Approximation theory for distant Bang calculus

Approximation semantics capture the observable behaviour of λ-terms, with Böhm Trees and Taylor Expansion standing as two central paradigms. Although conceptually different, these notions are related via the Commutation Theorem, which links the Taylor expansion of a term to that of its Böhm tree. These notions are well understood in Call-by-Name λ-calculus and have been more recently introduced in Call-by-Value settings. Since these two evaluation strategies traditionally require separate theories, a natural next step is to seek a unified setting for approximation semantics. The Bang-calculus offers exactly such a framework, subsuming both CbN and CbV through linear-logic translations while providing robust rewriting properties. However, its approximation semantics is yet to be fully developed. In this work, we develop the approximation semantics for dBang, the Bang-calculus with explicit substitutions and distant reductions. We define Böhm trees and Taylor expansion within dBang and establish their fundamental properties. Our results subsume and generalize Call-By-Name and Call-By-Value through their translations into Bang, offering a single framework that uniformly captures infinitary and resource-sensitive semantics across evaluation strategies.

💡 Research Summary

The paper develops a unified approximation semantics for the distant Bang calculus (dBang), an extension of the λ‑calculus equipped with explicit substitutions and “bang”/dereliction operators that encode delayed computation and its re‑activation. The authors first recall the syntax of dBang, its three reduction relations (surface →!s, full →! and internal →i), and the confluence and standardisation results that underpin its operational behaviour. They then introduce the notion of “meaningful” terms, a generalisation of solvability (for Call‑by‑Name) and scrutability (for Call‑by‑Value), showing that meaningfulness can be characterised by intersection types and is preserved under reduction.

The core contribution is the construction of two approximation frameworks within dBang: Böhm trees and Taylor expansion. To define Taylor expansion, the authors present a resource calculus δ‑Bang, whose terms are linear resources (variables, applications, λ‑abstractions, dereliction, explicit substitutions, and finite multisets of resources). Reduction in δ‑Bang is non‑deterministic but strongly normalising because each reduction strictly decreases the size of resource bags. They then define an approximation relation “m ◁! M” indicating that a resource term m is a multilinear approximation of a dBang term M. This relation is shown to be compatible with list and surface contexts.

Taylor expansion of a dBang term M is the set T(M) of all resource terms m such that m ◁! M. The Taylor normal form TNF(M) is obtained by taking the normal forms of all such m under the full δ‑Bang reduction and collecting them. Examples illustrate that the divergent term Ω has an empty Taylor normal form, while a call‑by‑name fixed‑point combinator Yₙ x yields an infinite family of normal forms built from the variable x and its resource bags.

Böhm approximants are defined by iterating surface reductions and unfolding “!”‑boxes, yielding an infinitary tree that captures the observable head‑normal‑form structure of a term. The authors prove a commutation theorem (Theorem 45) stating that normalising the Taylor expansion of a term yields exactly the Taylor expansion of its Böhm tree, thereby extending the classic Commutation Theorem from pure Call‑by‑Name and Call‑by‑Value λ‑calculi to the unified dBang setting.

Finally, the paper shows that the established semantics subsume the traditional CbN and CbV cases. Using Girard’s translations of intuitionistic implication into linear logic, any λ‑term can be encoded into dBang. The authors prove that the Böhm tree of a term in CbN (resp. CbV) coincides with the Böhm tree of its dBang encoding (Theorem 56), and similarly for Taylor expansions (Lemmas 51 and Corollary 52). Consequently, solvable CbN terms correspond exactly to meaningful dBang terms, and scrutable CbV terms correspond to meaningful dBang terms as well.

In summary, the work provides a comprehensive, resource‑sensitive infinitary semantics for dBang that unifies the approximation theories of Call‑by‑Name and Call‑by‑Value. It demonstrates that a single linear‑logic‑based calculus can host both evaluation strategies without loss of expressive power, offering a solid foundation for further investigations into typed extensions, connections with proof‑nets, and practical implementations of unified approximation semantics.