Oracle Computability and Turing Reducibility in the Calculus of Inductive Constructions

TL;DR

This paper develops synthetic notions of oracle computability and Turing reducibility within the Calculus of Inductive Constructions, enabling machine-checked proofs and advancing the theoretical understanding of computability in constructive type theory.

Contribution

It introduces a synthetic framework for oracle computability in CIC, proving key properties like the structure of Turing reducibility and its relation to semi-decidability, with formalization in Coq.

Findings

Turing reducibility forms an upper semilattice.

Turing reducibility is more expressive than truth-table reducibility.

Semi-decidability relative to an oracle implies Turing reducibility.

Abstract

We develop synthetic notions of oracle computability and Turing reducibility in the Calculus of Inductive Constructions (CIC), the constructive type theory underlying the Coq proof assistant. As usual in synthetic approaches, we employ a definition of oracle computations based on meta-level functions rather than object-level models of computation, relying on the fact that in constructive systems such as CIC all definable functions are computable by construction. Such an approach lends itself well to machine-checked proofs, which we carry out in Coq. There is a tension in finding a good synthetic rendering of the higher-order notion of oracle computability. On the one hand, it has to be informative enough to prove central results, ensuring that all notions are faithfully captured. On the other hand, it has to be restricted enough to benefit from axioms for synthetic computability, which usually concern first-order objects. Drawing inspiration from a definition by Andrej Bauer based on continuous functions in the effective topos, we use a notion of sequential continuity to characterise valid oracle computations. As main technical results, we show that Turing reducibility forms an upper semilattice, transports decidability, and is strictly more expressive than truth-table reducibility, and prove that whenever both a predicate $p$ and its complement are semi-decidable relative to an oracle $q$, then $p$ Turing-reduces to $q$.