Effect Systems Revisited—Control-Flow Algebra and Semantics

Abstract

Effect systems were originally conceived as an inference-based program analysis to capture program behaviour—as a set of (representations of) effects. Two orthogonal developments have since happened. First, motivated by static analysis, effects were generalised to values in an algebra, to better model control flow (e.g. for may/must analyses and concurrency). Second, motivated by semantic questions, the syntactic notion of set- (or semilattice-) based effect system was linked to the semantic notion of monads and more recently to graded monads which give a more precise semantic account of effects.

We give a lightweight tutorial explanation of the concepts involved in these two threads and then unify them via the notion of an effect-directed semantics for a control-flow algebra of effects. For the case of effectful programming with sequencing, alternation and parallelism—illustrated with music—we identify a form of graded joinads as the appropriate structure for unifying effect analysis and semantics.

A Issues Surrounding Monadic Strength

A monadic semantics for an effectful simply-typed $\lambda$-calculus requires that monads are strong. This captures the idea (implicit in the category of sets, but not in all categories) that a free variable may be captured by an outer $\lambda$-binding. Strong monads have an additional operation: $\mathsf {str}_{A,B} : A \times \mathsf {T}B \rightarrow \mathsf {T}(A \times B)$ satisfying various axioms (see [23, 24]) which amount to saying that the effects encoded in the result of $\mathsf {str}$ are the effects encoded by the second argument.

The monadic semantics for (bind) in Monad (shown in Sect. 2.3), omitting the type subscripts on $\mathsf {extend}$ and $\mathsf {str}$, is then:

The $\mathsf {str}$ operation turns an environment $\gamma : {\llbracket {\varGamma }\rrbracket }$ and a result $\mathsf {T}{\llbracket {\tau }\rrbracket }$ into $\mathsf {T}({\llbracket {\varGamma }\rrbracket } \times {\llbracket {\tau }\rrbracket }$) for composition with $\mathsf {extend} \; {\llbracket {\varGamma , x \!:\! \tau \vdash _{\!M}\ldots }\rrbracket } : \mathsf {T}({\llbracket {\varGamma }\rrbracket } \times {\llbracket {\tau }\rrbracket }) \rightarrow \mathsf {T}{\llbracket {\tau '}\rrbracket }$.

Graded monads can be similarly strong with operation $\mathsf {str}^F_{A,B} : A \times \mathsf {T}_F B \rightarrow \mathsf {T}_F (A \times B)$, satisfying analogous axioms to the usual strong monad axioms [16]. The graded semantics is then the analogous one to the above.

We might consider adding an effect operation $\mathsf {S} : \mathcal {F} \rightarrow \mathcal {F}$ corresponding to use of strength in the semantics e.g., $\mathsf {str}^F_{A,B} : A \times \mathsf {T}_F B \rightarrow \mathsf {T}_{\mathsf {S} F} (A \times B)$. However, an axiom of (non-graded) strong monads is that for all $x \in A, y \in \mathsf {T}B$ then $\mathsf {map} \, \mathsf {fst} \, (\mathsf {str}_{A,B} (x, y)) = y$ which for graded strength would imply that $\mathsf {S} \, F = F$. We accordingly exclude $\mathsf {S}$ from the effect algebra since it is necessarily identity.