Lambda lifting is a meta-process that restructures a computer program so that functions are defined independently of each other in a global scope. An individual lift transforms a local function (subroutine) into a global function. It is a two step process, consisting of:
Eliminating free variables in the function by adding parameters. Moving functions from a restricted scope to broader or global scope. The term "lambda lifting" was first introduced by Thomas Johnsson around 1982 and was historically considered as a mechanism for implementing programming languages based on functional programming. It is used in conjunction with other techniques in some modern compilers. Lambda lifting is not the same as closure conversion. It requires all call sites to be adjusted (adding extra arguments (parameters) to calls) and does not introduce a closure for the lifted lambda expression. In contrast, closure conversion does not require call sites to be adjusted but does introduce a closure for the lambda expression mapping free variables to values. The technique may be used on individual functions, in code refactoring, to make a function usable outside the scope in which it was written. Lambda lifts may also be repeated, to transform the program. Repeated lifts may be used to convert a program written in lambda calculus into a set of recursive functions, without lambdas. This demonstrates the equivalence of programs written in lambda calculus and programs written as functions. 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). Lambda lifting can be performed with time complexity linear in the size of the lifted program, which is at most O ( n 2 ) {\displaystyle O(n^{2})} . In the untyped lambda calculus, where the basic types are functions, lifting may change the result of beta reduction of a lambda expression. The resulting functions will have the same meaning, in a mathematical sense, but are not regarded as the same function in the untyped lambda calculus. See also intensional versus extensional equality. The reverse operation to lambda lifting is lambda dropping. 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. However it makes a function harder to re-use. A dropped function is tied to its context, and can only be used in a different context if it is first lifted.
Algorithm The following algorithm is one way to lambda-lift an arbitrary program in a language which doesn't support closures as first-class objects:
Rename the functions so that each function has a unique name. Replace each free variable with an additional argument to the enclosing function, and pass that argument to every use of the function. Replace every local function definition that has no free variables with an identical global function. Repeat steps 2 and 3 until all free variables and local functions are eliminated. 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.
Example The following OCaml program computes the sum of the integers from 1 to 100:
(The let rec declares sum as a function that may call itself.) The function f, which adds sum's argument to the sum of the numbers less than the argument, is a local function. Within the definition of f, n is a free variable. 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 versus closures Lambda lifting and closure are both methods for implementing block structured programs. Lifting implements block structure by eliminating it. All functions are lifted to the global level. Closure conversion provides a "closure" which links the current frame to other frames. Closure conversion takes less compile time. Recursive functions, and block structured programs, with or without lifting, may be implemented using a stack based implementation, which is simple and efficient. However a stack frame based implementation must be strict (eager). The stack frame based implementation requires that the life of functions be last-in, first-out (LIFO). That is, the most recent function to start its calculation must be the first to end. Some functional languages (such as Haskell) are implemented using lazy evaluation, which delays calculation until the value is needed. The lazy implementation strategy gives flexibility to the programmer. Lazy evaluation requires delaying the call to a function until a request is made to the value calculated by the function. One implementation is to record a reference to a "frame" of data describing the calculation, in place of the value. Later when the value is required, the frame is used to calculate the value, just in time for when it is needed. The calculated value then replaces the reference. The "frame" is similar to a stack frame, the difference being that it is not stored on the stack. Lazy evaluation requires that all the data required for the calculation be saved in the frame. If the function is "lifted", then the frame needs only record the function pointer, and the parameters to the function. 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.
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