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Static analysis of different non-strict functional programming languages makes use of set constants like Top, Inf, and Bot denoting all expressions, all lists without a last Nil as tail, and all non-terminating programs, respectively. We use a set language that permits union, constructors and recursive definition of set constants with a greatest fixpoint semantics. This paper proves decidability, in particular EXPTIMEcompleteness, of subset relationship of co-inductively defined sets by using algorithms and results from tree automata. This shows decidability of the test for set inclusion, which is required by certain strictness analysis algorithms in lazy functional programming languages.
This paper proves correctness of Nöcker's method of strictness analysis, implemented in the Clean compiler, 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 did on the correctness of the abstract reduction rules. Our method fully considers the cycle detection rules, which are the main strength of Nöcker's strictness analysis. Our algorithm SAL is a reformulation of Nöcker's strictness analysis algorithm in a higher-order call-by-need lambda-calculus with case, constructors, letrec, and seq, extended by set constants like Top or Inf, denoting sets of expressions. It is also possible to define new set constants by recursive equations with a greatest fixpoint semantics. The operational semantics is a small-step semantics. Equality of expressions is defined by a contextual semantics that observes termination of expressions. Basically, SAL is a non-termination checker. The proof of its correctness and hence of Nöcker's strictness analysis is based mainly on an exact analysis of the lengths of normal order reduction sequences. The main measure being the number of 'essential' reductions in a normal order reduction sequence. Our tools and results provide new insights into call-by-need lambda-calculi, the role of sharing in functional programming languages, and into strictness analysis in general. The correctness result provides a foundation for Nöcker's strictness analysis in Clean, and also for its use in Haskell.