Lambda lifting is a meta-process that restructures a computer program so that functions are defined independently of each other in a global scope.
It is a two step process, consisting of; The term "lambda lifting" was first introduced by Thomas Johnsson around 1982 and was historically considered as a mechanism for implementing functional programming languages.
[1] However it does not demonstrate the soundness of lambda calculus for deduction, as the eta reduction used in lambda lifting is the step that introduces cardinality problems into the lambda calculus, because it removes the value from the variable, without first checking that there is only one value that satisfies the conditions on the variable (see Curry's paradox).
[3] Lambda dropping may make the compilation of programs quicker for the compiler, and may also increase the efficiency of the resulting program, by reducing the number of parameters, and reducing the size of stack frames.
The following algorithm is one way to lambda-lift an arbitrary program in a language which doesn't support closures as first-class objects: If the language has closures as first-class objects that can be passed as arguments or returned from other functions, the closure will need to be represented by a data structure that captures the bindings of the free variables.
Start by converting the free variable to a parameter: Next, lift f into a global function: The following is the same example, this time written in JavaScript: Lambda lifting and closure are both methods for implementing block structured programs.
The stack frame based implementation requires that the life of functions be last-in, first-out (LIFO).
Some functional languages (such as Haskell) are implemented using lazy evaluation, which delays calculation until the value is needed.
One implementation is to record a reference to a "frame" of data describing the calculation, in place of the value.
Some modern languages use garbage collection in place of stack based allocation to manage the life of variables.
In a managed, garbage collected environment, a closure records references to the frames from which values may be obtained.
In contrast a lifted function has parameters for each value needed in the calculation.
Also, block structured programming languages like ALGOL and Pascal are similar in that they too allow the local definition of a function for use in a restricted scope.
This is because let expressions allow mutual recursion, which is, in a sense, more lifted than is supported in lambda calculus.
Lambda calculus does not support mutual recursion and only one function may be defined at the outermost global scope.
Conversion rules which describe translation without lifting are given in the Let expression article.
Such lifts may also be repeated, until the expression has no lambda abstractions, in order to transform the program.
is a meta function that returns the set of variables used in E. See de-lambda in Conversion from lambda to let expressions.
For example, The de-let meta function may then be used to convert the result back into lambda calculus.
The second, treats lambda abstractions which are applied to a parameter as defining a function.
This definition does not recognize a lambda abstractions with a parameter as defining a function.
This definition recognizes a lambda abstraction with an actual parameter as defining a function.
Reducing the number of parameters makes functions easier to comprehend.
Lambda lifting added parameters that were necessary so that a function can be moved out of its context.
In dropping, this process is reversed, and extra parameters that contain variables that are free may be removed.
In this case the parameter that is dropped is replaced by the expression in the body of the function definition.
The result of dropping the parameter is, For the main example, The definition of drop-params-tran is, where, For each abstraction that defines a function, build the information required to make decisions on dropping names.
Also receive any actual parameter list from the body of the expression, and return it as the actual parameter list from this expression Variable - A call to a function.
drop-formal removes formal parameters, based on the contents of the drop lists.
Its parameters are, drop-formal is defined as, Which can be explained as, Starting with the function definition of the Y-combinator, Which gives back the Y combinator,