Parallel beta reduction is not elementary recursive

@article{Asperti1998ParallelBR,
  title={Parallel beta reduction is not elementary recursive},
  author={Andrea Asperti and Harry G. Mairson},
  journal={Inf. Comput.},
  year={1998},
  volume={170},
  pages={49-80}
}
We analyze the inherent complexity of implementing Lévy's notion of optimal evaluation for the &lambda-calculus, where similar redexes are contracted in one step via so-called parallel ß-reduction. optimal evaluation was finally realized by Lamping, who introduced a beautiful graph reduction technology for sharing evaluation contexts dual to the sharing of values. His pioneering insights have been modified and improved in subsequent implementations of optimal reduction.We prove that the cost of… 
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References

SHOWING 1-10 OF 74 REFERENCES
Paths , Computations and Labels in the . \-calculus *
Cosimo Laneve INRJA Sophia-Antipolia We provide a new characterization of Levy's redex-families in the .\-calculus [11) as suitable paths in the initial term of the derivation. The idea ie that
On global dynamics of optimal graph reduction
TLDR
This global analysis of the finitary dynamics of optimal reduction is the first demonstration that a reasonable implementation-independent cost model for the λ-calculus is in fact realized by Lamping's abstract algorithm.
Optimality and inefficiency: what isn't a cost model of the lambda calculus?
We investigate the computational efficiency of the sharing graphs of Lamping [Lam90], Gonthier, Abadi, and Lévy [GAL92], and Asperti [Asp94], designed to effect so-called optimal evaluation,
What is an Efficient Implementation of the lambda-calculus?
TLDR
It is argued that any implementation of the lambda-calculus must have complexity Omega(nu), i.e. a linear lower bound, and it is shown that implementations based upon Turner Combinators of Hughes Super-combinators have complexities 2Omega(nu, i.
What is an Efficient Implementation of the λ-calculus ?
TLDR
It is argued that any implementation of the λcalculus must have complexity Ω(ν), i.e. a linear lower bound, and it is shown that implementations based upon Turner Combinators or Hughes Super-combinators have complexities 2, i.
An algorithm for optimal lambda calculus reduction
We present an algorithm for lambda expression reduction that avoids any copying that could later cause duplication of work. It is optimal in the sense defined by Lévy. The basis of the algorithm is a
On laziness and optimality in lambda interpreters: tools for specification and analysis
TLDR
It is shown that ACCL has properties not possessed by Curien's original combinators that make it particularly appropriate as the basis for implementation and analysis of a wide range of reduction schemes using shared environments, closures, or λ-terms.
Linear Logic, Comonads and Optimal Reduction
  • A. Asperti
  • Computer Science
    Fundam. Informaticae
  • 1995
TLDR
The paper relates the two “brackets” in [GAL92a] to the two operations associated with the comonad “e” of Linear Logic, which can be then understood as a “local implementation’ of naturality laws.
Linear logic without boxes
TLDR
An implementation of proof nets without boxes is described, which helps in understanding optimal reductions in the lambda -calculus and in the various programming languages inspired by linear logic.
...
...