Abstract
\(\lambda \)-calculi come with no fixed evaluation strategy. Different strategies may then be considered, and it is important that they satisfy some abstract rewriting property, such as factorization or normalization theorems. In this paper we provide simple proof techniques for these theorems. Our starting point is a revisitation of Takahashi’s technique to prove factorization for head reduction. Our technique is both simpler and more powerful, as it works in cases where Takahashi’s does not. We then pair factorization with two other abstract properties, defining essential systems, and show that normalization follows. Concretely, we apply the technique to four case studies, two classic ones, head and the leftmost-outermost reductions, and two less classic ones, non-deterministic weak call-by-value and least-level reductions.
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Notes
- 1.
The terminology at work in the literature on \(\lambda \)-calculus and the rewriting terminology often clash: the former calls parallel \(\beta \) reduction what the latter calls multi-step \(\beta \) reduction—parallel reduction in rewriting is something else.
- 2.
It can be obtained indirectly, as a corollary of standardization, proved by Takahashi [28] using the concrete structure of terms. Thus the proof is not of an abstract nature.
- 3.
In rewriting theory, a full reduction for \(\rightarrow _{}\) is called a reduction strategy for \(\rightarrow _{}\). We prefer not to use the term strategy because it has different meaning in the \(\lambda \)-calculus, where it is a deterministic, not necessarily full, reduction for \(\rightarrow _{}\).
- 4.
The difference between factorization and its terminal case is relevant for normalization: van Oostrom and Toyama [22, footnote 8] give an example of normalizing full reduction for a rewriting system in which factorization fails but terminal factorization holds.
- 5.
“\(t\) has a head normal form” is the usual formulation for “\(t\rightarrow _{\beta }^* s\) for some \(s\) that is head normal”. We prefer the latter to avoid the ambiguity of the former about the reduction leading from \(t\) to one of its head normal forms (\(\rightarrow _{\beta }^*\) or \(\rightarrow _{h}^*\)?).
- 6.
Namely, if \(s_1 \; _{\beta }{\Leftarrow }\ t\Rightarrow _{\beta }s_2\) then there exists \(u\) such that \(s_1 \Rightarrow _{\beta }u\; _{\beta }{\Leftarrow }\ s_2\).
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Acknowledgments
This work has been partially funded by the ANR JCJC grant COCA HOLA (ANR-16-CE40-004-01) and by the EPSRC grant EP/R029121/1 “Typed Lambda-Calculi with Sharing and Unsharing”.
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Accattoli, B., Faggian, C., Guerrieri, G. (2019). Factorization and Normalization, Essentially. In: Lin, A. (eds) Programming Languages and Systems. APLAS 2019. Lecture Notes in Computer Science(), vol 11893. Springer, Cham. https://doi.org/10.1007/978-3-030-34175-6_9
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