Functional pearl: every bit counts

We show how the binary encoding and decoding of typed data and typed programs can be understood, programmed, and verified with the help of question-answer games. The encoding of a value is determined by the yes/no answers to a sequence of questions about that value; conversely, decoding is the interpretation of binary data as answers to the same question scheme. We introduce a general framework for writing and verifying game-based codecs. We present games for structured, recursive, polymorphic, and indexed types, building up to a representation of well-typed terms in the simply-typed λ-calculus. The framework makes novel use of isomorphisms between types in the definition of games. The definition of isomorphisms together with additional simple properties make it easy to prove that codecs derived from games never encode two distinct values using the same code, never decode two codes to the same value, and interpret any bit sequence as a valid code for a value or as a prefix of a valid code.