### Non-Confluence of the Perturbative λ-Calculus

Over the last month, I have been trying to prove that the reduction relation defined in my MFPS paper “A Simply Typed Lambda-Calculus of Forward Automatic Differentiation” is confluent. Alas, this is not true. A counterexample is provided by the term $\mathsf{T}\,(\lambda x.\;(\lambda y.\;f x y) x)$, where $f$ is a free variable. There are two reduction sequences that produce two different normal forms:

$\displaystyle \begin{array}{cl} & \lambda x.\;\iota_1\,((\pi_1\,((\mathsf{T}\,f) x)) (\pi_2\,x)) \\ {} + & \lambda x.\;\iota_1\,(\pi_1\,((\mathsf{T}\,(f (\pi_2\,x))) x)) \\ {} + & \lambda x.\;\iota_2 ((f (\pi_2\,x)) (\pi_2\,x)), \\ \neq & \lambda x.\;\iota_1\,((\pi_1\,((\mathsf{T}\,f) x)) (\pi_2\,x)) \\ {} + & \lambda x.\;\iota_1\,(\pi_1\,((\mathsf{T}\,(f (\pi_2\,x))) (\iota_2\,(\pi_2\,x)))) \\ {} + & \lambda x.\;\iota_1\,(\pi_1\,((\mathsf{T}\,(f (\pi_2\,x))) x)) \\ {} + & \lambda x.\;\iota_1\,((\pi_1\,((\mathsf{T}\,f) (\iota_2\,(\pi_2\,x)))) (\pi_2\,x)) \\ {} + & \lambda x.\;\iota_2\,((f (\pi_2\,x)) (\pi_2\,x)). \end{array}$

The latter contains terms of the form $\pi_1\,((\mathsf{T}\,M) (\iota_2\,N))$, which are semantically zero but don’t reduce. The reduction sequences have been discovered with the help of Redex, which proved to be an extremely valuable tool. An encoding of the perturbative λ-calculus in Redex is available here.