The content discusses the complexity of normalizing planar lambda-terms, which are a restricted class of linear lambda-terms where the syntax tree with binding edges is planar.
The key points are:
For general linear lambda-terms, the problem of deciding beta-convertibility (i.e., whether two terms have the same normal form) is P-complete.
The authors previously claimed that this problem is also P-complete for planar lambda-terms, but the proposed proof contained a flaw.
The authors outline a new attempt to reduce the Circuit Value Problem (CVP), which is a P-complete problem, to the normalization problem for planar lambda-terms.
The main challenge is finding a planar lambda-term that can "copy" a boolean value, which is crucial for encoding boolean circuits in the planar lambda-calculus. The authors have not been able to find such a term, leading to a gap in their previous reduction attempt.
The authors then describe a new encoding of the Topologically Ordered Circuit Value Problem (TopCVP) using planar lambda-terms representing bit vectors and operations on them, such as negation, conjunction, and disjunction. This is a step towards a potential reduction from CVP to planar normalization.
The content concludes that the complexity of normalizing planar lambda-terms remains an open problem, despite these new efforts.
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by Anup... lúc arxiv.org 04-09-2024
https://arxiv.org/pdf/2404.05276.pdfYêu cầu sâu hơn