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A callbyneed lambdacalculus with locally bottomavoiding choice: Context lemma and correctness of transformations
 MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE
, 2008
"... We present a higherorder callbyneed lambda calculus enriched with constructors, caseexpressions, recursive letrecexpressions, a seqoperator for sequential evaluation and a nondeterministic operator amb that is locally bottomavoiding. We use a smallstep operational semantics in form of a sin ..."
Abstract

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We present a higherorder callbyneed lambda calculus enriched with constructors, caseexpressions, recursive letrecexpressions, a seqoperator for sequential evaluation and a nondeterministic operator amb that is locally bottomavoiding. We use a smallstep operational semantics in form of a singlestep rewriting system that defines a (nondeterministic) normal order reduction. This strategy can be made fair by adding resources for bookkeeping. As equational theory we use contextual equivalence, i.e. terms are equal if plugged into any program context their termination behaviour is the same, where we use a combination of may as well as mustconvergence, which is appropriate for nondeterministic computations. We show that we can drop the fairness condition for equational reasoning, since the valid equations w.r.t. normal order reduction are the same as for fair normal order reduction. We evolve different proof tools for proving correctness of program transformations, in particular, a context lemma for may as well as mustconvergence is proved, which restricts the number of contexts that need to be examined for proving contextual equivalence. In combination with socalled complete sets of commuting and forking diagrams we show that
all the deterministic reduction rules and also some additional transformations preserve contextual equivalence.We also prove a standardisation theorem for fair normal order reduction. The structure of the ordering <= c is also analysed: Ω is not a least element, and <=c already implies contextual equivalence w.r.t. mayconvergence.