On Encoding LF in a Predicate Logic over Simply-Typed Lambda Terms

TL;DR

This paper critically examines a proposed encoding of the dependently typed lambda calculus LF into hereditary Harrop formulas, revealing its non-faithfulness due to non-injective mappings and ambiguity issues.

Contribution

It identifies fundamental flaws in a prior encoding approach, providing a counterexample and analyzing the reasons for its non-faithfulness.

Findings

The encoding is not faithful due to non-injective mappings.

Counterexample demonstrates ambiguity in inverse transformation.

Implications for related encoding schemes are discussed.

Abstract

Felty and Miller have described what they claim to be a faithful encoding of the dependently typed lambda calculus LF in the logic of hereditary Harrop formulas, a sublogic of an intuitionistic variant of Church's Simple Theory of Types. Their encoding is based roughly on translating object expressions in LF into terms in a simply typed lambda calculus by erasing dependencies in typing and then recapturing the erased dependencies through the use of predicates. Unfortunately, this idea does not quite work. In particular, we provide a counterexample to the claim that the described encoding is faithful. The underlying reason for the falsity of the claim is that the mapping from dependently typed lambda terms to simply typed ones is not one-to-one and hence the inverse transformation is ambiguous. This observation has a broad implication for other related encodings.