On Jumps, Interactions, and Intersection Types

2026-06-25

AI Summary

This work investigates the precise correspondence between abstract machine execution steps and type systems to establish a sound computational complexity model. To this end, it introduces the Parameterized Jumping Abstract Machine (PaJAM), which unifies existing abstract machines such as JAM, PAM, and IAM, and couples it with a non-idempotent intersection type system. For the first time, a tight correspondence is revealed between typing derivations in this system and PaJAM execution steps. The main contribution lies in proving that, for any fixed finite backtracking depth, the number of PaJAM execution steps is polynomially related to the number of weak head β-reduction steps in the λ-calculus, thereby providing a rigorous theoretical foundation for PaJAM as a reasonable cost model of computation.

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 $\textit{et al}$. 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 $\textit{polynomial}$ in the number of weak head $β$ steps, giving rise to a $\textit{reasonable}$ cost model, for each $\textit{finite}$ bound on the backtracking depth.

Problem

• Jumping Abstract Machine
• Intersection Types
• Backtracking Depth
• Cost Model
• Lambda Calculus

Innovation

• Parametric Jumping Abstract Machine
• non-idempotent intersection types
• reasonable cost model
• backtracking depth
• polynomial complexity