Full Abstraction for the Resource Lambda Calculus with Tests, through Taylor Expansion

We study the semantics of a resource-sensitive extension of the lambda calculus in a canonical reflexive object of a category of sets and relations, a relational version of Scott’s original model of the pure lambda calculus. This calculus is related to Boudol’s resource calculus and is derived from Ehrhard and Regnier’s differential extension of Linear Logic and of the lambda calculus. We extend it with new constructions, to be understood as implementing a very simple exception mechanism, and with a “must” parallel composition. These new operations allow to associate a context of this calculus with any point of the model and to prove full abstraction for the finite sub-calculus where ordinary lambda calculus application is not allowed. The result is then extended to the full calculus by means of a Taylor Expansion formula. As an intermediate result we prove that the exception mechanism is not essential in the finite sub-calculus.

💡 Research Summary

The paper investigates a resource‑sensitive extension of the λ‑calculus (the Resource Lambda Calculus, RLC) within a canonical relational model – a reflexive object in the category of sets and relations that mirrors Scott’s original model for the pure λ‑calculus. Starting from Ehrhard and Regnier’s differential linear logic and its differential λ‑calculus, the authors adapt the framework to incorporate explicit resource management (duplication and erasure) while preserving the usual λ‑abstraction and variable binding.

Two new constructs are introduced: a very simple exception mechanism and a “must” parallel composition. The exception is modeled by a distinguished element ⊥ that forces immediate termination of a computation; the must operator ⊗ requires both of its sub‑computations to succeed for the whole to succeed, thereby encoding a strict form of parallelism. These operators are not part of the original Boudol resource calculus but are added to enable the definition of observation contexts that correspond directly to points of the relational model.

The authors first restrict the language to a finite sub‑calculus that excludes ordinary λ‑application. This fragment contains only variables, λ‑abstractions, the resource primitives, the exception, and the must operator. In this setting every term can be interpreted as a finite relational element, and there is a one‑to‑one correspondence between terms and points of the model. For each model point p a context C(p) is constructed; plugging a term M into C(p) yields a closed term whose evaluation either terminates with a distinguished success value or diverges to ⊥. The key theorem shows that two terms are observationally equivalent (i.e., they behave identically in every C(p)) if and only if they denote the same point in the relational model. This establishes full abstraction for the finite fragment.

An intermediate result demonstrates that the exception mechanism is not essential for full abstraction in the finite fragment: the same proof goes through when the exception construct is removed, indicating that exceptions serve only as a convenient syntactic tool for building observation contexts.

To lift the result to the full RLC, which does contain ordinary λ‑application, the paper employs a Taylor expansion technique. Every term P is expanded into an infinite formal series

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