Monadic Refinements for Relational Cost Analysis

Formal frameworks for cost analysis of programs have been widely studied in the unary setting and, to a limited extent, in the relational setting. However, many of these frameworks focus only on the cost aspect, largely side-lining functional properties that are often a prerequisite for cost analysis, thus leaving many interesting programs out of their purview. In this paper, we show that elegant, simple, expressive proof systems combining cost analysis and functional properties can be built by combining already known ingredients: higher-order refinements and cost monads. Specifically, we derive two syntax-directed proof systems, U-C and R-C, for unary and relational cost analysis, by adding a cost monad to a (syntax-directed) logic of higher-order programs. We study the metatheory of the systems, show that several nontrivial examples can be verified in them, and prove that existing frameworks for cost analysis (RelCost and RAML) can be embedded in them.