A Theory of Nested Cascading in Directed Logic
Abstract
In directed logic (DL), electronically controlled optical elements serve as photonic gates. Such electro-optical elements are of mixed nature: they have two inputs--electronic and optical--but only one, optical, output. This makes cascading such gates without repeated conversion between optical and electronic representations cumbersome. This problem can be largely overcome using the nested cascading scheme proposed by Shamir and Hardy [Opt. Express, 17, 150 (2007)]. Although promising as a solution to the cascading problem, the Shamir-Hardy scheme has so far been proved or implemented only for a few simplest cases. Here, we develop a general rigorous theory of nested cascading, valid for an arbitrary number of gates. We propose a variant of the algorithm which easily extendable to large number of gates, and rigorously prove its validity. Furthermore, we analyze how nested cascading scales with the size of the corresponding Boolean formula. We show that good (linear) scalability is guaranteed in many important cases, while the average scaling with respect to the corresponding Boolean formulas is only moderately polynomial, with an exponent of approximately 3/2. Yet, the worst-case scaling remains exponential with respect to more general Boolean circuits allowing sharing and reuse of intermediate results.
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BibTeX
@misc{babushkin2026theory,
title = {{A Theory of Nested Cascading in Directed Logic}},
author = {Ihar Babushkin and Oliver Melchert and Ayhan Demircan and Uwe Morgner},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02930},
eprint = {2610.02930},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02930},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Theory of Nested Cascading in Directed Logic
AU - Ihar Babushkin
AU - Oliver Melchert
AU - Ayhan Demircan
AU - Uwe Morgner
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02930
UR - https://arxiv.org/abs/2610.02930
AB - In directed logic (DL), electronically controlled optical elements serve as photonic gates. Such electro-optical elements are of mixed nature: they have two inputs--electronic and optical--but only one, optical, output. This makes cascading such gates without repeated conversion between optical and electronic representations cumbersome. This problem can be largely overcome using the nested cascading scheme proposed by Shamir and Hardy [Opt. Express, 17, 150 (2007)]. Although promising as a solution to the cascading problem, the Shamir-Hardy scheme has so far been proved or implemented only for a few simplest cases. Here, we develop a general rigorous theory of nested cascading, valid for an arbitrary number of gates. We propose a variant of the algorithm which easily extendable to large number of gates, and rigorously prove its validity. Furthermore, we analyze how nested cascading scales with the size of the corresponding Boolean formula. We show that good (linear) scalability is guaranteed in many important cases, while the average scaling with respect to the corresponding Boolean formulas is only moderately polynomial, with an exponent of approximately 3/2. Yet, the worst-case scaling remains exponential with respect to more general Boolean circuits allowing sharing and reuse of intermediate results.
ER -