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DOI: 10.4230/LIPIcs.MFCS.2026.64

On Jumps, Interactions, and Intersection Types

Authors: Stefano Catozi, Ugo Dal Lago, and Gabriele Vanoni

Published in: LIPIcs, Volume 386, 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)

Abstract

The Jumping Abstract Machine (JAM), an evaluation mechanism for the λ-calculus, was introduced by Danos and Regnier as an optimization of the Interaction Abstract Machine (IAM), itself an operational counterpart to Girard’s Geometry of Interaction and Abramsky et al. game semantics. Moreover, the JAM is isomorphic to the Pointer Abstract Machine (PAM), the syntactical counterpart of Hyland and Ong’s game semantics. We study a generalization of the JAM, that we call the Parametric Jumping Abstract Machine (PaJAM) and show that there is a tight correspondence between the PaJAM and non-idempotent intersection types: given a normalizing term t, the number of steps taken by the PaJAM when evaluating t can be extracted from its non-idempotent intersection type derivation. Remarkably, fixing the backtracking depth of the PaJAM, one can easily recover both the JAM/PAM, when the depth is constrained to be zero, and the IAM, when it is instead unconstrained. Exploiting type-theoretic machinery, we analyze the complexity of the PaJAM, showing that it is polynomial in the number of weak head β steps, giving rise to a reasonable cost model, for each finite bound on the backtracking depth.

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Stefano Catozi, Ugo Dal Lago, and Gabriele Vanoni. On Jumps, Interactions, and Intersection Types. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 64:1-64:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{catozi_et_al:LIPIcs.MFCS.2026.64,
  author =	{Catozi, Stefano and Dal Lago, Ugo and Vanoni, Gabriele},
  title =	{{On Jumps, Interactions, and Intersection Types}},
  booktitle =	{51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
  pages =	{64:1--64:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-442-0},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{386},
  editor =	{Kouck\'{y}, Michal and Petrișan, Daniela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.64},
  URN =		{urn:nbn:de:0030-drops-274468},
  doi =		{10.4230/LIPIcs.MFCS.2026.64},
  annote =	{Keywords: lambda-calculus, geometry of interaction, intersection types, abstract machines}
}

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On Jumps, Interactions, and Intersection Types

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