Unification Source-Tracking with Application to Diagnosis of Type Inference

Abstract

Prior diagnoses in unification-based type reconstruction systems have either missed information that is relevant, presented irrelevant details, or both. We use a framework based on the Unification Logics of Le~Chenadec to define, derive and simplify proof-based source-tracking for term unification. The objects of source-tracking are proofs in this deduction system, and correspond to path expressions over a unification graph whose labels form a semi-Dyck language of balanced parentheses. Simplification of source-tracking information is implemented as proof normalization in the rewrite system for free groups. Subject-reduction properties guarantee that normalization preserves the semantics of deductions. The presentation of the logic facilitates proof construction by a simple extension to standard unification algorithms.

We apply unification source-tracking to type inference in the Curry-Hindley type system. Programs are represented as systems of syntax equations. Program slices correspond to weakenings of syntax and type equations. A constraint generation function maps weakenings of type equations to weakenings of syntax equations. Source-tracking information is defined in terms of the inverse of this generating function.

Unification is central to many applications of symbolic computation and artificial intelligence, including computer algebra, automated theorem proving, expert systems, and programming language type systems. Source-tracking is a debugging technique based on tracing the execution of a program to identify those subparts that contribute to the result of the execution.