Ill-Typed Programs Don't Evaluate

TL;DR

This paper introduces two-sided type systems that enable reasoning about program correctness by ensuring well-typed programs don't go wrong and ill-typed programs don't evaluate, enhancing higher-order program verification.

Contribution

It presents a novel two-sided sequent calculus for types, supporting refined notions of well-typing and ill-typing, and explores internalising negation for improved semantics.

Findings

Supports more precise data-flow typing with constructors and pattern matching

Guarantees well-typed programs don't go wrong and ill-typed programs don't evaluate

Proposes an alternative semantics internalising meta-level negation

Abstract

We introduce two-sided type systems, which are sequent calculi for typing formulas. Two-sided type systems allow for hypothetical reasoning over the typing of compound program expressions, and the refutation of typing formulas. By incorporating a type of all values, these type systems support more refined notions of well-typing and ill-typing, guaranteeing both that well-typed programs don't go wrong and that ill-typed programs don't evaluate - that is, reach a value. This makes two-sided type systems suitable for incorrectness reasoning in higher-order program verification, which we illustrate through an application to precise data-flow typing in a language with constructors and pattern matching. Finally, we investigate the internalisation of the meta-level negation in the system as a complement operator on types. This motivates an alternative semantics for the typing judgement, which guarantees that ill-typed programs don't evaluate, but in which well-typed programs may yet go wrong.