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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.
Divergenceleast semantics of amb is Hoare
 Short presentation at the APPSEM II workshop, Frauenchiemsee
, 2005
"... Abstract This note strengthens the hoary observation that McCarthy’s amb is not monotone with respect to the Smyth and Plotkin powerdomains. It shows that there is no least fixpoint semantics for amb that is sensitive to divergence. This paper is concerned with an erratic choice operator MM ′ , and ..."
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Abstract This note strengthens the hoary observation that McCarthy’s amb is not monotone with respect to the Smyth and Plotkin powerdomains. It shows that there is no least fixpoint semantics for amb that is sensitive to divergence. This paper is concerned with an erratic choice operator MM ′ , and an ambiguous choice operator M amb M ′. Recall that MM ′ means: either evaluate M or evaluate M ′. And M amb M ′ means: evaluate both M and M ′ on an arbitrary fair scheduler, and return whatever answer you get first. We defer the study of ambiguous choice until Sect. 2. 1 Erratic Choice Suppose we have a language L containing the following: – a boolean type bool, equipped with constants t and f, and a conditional operator if M then N else N ′ at every type – a natural number type nat, equipped with a constant n for each n ∈ N, and an equality operator N = N ′ – a term d (short for diverge) at every type