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A Constructive Proof of Dependent Choice, Compatible with Classical Logic
, 2012
"... Abstract—MartinLöf’s type theory has strong existential elimination (dependent sum type) that allows to prove the full axiom of choice. However the theory is intuitionistic. We give a condition on strong existential elimination that makes it computationally compatible with classical logic. With thi ..."
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Abstract—MartinLöf’s type theory has strong existential elimination (dependent sum type) that allows to prove the full axiom of choice. However the theory is intuitionistic. We give a condition on strong existential elimination that makes it computationally compatible with classical logic. With this restriction, we lose the full axiom of choice but, thanks to a lazilyevaluated coinductive representation of quantification, we are still able to constructively prove the axiom of countable choice, the axiom of dependent choice, and a form of bar induction in ways that make each of them computationally compatible with classical logic. KeywordsDependent choice; classical logic; constructive logic; strong existential I.
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"... Classical callbyneed sequent calculi: The unity of semantic artifacts ..."
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Author manuscript, published in "FLOPS 2012 11th International Symposium on Functional and Logic Programming (2012)" DOI: 10.1007/9783642298226 Classical callbyneed sequent calculi: The unity of semantic artifacts
, 2012
"... Abstract. We systematically derive a classical callbyneed sequent calculus, which does not require an unbounded search for the standard redex, by using the unity of semantic artifacts proposed by Danvy et al. The calculus serves as an intermediate step toward the generation of an environmentbased ..."
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Abstract. We systematically derive a classical callbyneed sequent calculus, which does not require an unbounded search for the standard redex, by using the unity of semantic artifacts proposed by Danvy et al. The calculus serves as an intermediate step toward the generation of an environmentbased abstract machine. The resulting abstract machine is contextfree, so that each step is parametric in all but one component. The contextfree machine elegantly leads to an environmentbased CPS transformation. This transformation is observationally di erent from a natural classical extension of the transformation of Okasaki et al., due to duplication of unevaluated bindings.
A Synthetic Operational Account of CallbyNeed Evaluation
"... We present the first operational account of call by need that connects syntactic theory and implementation practice. Syntactic theory: the storeless operational semantics using syntax rewriting to account for demanddriven computation and for caching intermediate results. Implementational practic ..."
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We present the first operational account of call by need that connects syntactic theory and implementation practice. Syntactic theory: the storeless operational semantics using syntax rewriting to account for demanddriven computation and for caching intermediate results. Implementational practice: the storebased operational technique using memothunks to implement demanddriven computation and to cache intermediate results for subsequent sharing. The implementational practice was initiated by Landin and Wadsworth and is prevalent today to implement lazy programming languages such as Haskell. The syntactic theory was initiated by Ariola, Felleisen, Maraist, Odersky and Wadler and is prevalent today to reason equationally about lazy programs, on par with Barendregt et al.’s term graphs. Nobody knows, however, how the theory of call by need compares to the practice of call by need: all that is