An Introduction to Logical Relations

Logical relations (LR) have been around for many years, and today they are used in many formal results. However, it can be difficult to LR beginners to find a good place to start to learn. Papers often use highly specialized LRs that use the latest advances of the technique which makes it impossible to make a proper presentation within the page limit. This note is a good starting point for beginners that want to learn about LRs. Almost no prerequisite knowledge is assumed, and the note starts from the very basics. The note covers the following: LRs for proving normalization and type safety of simply typed lambda calculus, relational substitutions for reasoning about universal and existential types, step-indexing for reasoning about recursive types, and worlds for reasoning about references.