Linear Call-by-Push-Value Language

Updated 29 October 2025

LCBPV is a computational model that extends CBPV by integrating linear logic to enforce precise resource management and a clear distinction between values and computations.
It utilizes advanced type systems and denotational semantics to derive recurrence relations, providing rigorous analysis of computational effects and resource usage.
LCBPV bridges functional and imperative paradigms, fostering innovations in resource-aware programming and enabling robust simulation of complex effects.

The Linear Call-by-Push-Value (LCBPV) Language is a computational model extending Paul Blain Levy’s Call-by-Push-Value paradigm to incorporate linear types, leveraging the structure and semantics of linear logic. This article explores the core principles, historical context, methodologies, semantic frameworks, and implications of LCBPV, demonstrating its role as a bridge between functional and imperative programming while supporting advanced type systems and computational effects.

1. Overview and Core Principles

Linear Call-by-Push-Value (LCBPV) extends Levy’s Call-by-Push-Value (CBPV) to include linear logic principles, highlighting the separation between computations that consume resources and values that are reusable. CBPV decomposes computation into distinct stages, utilizing value types and computation types for clear semantic representation.

Core Structure: LCBPV utilizes value types to describe data that can be duplicated or discarded and computation types to represent stateful or effect-producing computations.
Linear Logic Influence: By integrating linear logic, LCBPV mandates that resources are linearly consumed unless explicitly duplicated using linear logic's control operators such as the ! (bang) modality, which denotes duplicability.

2. Theoretical Foundations

The expansion of CBPV into LCBPV takes inspiration from linear logic, enhancing the conventional CBPV model to allow for more precise resource management.

Type System: Incorporates resource-sensitive typing disciplines that accurately track resource usage.
Operational semantics: LCBPV provides a fine-grained semantic framework to rigorously model computational effects and resource management.

3. Applications and Methodologies

Denotational Semantics and Recurrence Extraction: LCBPV provides a robust framework to derive recurrence relations for approximating program complexity, as seen in "Recurrence Extraction for Functional Programs through Call-by-Push-Value" (Kavvos et al., 2019). This approach advocates for the use of domain theory and denotational semantics to define operations on CBPV terms, aiding computational cost analysis.

Domain Theory: Utilized for denotational semantics, modeling the runtime characteristics of higher-order functions.
Recurrence Extraction: Supports the automatic derivation of cost recurrences from program syntax.

4. Connections to Linear Logic

Incorporating Girard’s linear logic, LCBPV’s semantic model accommodates resource-bound scenarios efficiently.

Bang Calculus: As outlined in works such as "The Bang Calculus and the Two Girard's Translations" (Guerrieri et al., 2019), LCBPV utilizes Girard’s translations to unify call-by-name and call-by-value strategies within a single syntactical framework.

5. Advanced Computational Effects

LCBPV effectively models complex computational features, enabling sophisticated effect simulations and concurrent processes.

Categorical Semantics: Leveraging enriched adjunctions and internal fibrations, as highlighted in (Amorim et al., 20 May 2025), LCBPV is particularly advantageous for simulating effects within a structured categorical context.
Effect Handling: Supports the integration of a wide range of computational effects through the state-passing and linear usage of state techniques (Møgelberg et al., 2014).

6. Implications and Future Directions

LCBPV serves as a powerful tool for further exploration in resource-aware programming paradigms and advanced type systems. The framework’s separation of computation from value provides extensive flexibility and modularity, positioning LCBPV at the intersection of functional and imperative programming paradigms.

Research and Development: Continued exploration might involve extending LCBPV to support richer LLMs such as linear session types and probabilistic modeling (Ehrhard et al., 2016).
Potential Innovations: As indicated in (Ângelo et al., 8 Apr 2024), LCBPV’s structured handling of resources may lead to advancements in metaprogramming, enabling greater flexibility in program execution contexts and allowing for secure and efficient code execution.

7. Concluding Remarks

The essentially modular nature of Linear Call-by-Push-Value allows it to serve as a foundational framework in complex computational environments, effectively bridging distinct paradigms and facilitating comprehensive resource management. LCBPV’s rich theoretical underpinnings provide a versatile platform for developing efficient, scalable, and effectual programming languages. By maintaining a clear semantic delineation of values and computations, LCBPV advances the potentials for innovations in both pure research and practical language design, illustrating its significance in the field of computational logic and programming language theory.