Beta reduction is invariant, indeed

@article{Accattoli2014BetaRI,
  title={Beta reduction is invariant, indeed},
  author={Beniamino Accattoli and Ugo Dal Lago},
  journal={Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)},
  year={2014}
}
  • Beniamino Accattoli, Ugo Dal Lago
  • Published 14 July 2014
  • Mathematics
  • Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
Slot and van Emde Boas' weak invariance thesis states that reasonable machines can simulate each other within a polynomially overhead in time. Is λ-calculus a reasonable machine? Is there a way to measure the computational complexity of a λ-term? This paper presents the first complete positive answer to this long-standing problem. Moreover, our answer is completely machine-independent and based over a standard notion in the theory of λ-calculus: the length of a leftmost-outermost derivation to… 

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Beta reduction is invariant, indeed
TLDR
The main technical contribution of the paper is indeed the definition of useful reductions and the thorough analysis of their properties, and the first complete positive answer to this long-standing problem of λ-calculus.
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TLDR
This paper presents the first complete positive answer to this long-standing problem, completely machine-independent and based over a standard notion in the theory of $\lambda$-calculus: the length of a leftmost-outermost derivation to normal form is an invariant cost model.
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