Remarks on the equational theory of non-normalizing pure type systems

Abstract

Pure Type Systems (PTS) come in two flavours: domain-free systems with untyped λ-abstractions (i.e. of the form λx . M); and domain-free systems with typed λ-abstractions (i.e. of the form λx : A . M). Both flavours of systems are related by an erasure function |.| that removes types from λ-abstractions. Preservation of Equational Theory, which states the equational theories of both systems coincide through the erasure function, is a property of functional and normalizing PTSs. In this paper we establish that Preservation of Equational Theory fails for some non-normalizing PTSs, including the PTS with * : * . The gist of our argument is to exhibit a typable expression Y H whose erasure |Y | is a fixpoint combinator, but which is not a fixpoint combinator itself.

Pure type systems with explicit substitutions

“…There are, however, several details to be sorted out, beginning with the formulation of the η rule in the untyped setting. As far as we know, the equivalence is known only for functional, normalising PTSs (Geuvers & Werner, 1994) and for a restricted η-reduction in the setting of domain-free PTSs (Barthe & Coquand, 2006).…”

“…PTSs with explicit substitutions were considered by Bloo (2001); Muñoz (2001) studied one particular PTSs with explicit substitutions and de Bruijn indices. Barthe and Sørensen (2000) introduced domain-free PTSs and studied their meta-theory, which was further extended by Barthe and Coquand (2006). Geuvers (1993) conjectured that the presentation of PTSs with external equality is equivalent to the PTSs with judgemental equality.…”