Existence of a planar copy combinator for the planar boolean encoding

Determine whether there exists a planar linear untyped λ-term copy such that for all boolean terms t encoded as planar linear λ-terms (false = λk. λf. k f (λx. x) and true = λk. λf. k (λx. x) f) and for all functions f, the β-reduction of copy f t yields f t t. The λ-term copy must be planar in the sense that its derivation in the linear λ-calculus does not use the exchange (exc) rule.

Background

The paper investigates whether planarity lowers the complexity of normalizing linear λ-terms, aiming to reduce the Topologically Ordered Circuit Value Problem (TopCVP) to β-convertibility for planar λ-terms. Mairson’s reduction for general linear λ-terms relies on a duplicating combinator (akin to the W combinator) to reuse boolean values, but his boolean encoding is non-planar.

To adapt Mairson’s approach to the planar setting, the authors use a planar encoding of booleans and require a duplicator that respects the planarity constraints (no use of the exchange rule). They can define a related term copy' with the reversed argument order (copy' t f =β f t t), which does not suffice in the planar context for their reduction strategy, highlighting a key obstacle.

References

However, to transpose Mairson's methodology for encoding boolean circuits to the planar linear setting, we would need a planar λ-term copy such that (similarly to the W combinator in Curry's BCKW) ∀ t∈{true,false}, copy f t =β f t t We have not been able to find such a term; we did manage to define a planar λ-term copy' that satisfies copy' t f =β f t t, but this is significantly different in a planar setting. Hence the gap in our previous attempt at reducing CVP to β-convertibility of planar λ-terms.

— On the complexity of normalization for the planar $λ$-calculus  (2404.05276 - Das et al., 2024) in Section 3 (Planar booleans do not suffice)