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The curiously recurring template pattern (CRTP) is an idiom, originally in C++, in which a class X derives from a class template instantiation using X itself as a template argument. [1] More generally it is known as F-bound polymorphism , and it is a form of F -bounded quantification .
so that FIX G = G (FIX G) = (λn.(1, if n = 0; else n × ((FIX G) (n−1)))) Given a lambda term with first argument representing recursive call (e.g. G here), the fixed-point combinator FIX will return a self-replicating lambda expression representing the recursive function (here, F). The function does not need to be explicitly passed to ...
A class is used to contain the fix function, called fixer. The function to be fixed is contained in a class that inherits from fixer. The fix function accesses the function to be fixed as a virtual function. As for the strict functional definition, fix is explicitly given an extra parameter x, which means that lazy evaluation is not needed.
In the Lambda lifting section, a meta function for first lifting and then converting the resulting lambda expression into recursive equation was described. The Lambda Drop meta function performs the reverse by first converting recursive equations to lambda abstractions, and then dropping the resulting lambda expression, into the smallest scope ...
The variadic template feature of C++ was designed by Douglas Gregor and Jaakko Järvi [1] [2] and was later standardized in C++11. Prior to C++11, templates (classes and functions) could only take a fixed number of arguments, which had to be specified when a template was first declared.
This is done by adding another parameter to the basic recursive function and using this parameter as the function for the recursive call. This creates a higher-order function, and passing this higher function itself allows anonymous recursion within the actual recursive function.
A central property of the lambda calculus is that recursive definitions are not directly available, but can instead be expressed with a fixed point combinator. But unfortunately, the fixpoint combinator cannot be formulated in a typed version of the lambda calculus without having a disastrous effect on the system as outlined below.
A typed lambda calculus is a typed formalism that uses the lambda-symbol to denote anonymous function abstraction.In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a type depends on the calculus considered (see kinds below).