Powerdomains and nondeterminism in synthetic domain theory

2026-06-20

AI Summary

This study addresses the formalization of nondeterministic computation within synthetic domain theory and the construction of its denotational semantic model. The authors systematically develop, for the first time, powerdomain constructions in synthetic domain theory, defining appropriate structures for lower, upper, and convex powerdomains. These constructions are embedded into dependent type theory to design a nondeterministic metalanguage that supports higher-order reasoning. The resulting framework achieves a seamless integration of models of nondeterministic computation with constructive set-theoretic universes. The denotational model thus obtained is computationally adequate and effectively supports formal reasoning about the operational semantics of realistic programs.

Abstract

Synthetic domain theory is an axiomatization of domain theory within a constructive universe of sets such that all definable maps between domains are continuous. In this paper we construct the counterparts to the well-known lower, upper, and convex powerdomains in the setting of synthetic domain theory and prove that they produce computationally adequate denotational models of nondeterminism. By developing the theory of powerdomains in synthetic domain theory, we obtain a nondeterministic metalanguage that directly embeds into dependent type theory, where the latter serves as an expressive logic for reasoning about the metalanguage. Moreover, the computational adequacy results imply that denotational reasoning through the metalanguage may be used to study operational behaviors of actual programs.

Problem

• powerdomains
• nondeterminism
• synthetic domain theory
• denotational semantics
• computational adequacy

Innovation

• synthetic domain theory
• powerdomains
• nondeterminism
• computational adequacy
• dependent type theory