Cost-analysis: How do monads and comonads differ?

Monads are a de facto standard for the type-based analysis of impure aspects of programs, such as runtime cost [9, 5]. Recently, the logical dual of a monad, the comonad, has also been used for the cost analysis of programs, in conjunction with a linear type system [6, 8]. The logical duality of monads and comonads extends to cost analysis: In monadic type systems, costs are (side) effects, whereas in comonadic type systems, costs are coeffects. However, it is not clear whether these two methods of cost analysis are related and, if so, how. Are they equally expressive? Are they equally well-suited for cost analysis with all reduction strategies? Are there translations from type systems with effects to type systems with coeffects and viceversa? The goal of this work-in-progress paper is to explore some of these questions in a simple context — the simply typed lambda-calculus (STLC). As we show, even this simple context is already quite interesting technically and it suffices to bring out several key points.