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Quantitative Weak Linearisation

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Theoretical Aspects of Computing – ICTAC 2022 (ICTAC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13572))

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Abstract

Weak linearisation was defined years ago through a static characterization of the intuitive notion of virtual redex, based on (legal) paths computed from the (syntactical) term tree. Weak-linear terms impose a linearity condition only on functions that are applied (consumed by reduction) and functions that are not applied (therefore persist in the term along any reduction) can be non-linear. This class of terms was shown to be strongly normalising with deciding typability in polynomial time. We revisit this notion through non-idempotent intersection types (also called quantitative types). By using an effective characterisation of minimal typings, based on the notion of tightness, we are able to distinguish between “consumed” and “persistent” term constructors, which allows us to define an expansion relation, between general \(\lambda \)-terms and weak-linear \(\lambda \)-terms, whilst preserving normal forms by reduction.

First author supported by National Funds through the Portuguese funding agency, FCT - Fundação para a Ciência e a Tecnologia, within project LA/P/0063/2020. Second author partially supported by CNPq Universal 430667/2016-7 grant.

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Notes

  1. 1.

    A notion of reduction used in functional programming languages implementation, which does not evaluate inside values, i.e. under abstractions.

  2. 2.

    Both deterministic and non-deterministic strategies have the same normal forms.

  3. 3.

    Note that \((A ; \{y : [y_1:\sigma _1,\dots ,y_m:\sigma _m]\})+ A_1 + \dots + A_n = A + A_1 + \dots + A_n; \{y : [y_1:\sigma _1,\dots ,y_m:\sigma _m]\}\) since by BVC \(y \notin \texttt{fv}(u)\) thus \(y \notin \texttt{dom}(A_i)\) for each \(1\le i\le n\).

  4. 4.

    \(y \notin \texttt{fv}(t')\) and by BVC \(y \notin \texttt{fv}(u)\) thus \(y \notin \texttt{fv}(t'\{u/x\})\).

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Authors and Affiliations

  1. DCC-FCUP & CRACS, Univ. Porto, Porto, Portugal

    Sandra Alves

  2. INF, Univ. Federal de Goiás, Goiânia, Brazil

    Daniel Ventura

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  1. Sandra Alves
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Correspondence to Sandra Alves .

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Editors and Affiliations

  1. TU München, München, Germany

    Helmut Seidl

  2. Southwest University, Chongqing, China

    Zhiming Liu

  3. NASA Ames Research Center, Moffett Field, CA, USA

    Corina S. Pasareanu

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Alves, S., Ventura, D. (2022). Quantitative Weak Linearisation. In: Seidl, H., Liu, Z., Pasareanu, C.S. (eds) Theoretical Aspects of Computing – ICTAC 2022. ICTAC 2022. Lecture Notes in Computer Science, vol 13572. Springer, Cham. https://doi.org/10.1007/978-3-031-17715-6_7

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  • DOI: https://doi.org/10.1007/978-3-031-17715-6_7

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