The CPS/ANF Saga

The CPS transformation dates back to the early 1970’s, where it arose as a technique to represent the control flow of programs in λ-calculus based programming languages as λ-terms in the full (free β) λ-calculus. Toward the end of that decade and afterward, compiler writers found that terms in continuation passing style represented a good intermediate language between their corresponding direct-style (pre-CPS) terms and machine code. Subsequent study of the transformation’s theoretical and pragmatic properties led to various refinements on the core algorithm, as well as several “inverse” transformations to produce direct-style terms from CPS terms. These developments led to the investigation of the “essence” of the CPS transform: a set of reduction axioms on source terms, A. Terms in A-Normal Form correspond formally to CPS terms, enabling compiler writers to use optimizations and code generators written for CPS terms on source terms without first having to do a CPS transform. 1 Reasoning about programs in continuation passing style A. Sabry and M. Felleisen. Reasoning about programs in continuation-passing style. Lisp and Symbolic Computation, 6(3):289–360, 1993 Paraphrasing the introduction, the paper gives an answer to the question: Is there a set of axioms A which may be added to the β-value rule such that two λ-terms, when CPS’d, are full-β-equal if and only if they were equal in the extended calculus. Solving this problem (and indeed, using CPS in a compiler at all) is complicated by redexes generated during the CPS transformation which do not correspond to any in the source terms (administrative redexes). A two-pass CPS translation which also reduces administrative redexes is given. This is then shown to be equivalent to a one-pass compactifying transformation, which is used alongside an “unCPS” transformation. The “inverse” unCPS takes CPS’d terms to canonical representations of equivalence classes of source terms. The equivalence relation unCPS induces on terms is shown to be generated by a finite set of rules (including βv), and these are used as the axioms A.