Initial Algebras of Domains via Quotient Inductive-Inductive Types

Domain theory has been developed as a mathematical theory of computation and to give a denotational semantics to programming languages. It helps us to fix the meaning of language concepts, to understand how programs behave and to reason about programs. At the same time it serves as a great theory to model various algebraic effects such as non-determinism, partial functions, side effects and numerous other forms of computation. In the present paper, we present a general framework to construct algebraic effects in domain theory, where our domains are DCPOs: directed complete partial orders. We first describe so called DCPO algebras for a signature, where the signature specifies the operations on the DCPO and the inequational theory they obey. This provides a method to represent various algebraic effects, like partiality. We then show that initial DCPO algebras exist by defining them as so called Quotient Inductive-Inductive Types (QIITs), known from homotopy type theory. A quotient inductive-inductive type allows one to simultaneously define an inductive type and an inductive relation on that type, together with equations on the type. We illustrate our approach by showing that several well-known constructions of DCPOs fit our framework: coalesced sums, smash products and free DCPOs (partiality and power domains). Our work makes use of various features of homotopy type theory and is formalized in Cubical Agda.

💡 Research Summary

The paper presents a novel, predicative construction of initial algebras for directed complete partial orders (DCPOs) by leveraging Quotient Inductive‑Inductive Types (QIITs) from homotopy type theory (HoTT). The authors first introduce the notion of a “DCPO signature”, which consists of a collection of operation symbols together with an inequational theory specifying the ordering constraints that these operations must satisfy. This abstraction captures a wide range of algebraic effects—partiality, nondeterminism, state, exceptions, etc.—in a uniform way: a DCPO algebra for a given signature is simply a DCPO equipped with the prescribed operations obeying the stipulated inequalities.

The central technical contribution is the use of QIITs to construct, for any signature, an initial DCPO algebra. A QIIT simultaneously defines an inductive type (the carrier of the domain) and an inductively defined relation (the order ⊑) on that type, together with path constructors that encode the required inequational axioms. By allowing constructors of the carrier and of the relation to refer to each other, QIITs can express the Scott‑continuity conditions that every directed family must have a least upper bound. The authors give a concrete QIIT specification: point constructors introduce elements (e.g., a bottom element ⊥, operation symbols applied to existing elements), while path constructors impose the ordering axioms (e.g., ⊥ ⊑ a, monotonicity of operations, associativity/commutativity/idempotence for union in a power domain). Crucially, the path constructors are written as inequalities rather than equalities, reflecting the order‑theoretic nature of DCPOs.

To prove initiality, the paper exploits the elimination principle of QIITs: any algebra satisfying the signature’s operations and inequations admits a unique homomorphism from the QIIT‑defined algebra. This homomorphism preserves the order and the operations, establishing that the QIIT construction yields the free (initial) DCPO algebra for the signature. The approach avoids impredicative constructions such as power‑set based completions, which would otherwise require large universes or the axiom of choice. Instead, the construction stays within predicative type theory, with universe levels made explicit in the formalisation.

The authors demonstrate the expressive power of their framework by re‑deriving several classic domain constructions:

• Coalesced sums – modelling a binary choice with a distinguished bottom.
• Smash products – forming a product domain where the bottom element propagates.
• Free DCPOs for partiality – the partiality monad, previously built via QIITs by Altenkirch et al., is recovered as a special case.
• Power domains – the Plotkin power domain is obtained by adding a binary union operation together with the usual lattice axioms (associativity, commutativity, idempotence) expressed as order inequalities.

Each example is presented as a concrete instance of a signature and the corresponding QIIT, showing that the same underlying machinery suffices for a broad spectrum of effects.

The paper also discusses alternative constructions of DCPOs, namely presentations (preorders with a covering relation) and rounded‑ideal completions. While presentations can also yield free DCPOs, they typically live in a higher universe and rely on impredicative reasoning. The QIIT approach, by contrast, yields a direct, constructive definition of the domain together with its order, and the authors formalise all definitions and theorems in Cubical Agda. The Agda code is provided as an artifact, demonstrating that the QIIT specifications are not only theoretically sound but also practically implementable.

In terms of foundations, the work assumes HoTT with function extensionality but does not require the univalence axiom. The reliance on QIITs is justified by recent developments showing that they can be interpreted in cubical models, making the construction compatible with existing proof assistants.

Overall, the paper makes three significant contributions:

1. A uniform signature‑based framework for describing algebraic effects in domain theory.
2. A predicative construction of initial DCPO algebras using QIITs, avoiding impredicative power‑set methods.
3. A comprehensive formalisation in Cubical Agda, covering classic domain constructions and demonstrating the practicality of the approach.

These results open the door to integrating domain‑theoretic semantics more tightly with modern type‑theoretic proof assistants, facilitating verified semantics for effectful programming languages within a predicative, constructive setting.