Since the creation of Bitcoin 15 years ago, there has been an explosion in the number of permissionless blockchains. Each of these blockchains provides an open ledger that anyone can read from and write to. In this multi-chain world, an important question emerges: how can we build a more secure overlay blockchain by reading from and writing to a given set of blockchains? Drawing an analogy with switching circuits, we approach the problem by defining two basic compositional operations between blockchains, serial and triangular compositions, and use these operations as building blocks to construct general overlay blockchains. Under the partially synchronous setting, we have the following results: 1) the serial composition, between two blockchains, yields an overlay blockchain that is safe if at least one of the two underlay blockchains is safe and that is live if both underlay blockchains are live; 2) the triangular composition between three blockchains, akin to parallel composition of switching circuits, yields an overlay blockchain that is safe if all underlay blockchains are safe and that is live if at least half of them are live; 3) repeated composition of these two basic operations can yield all possible tradeoffs of safety and liveness for an overlay blockchain built on arbitrary number of underlay chains. The results are also extended to the synchronous setting.

翻译：自15年前比特币诞生以来，无许可区块链的数量呈爆炸式增长。每条区块链都提供一个任何人都可读写公开账本。在多链世界中，一个重要问题随之浮现：如何通过读写给定区块链集合来构建更安全的上层区块链？类比开关电路，我们通过定义区块链间的两种基本组合操作（串联组合与三角组合）来研究该问题，并将这些操作作为构建通用上层区块链的基本单元。在部分同步环境下，我们得到以下结果：（1）两条区块链的串联组合可产生上层区块链，当至少一条底层区块链安全时该上层链安全，当两条底层链均存活时该上层链存活；（2）三条区块链的三角组合（类似于开关电路的并联组合）可产生上层区块链，当所有底层链安全时该上层链安全，当至少半数底层链存活时该上层链存活；（3）对这两种基本操作的重复组合，可为基于任意数量底层链构建的上层区块链实现安全性与存活性所有可能的权衡。该结果也可推广至同步环境。