Approximation theory for distant Bang calculus
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915969578827776 |
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| author | Chardonnet, Kostia Chouquet, Jules Kerinec, Axel |
| author_facet | Chardonnet, Kostia Chouquet, Jules Kerinec, Axel |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_05199 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Approximation theory for distant Bang calculus Chardonnet, Kostia Chouquet, Jules Kerinec, Axel Logic in Computer Science 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. |
| title | Approximation theory for distant Bang calculus |
| topic | Logic in Computer Science |
| url | https://arxiv.org/abs/2601.05199 |