Safety of Nöcker's strictness analysis

Abstract

Abstract. This paper proves correctness of Nöcker's method of strictness analysis, implemented for Clean, which is an effective way for strictness analysis in lazy functional languages based on their operational semantics. We improve upon the work of Clark, Hankin and Hunt, which addresses correctness of the abstract reduction rules. Our method also addresses the cycle detection rules, which are the main strength of Nöcker's strictness analysis. We reformulate Nöcker's strictness analysis algorithm in a higherorder lambda-calculus with case, constructors, letrec, and a nondeterministic choice operator ⊕ used as a union operator. Furthermore, the calculus is expressive enough to represent abstract constants like Top or Inf . The operational semantics is a small-step semantics and equality of expressions is defined by a contextual semantics that observes termination of expressions. The correctness of several reductions is proved using a context lemma and complete sets of forking and commuting diagrams. The proof is based mainly on an exact analysis of the lengths of normal order reductions. However, there remains a small gap: Currently, the proof for correctness of strictness analysis requires the conjecture that our behavioral preorder is contained in the contextual preorder. The proof is valid without referring to the conjecture, if no abstract constants are used in the analysis.

Conservative Concurrency in Haskell

“…contextual equivalence) to a usual extended call-by-need letrec-calculus as e.g. the calculi in [28,27].…”

“…This also justifies to use well-understood (also denotational) semantics for the pure subcalculus, for example the free theorems in the presence of seq [8], or results from call-by-need lambda calculi (e.g. [13,27]) for reasoning on pure expressions inside Concurrent Haskell. The proof of the main results appears to be impossible by a direct attack.…”

A Finite Simulation Method in a Non-deterministic Call-by-Need Lambda-Calculus with Letrec, Constructors, and Case

“…The argument is completely analogous to that in [SSS06,SSSS08]: if any of these reductions appears in a reduction context, then it must be a normal order reduction. Since the normal-order reduction is unique for these reductions, the context lemma 3.2 is applicable and implies correctness.…”

“…In Subsection 2.1 we introduce the syntax of L, followed by Subsection 2.2 where we define the normal order reduction for L. Based on the notion of termination we introduce contextual equivalence in Subsection 2.2.2. The calculus is the same as the one considered in [SSSS04] and extension by choice of the one in [SSSS08].…”

“…the appendix of [SSSS08]) using the context lemma and complete sets of diagrams. However, the correctness proof for (case) requires further tools, in particular the extra transformations defined below in the next subsection.…”