

In contrast, a person only wanting to apply a fixed-point combinator to some general programming task may see it only as a means of implementing recursion. Every expression has one value. This is true in general mathematics and it must be true in lambda calculus. This means that in lambda calculus, applying a fixed-point combinator to a function gives you an expression whose value is the fixed point of the function.Take function Math.

Sqrt 1. Similarly, the fixed point combinator Y is defined as if Y f is the fixed point of f:. The following Y combinator is an implementation of fixed point combinator, discovered by Haskell Curry:. So Y can be used to implement recursion. As fore mentioned, in lambda calculus, a function cannot directly apply it self in its body.

Take the factorial function as example, the factorial of n is defined recursively:. However, in lambda calculus the above definition is illegal, because the self reference does not work anonymously:.

Now with the power of Y combinator, the recursion can be implemented, but still in the anonymous way. If the above function is called FactorialHelper, then the Factorial function can be implemented as:.

Here Y FactorialHelper can be substituted by Factorial, according to the definition. So FactorialHelper is called with Factorial and n, exactly as expected. The normal order Y combinator does not work with applicative order reduction.

In applicative order, here FactorialHelper is applied with Y FactorialHelperso the right most argument Y FactorialHelper should be reduced first, which leads to infinite reduction:. The normal order Y combinator only works with normal order.

In normal order, here FactorialHelper is applied with Y FactorialHelperso the left most function FactorialHelper should be reduced first:. So the Y f infinite reduction is blocked in in normal order reduction.

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First, Y f is reduced to f Y fthen the next reduction is to reduce leftmost expression f, not the the rightmost Y f. In the above example Y FactorialHelper n:. Y combinator is easy to implement in C. Y can be viewed as accepting helper function and returns f.

Calling above Factorial always throws StackOverflowException, because in C executes in applicative order. When Factorial is called, it calls normal order Y in applicative order, which causes infinite execution.

The above Y combinator does not work in C. When reducing Y f in applicative order, the self application in expression f g g leads to infinite reduction, which need to be blocked. So the applicative order fixed point combinator is:. Still take factorial as example:. In CZ combinator can be implemented in the same pattern. The types of the elements in above lambda expression are the same as in Y combinator, and x is of type T.

Another recursion example is Fibonacci number. The nth Fibonacci number is defined recursively:.

Many faces of the fixed-point combinator

Again, the above recursive definition is illegal in lambda calculus, because the self reference does not work anonymously:. Previously, in the Church numeral arithmeticthe following illegal DivideBy with self reference was temporarily used:.

Finally, with Z, an legal DivideBy in lambda calculus can be defined, following the same helper function pattern:. Is it possible to round a square from the second digit after the dot? I tried to do a calculation that tells people how much the service will cost. I could not get to the circle of numbers. Maybe there are more ideas? Toggle navigation Microsoft Dixin's Blog.An introduction to fixed-point combinators and lambda calculus with real-world JavaScript examples showing their power and beauty.

Fixed-point combinators are higher-order functions such that:. This expands as follows:. This expansion is recursion. With the use of combinators we can define self-referencing anonymous functions avoiding the use of variables.

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The U combinator is the most simple fixed-point combinator. In practice we want our recursions to halt so we can calculate something. With currying we can introduce a halting condition.

fixed point combinator

Haskell B. Curry defined the Y combinator as follows. The Y combinator could be used even in this form but we can simplify it. The reduced form reveals that this is a fixed point combinator:. The Y combinator works only in lazy languages such as Haskell. In non-lazy languages like JavaScript the intuitive implementation expands until stack overflow or never halts in case of tail call optimization.

This way we can avoid automatic expansion. The difference to the previous implementation is that there is no need to pass g to git is already bound.

There is an extra v argument and an extra function application which is not present in the case of Y combinator definition.

An extra v parameter and function application to the Y combinator gives us the Z combinator. Because of the extra parameter v we do not need to construct a lazy recursion in JavaScript. Define the sum function with the Z combinator:. We can see that there is no extra laziness implemented because the function application is lazy in itself.

Let us consider the JavaScript implementations. Here are some real-world examples using fixed point combinators. We would like users to accept terms. Ask them until they accept it.

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It can be done without naming a function for the recursion.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. Most programmers don't know much at all about combinators, if they've even heard about them. If you're ready for a long read, Mike Vanier has a great explanation. Long story short, it allows you to implement recursion in a language that doesn't necessarily support it natively.

A Y-combinator is a "functional" a function that operates on other functions that enables recursion, when you can't refer to the function from within itself. In computer-science theory, it generalizes recursionabstracting its implementation, and thereby separating it from the actual work of the function in question. The benefit of not needing a compile-time name for the recursive function is sort of a bonus. This is applicable in languages that support lambda functions.

The expression -based nature of lambdas usually means that they cannot refer to themselves by name. And working around this by way of declaring the variable, refering to it, then assigning the lambda to it, to complete the self-reference loop, is brittle. The lambda variable can be copied, and the original variable re-assigned, which breaks the self-reference. Y-combinators are cumbersome to implement, and often to use, in static-typed languages which procedural languages often arebecause usually typing restrictions require the number of arguments for the function in question to be known at compile time.

This means that a y-combinator must be written for any argument count that one needs to use. Using a Y-combinator involves an "unusual" way of constructing a recursive function. First you must write your function as a piece of code that calls a pre-existing function, rather than itself:. Then you turn that into a function that takes a function to call, and returns a function that does so. This is called a functional, because it takes one function, and performs an operation with it that results in another function.

Now you have a function that takes a function, and returns another function that sort of looks like a factorial, but instead of calling itself, it calls the argument passed into the outer function.

How do you make this the factorial? Pass the inner function to itself. The Y-Combinator does that, by being a function with a permanent name, which can introduce the recursion.

Rather than the factorial calling itself, what happens is that the factorial calls the factorial generator returned by the recursive call to Y-Combinator.


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