Notions of Stack-manipulating Computation and Relative Monads (Extended Version)

Abstract: Monads provide a simple and concise interface to user-defined computational effects in functional programming languages. This enables equational reasoning about effects, abstraction over monadic interfaces and the development of monad transformer stacks to combine different effects. Compiler implementors and assembly code programmers similarly virtualize effects, and would benefit from similar abstractions if possible. However, the implementation details of effects seem disconnected from the high-level monad interface: at this lower level much of the design is in the layout of the runtime stack, which is not accessible in a high-level programming language. We demonstrate that the monadic interface can be faithfully adapted from high-level functional programming to a lower level setting with explicit stack manipulation. We use a polymorphic call-by-push-value (CBPV) calculus as a setting that captures the essence of stack-manipulation, with a type system that allows programs to define domain-specific stack structures. Within this setting, we show that the existing category-theoretic notion of a relative monad can be used to model the stack-based implementation of computational effects. To demonstrate generality, we adapt a variety of standard monads to relative monads. Additionally, we show that stack-manipulating programs can benefit from a generalization of do-notation we call "monadic blocks" that allow all CBPV code to be reinterpreted to work with an arbitrary relative monad. As an application, we show that all relative monads extend automatically to relative monad transformers, a process which is not automatic for monads in pure languages.

Overview of Notions of Stack-Manipulating Computation and Relative Monads

The paper presents a computational model exploring the intersection of stack manipulation and monadic interfaces through the framework of Call-by-Push-Value (CBPV) calculus, emphasizing the application of relative monads. This approach roots in providing a low-level representation of stack-based computation while preserving the abstraction benefits of monads typically found in high-level functional programming.

Stack-based Computation and Monadic Interfaces

Monads, since the introduction by Moggi and subsequent popularization by Wadler, provide a structured way to handle effects in functional programming, enabling abstraction and composability of effects via monad transformers. The paper notes the apparent disconnect between this high-level usage of monads and their implementation at the low-level of stack manipulation—especially in environments like assembly code or compiler implementations where stack layout is crucial.

The authors propose that relative monads can bridge this gap by acting as an interface between high-level abstractions and low-level virtualized effect implementations. Importantly, they explore the adaptation of monadic interfaces to stack operations typical in languages that are closer to the machine level, such as the CBPV calculus. This calculus is significant as it allows for explicit manipulation of the stack while also having a type system robust enough to define domain-specific stack structures.

Adapting Monads to Stack Manipulations

A prominent feature of the paper is demonstrating how relative monads can capture and adapt the stack-based implementation of computational effects by extending the CBPV framework. Relative monads permit the continuation of existing stack frames through stack manipulation, effectively adapting the high-level monadic constructs for use in low-level computation where stack management is critical.

A highlighted advancement is the extension from monads to relative monad transformers. Unlike monads which do not naturally extend to monad transformers, all relative monads automatically extend to relative transformers in this CBPV-enhanced view. This transformation supports incrementally adding effects in a modular stack-based style, akin to building monadic stacks.

Generalizing Programming Constructs

An innovative outcome of this approach is a generalized do-notation termed “monadic blocks.” These blocks enhance CBPV by mapping all its computational constructs to work seamlessly with any relative monad, thereby broadening the embedability of effectful programming by overloading the execution of CBPV code to work with arbitrary user-defined effects. This overcomes a significant limitation in traditional do-notation by embedding higher-order programming within the stack-manipulation model itself.

Implications and Future Directions

Practically, this model enriches the capability of compiler implementations by offering a systematic approach to handle effects that traditionally do not fit well with stack layouts in low-level code. Theoretically, the work offers a foundation for exploring how relative monads can be utilized to manage computational effects gracefully in environments where direct stack manipulation is requisite.

For future directions, the model suggests potential expansion into verified compilation pipelines that inherently support effects by utilizing CBPV's stack-manipulation abstractions. Additionally, it opens avenues for further exploration into the dual of this framework—leveraging relative comonads to model computational contexts in effectful environments, enriching the broader conversation about how programming paradigms can evolve with increasingly complex computational needs.

In essence, the paper invites a rethinking of how monadic abstractions can be implemented directly in the stack-based computation field using relative monads, presenting a coherent strategy for unifying high-level abstractions with low-level computational models.