# Undecidability of D_<: and Its Decidable Fragments

Dependent Object Types (DOT) is a calculus with path dependent types, intersection types, and object self-references, which serves as the core calculus of Scala 3. Although the calculus has been proven sound, it remains open whether type checking in DOT is decidable. In this paper, we establish undecidability proofs of type checking and subtyping of D_<:, a syntactic subset of DOT. It turns out that even for D_<:, undecidability is surprisingly difficult to show, as evidenced by counterexamples for past attempts. To prove undecidability, we discover an equivalent definition of the D_<: subtyping rules in normal form. Besides being easier to reason about, this definition makes the phenomenon of bad bounds explicit as a single inference rule. After removing this rule, we discover two decidable fragments of D_<: subtyping and identify algorithms to decide them. We prove soundness and completeness of the algorithms with respect to the fragments, and we prove that the algorithms terminate. Our proofs are mechanized in a combination of Coq and Agda.

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