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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 ..."
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Cited by 15 (10 self)
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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.
Correctness of Copy in Calculi with Letrec,
, 2007
"... Abstract. This paper extends the internal frank report 28 as follows: It is shown that for a callbyneed lambda calculus LRCCPλ extending the calculus LRCCλ by por, i.e in a lambdacalculus with letrec, case, constructors, seq and por, copying can be done without restrictions, and also that callby ..."
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Cited by 5 (3 self)
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Abstract. This paper extends the internal frank report 28 as follows: It is shown that for a callbyneed lambda calculus LRCCPλ extending the calculus LRCCλ by por, i.e in a lambdacalculus with letrec, case, constructors, seq and por, copying can be done without restrictions, and also that callbyneed and callbyname strategies are equivalent w.r.t. contextual equivalence. 1
Equivalence of callbyname and callbyneed for lambdacalculi with letrec
, 2007
"... We develop a proof method to show that in a (deterministic) lambda calculus with letrec and equipped with contextual equivalence the callbyname and the callbyneed evaluation are equivalent, and also that the unrestricted copyoperation is correct. Given a letbinding x = t, the copyoperation re ..."
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Cited by 3 (1 self)
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We develop a proof method to show that in a (deterministic) lambda calculus with letrec and equipped with contextual equivalence the callbyname and the callbyneed evaluation are equivalent, and also that the unrestricted copyoperation is correct. Given a letbinding x = t, the copyoperation replaces an occurrence of the variable x by the expression t, regardless of the form of t. This gives an answer to unresolved problems in several papers, it adds a strong method to the tool set for reasoning about contextual equivalence in higherorder calculi with letrec, and it enables a class of transformations that can be used as optimizations. The method can be used in different kind of lambda calculi with cyclic sharing. Probably it can also be used in nondeterministic lambda calculi if the variable x is “deterministic”, i.e., has no interference with nondeterministic executions. The main technical idea is to use a restricted variant of the infinitary lambdacalculus, whose objects are the expressions that are unrolled w.r.t. let, to define the infinite developments as a reduction calculus on the infinite trees and showing a standardization theorem.