# Inference in typed lambda calculus theories

I'd like to do automated inference, say solving word problems or reducing to normal form, in an equational theory of the typed lambda calculus (with product and unit types). Equivalently, in category-theoretic language: I'd like to do inference in a cartesian closed category presented by a finite set of generators and relations. I would be interested in any references on this topic.

I understand that inference in this setting is generally undecidable. However, I wonder whether there is any literature that either gives useful conditions under which exact inference is possible or proposes heuristic algorithms for approximate inference.

• The Seven Virtues of Simple Type Theory by William M. Farmer is a very good introduction, if you haven't read it or something similar yet. The issue of undecidability is handled there by using "general models" (Henkin semantics) instead of "standard models". Section 6 "Provability" gives a proof system for SST (which has the same consistency strength as bounded Zermelo set theory aka Mac Lane set theory). – Thomas Klimpel May 20 '17 at 10:48

At least the problem of whether 2 terms are equal modulo the theory of Cartesian Closed Categories (or $\beta\eta$ conversion) is decidable, because (in part) of the normalization property.