Algebraic Temporal Effects: Temporal Verification of Recursively Typed Higher-Order Programs

We present a general form of temporal effects for recursive types. Temporal effects have been adopted by effect systems to verify both linear-time temporal safety and liveness properties of higher-order programs with recursive functions. A challenge in a generalization to recursive types is that recursive types can easily cause unstructured loops, which obscure the regularity of the infinite behavior of computation and make it harder to statically verify liveness properties. To solve this problem, we introduce temporal effects with a later modality, which enable us to capture the behavior of non-terminating programs by stratifying obscure loops caused by recursive types. While temporal effects in the prior work are based on certain concrete formal forms, such as logical formulas and automata-based lattices, our temporal effects, which we call algebraic temporal effects , are more abstract, axiomatizing temporal effects in an algebraic manner and clarifying the requirements for temporal effects that can reason about programs soundly. We formulate algebraic temporal effects, formalize an effect system built on top of them, and prove two kinds of soundness of the effect system: safety and liveness soundness. We also introduce two instances of algebraic temporal effects: one is temporal regular effects , which are based on ω -regular expressions, and the other is temporal fixpoint effects , which are based on a first-order fixpoint logic. Their usefulness is demonstrated via examples including concurrent and object-oriented programs.