Parametric Church's Thesis: Synthetic Computability without Choice

Abstract

In synthetic computability, pioneered by Richman, Bridges, and Bauer, one develops computability theory without an explicit model of computation. This is enabled by assuming an axiom equivalent to postulating a function $\phi$ to be universal for the space $\mathbb{N}\to\mathbb{N}$ ($\mathsf{CT}_\phi$, a consequence of the constructivist axiom $\mathsf{CT}$), Markov's principle, and at least the axiom of countable choice. Assuming $\mathsf{CT}$ and countable choice invalidates the law of excluded middle, thereby also invalidating classical intuitions prevalent in textbooks on computability. On the other hand, results like Rice's theorem are not provable without a form of choice. In contrast to existing work, we base our investigations in constructive type theory with a separate, impredicative universe of propositions where countable choice does not hold and thus a priori $\mathsf{CT}_{\phi}$ and the law of excluded middle seem to be consistent. We introduce various parametric strengthenings of $\mathsf{CT}_{\phi}$, which are equivalent to assuming $\mathsf{CT}_\phi$ and an $S^m_n$ operator for $\phi$ like in the $S^m_n$ theorem. The strengthened axioms allow developing synthetic computability theory without choice, as demonstrated by elegant synthetic proofs of Rice's theorem. Moreover, they seem to be not in conflict with classical intuitions since they are consequences of the traditional analytic form of $\mathsf{CT}$. Besides explaining the novel axioms and proofs of Rice's theorem we contribute machine-checked proofs of all results in the Coq proof assistant.