Patterns for computational effects arising from a monad or a comonad

TL;DR

This paper develops equational logics for reasoning about programming language effects using monads and comonads, providing dual inference patterns that unify effect reasoning and are applicable to various effects like exceptions and state.

Contribution

It introduces two dual inference system patterns for monads and comonads, enabling systematic proof of properties in effectful programming languages.

Findings

Inference system for states is Hilbert-Post complete

Logic can be instantiated with monads or comonads for different effects

Applicable to languages with exceptions and state effects

Abstract

This paper presents equational-based logics for proving first order properties of programming languages involving effects. We propose two dual inference system patterns that can be instanciated with monads or comonads in order to be used for proving properties of different effects. The first pattern provides inference rules which can be interpreted in the Kleisli category of a monad and the coKleisli category of the associated comonad. In a dual way, the second pattern provides inference rules which can be interpreted in the coKleisli category of a comonad and the Kleisli category of the associated monad. The logics combine a 3-tier effect system for terms consisting of pure terms and two other kinds of effects called 'constructors/observers' and 'modifiers', and a 2-tier system for 'up-to-effects' and 'strong' equations. Each pattern provides generic rules for dealing with any monad (respectively comonad), and it can be extended with specific rules for each effect. The paper presents two use cases: a language with exceptions (using the standard monadic semantics), and a language with state (using the less standard comonadic semantics). Finally, we prove that the obtained inference system for states is Hilbert-Post complete.