Lambda - Dropping : Transforming Recursive Equations into Programswith Block

Lambda-lifting a functional program transforms it into a set of recursive equations. We present the symmetric transformation: lambda-dropping. Lambda-dropping a set of recur-sive equations restores block structure and lexical scope. For lack of scope, recursive equations must carry around all the parameters that any of their callees might possibly need. Both lambda-lifting and lambda-dropping thus require one to compute a transitive closure over the call graph: for lambda-lifting: to establish the Def/Use path of each free variable (these free variables are then added as parameters to each of the functions in the call path); for lambda-dropping: to establish the Def/Use path of each parameter (parameters whose use occurs in the same scope as their deenition do not need to be passed along in the call path). Without free variables, a program is scope-insensitive. Its blocks are then free to oat (for lambda-lifting) or to sink (for lambda-dropping) along the vertices of the scope tree. recursive equations block sinking lambda dropping block oating O O parameter dropping block-structured program lambda lifting G G parameter lifting O O We believe lambda-lifting and lambda-dropping are interesting per se, both in principle and in practice, but our prime application is partial evaluation: except for Malmkjjr and rbbk's case study presented at PEPM'95, most poly-variant specializers for procedural programs operate on re-cursive equations. To this end, in a pre-processing phase, they lambda-lift source programs into recursive equations. As a result, residual programs are also expressed as recur-sive equations, often with dozens of parameters, which most compilers do not handle eeciently. Lambda-dropping in a post-processing phase restores their block structure and lexical scope thereby signiicantly reducing both the compile time and the run time of residual programs. source block-structured program lambda-lifting / / source recursive equations partial evaluation residual block-structured program residual recursive equations lambda-dropping o o 1 Introduction and Motivation Block structure and lexical scope stand at the foundation of functional programming, but are they so much in everyday use? Evidence says that they are not. Consider the standard append function deened as a recursive equation: