Representing layered monads

Abstract

There has already been considerable research on constructing modular, monad-based specifications of computational effects (state, exceptions, nondeterminism, etc.) in programming languages. We present a simple framework in this tradition, based on a Church-style effect-typing system for an ML-like language.The semantics of this language is formally defined by a series of monadic translations, each one expanding away a layer of effects. Such a layered specification is easy to reason about, but its direct implementation (whether by parameterized interpretation or by actual translation) is often prohibitively inefficient.By exploiting deeper semantic properties of monads, however, it is also possible to derive a vastly more efficient implementation: we show that each layer of effects can be uniformly simulated by continuation-passing, and further that multiple such layers can themselves be simulated by a standard semantics for call/cc and mutable state. Thus, even multi-effect programs can be executed in Scheme or SML/NJ at full native speed, generalizing an earlier single-effect result. As an example, we show how a simple resumptionbased semantics of concurrency allows us to directly simulate a shared-state program across all possible dynamic interleavings of execution threads.

The F# Computation Expression Zoo

“…There are multiple ways of composing or layering monads [11,13,3]. Monad transformers are perhaps the most widely known technique.…”

“…Aside from monads, other ways of encoding effectful computations include effect handlers [18] and continuations [3]. Providing syntactic support for these encodings may provide an interesting alternative to our encoding using computation builder.…”

The sequential semantics of producer effect systems

“…Layered Monads Filinski introduced the semantic technique of layered monads [5]. A layering of a monad M , map , unit , join over another monad M, map, unit, join is a natural transformation layer : M • M → M satisfying certain equational properties.…”

Relating Computational Effects by ⊤ ⊤-Lifting

“…Filinski is one of the pioneers in logical relations for monads [4], and developed various techniques to establish relationships between semantics of higher-order languages with effects [3][4][5][6][7]. In [7,Proposition 3.7] three methods to construct logical relations for monads are mentioned: 1) When T is a monad constructed from the standard type constructors, such as state monad and continuation monad, then define a logical relationṪ for T in the same way as T is constructed.…”