  Higher inductive-inductive types (HIITs) generalize inductive types of
dependent type theories in two ways. On the one hand they allow the
simultaneous definition of multiple sorts that can be indexed over each other.
On the other hand they support equality constructors, thus generalizing higher
inductive types of homotopy type theory. Examples that make use of both
features are the Cauchy real numbers and the well-typed syntax of type theory
where conversion rules are given as equality constructors. In this paper we
propose a general definition of HIITs using a small type theory, named the
theory of signatures. A context in this theory encodes a HIIT by listing the
constructors. We also compute notions of induction and recursion for HIITs, by
using variants of syntactic logical relation translations. Building full
categorical semantics and constructing initial algebras is left for future
work. The theory of HIIT signatures was formalised in Agda together with the
syntactic translations. We also provide a Haskell implementation, which takes
signatures as input and outputs translation results as valid Agda code.
