g., Bitcoin script supports adding 32-bit numbers, so we need no binary circuit for that. We could also have larger bit commitments, e.g. it is possible to commit to 32 bits in a single hash. Additionally, scripts can be up to about 4 MB in size. Thus, we can implement substantially more than a single NAND instruction per leaf script. 在简单的NAND电路中表达函数是低效的。通过使用更高级的操作码，程序可以更有效地表达。例如，比特币脚本支持添加32位数字，因此我们无需为此使用二进制电路。我们还可以拥有更大的比特承诺，例如，可以在单个哈希中承诺32位。此外，脚本的大小可以达到约4 MB。因此，我们可以在每个叶子脚本中实现远远超过一个NAND指令。 The model proposed here is limited to two parties. However, it might be possible to have two-way channels, and chain them to form a network similar to Lightning. Exploring the two-party setting might yield interesting possibilities for generalization. For example, we can explore a 1-to-n star topology for the network. Another research question is if we can apply our model to the n-of-n setting and create more sophisticated channel factories. Furthermore, we could combine our system with different off-chain protocols, e.g., the Lightning Network or rollups. 这里提出的模型仅限于两方。然而，可能可以建立双向通道，并将它们链式连接以形成类似于闪电网络的网络。探索双方设置可能会产生一些有趣的泛化可能性。例如，我们可以探索网络的1对n星型拓扑结构。另一个研究问题是我们是否可以将我们的模型应用于n-of-n设置，并创建更复杂的通道工厂。此外，我们还可以将我们的系统与不同的链下协议结合使用，例如闪电网络或rollups。 Other directions of research include cross-application memory, how to make statements about arbitrary data inscribed into the chain, or off-chain programmable circuits, i.e. an off-chain VM. It also might be possible to apply more sophisticated sampling techniques, similar to STARKs, to check a circuit in a single round. 其他研究方向包括跨应用内存、如何对刻在链上的任意数据进行陈述，或链下可编程电路，即链下虚拟机。还有可能应用更复杂的采样技术，类似于STARKs，以在单一轮中检查电路。 The next major milestone is to complete a design and an implementation of a concrete BitVM and also of Tree++, a high-level language to write and debug Bitcoin contracts. 下一个重要的里程碑是完成具体的BitVM设计和实现，以及Tree++，一个用于编写和调试比特币合约的高级语言。 9 结论(Conclusion) Bitcoin is Turing-complete in the sense that encoding fraud proofs in large Taptrees allows to verify the execution of any program. A major constraint of the model outlined here is that it is limited to the two-party setting. Hopefully, this can be generalized in further works. 比特币在编码欺诈证明的大型 Taptrees 中可以验证任何程序的执行，因此在某种意义上是图灵完备的。这里概述的模型的一个主要限制是它仅限于两方设置。希望这可以在后续工作中得到泛化。 致谢(Acknowledgments) Special thanks to Super Testnet and Sam Parker, who always kept refusing to believe that Bitcoin would not be Turing-complete. 特别感谢Super Testnet和Sam Parker，他始终拒绝相信比特币不会是图灵完备的。 参考文献(References) [1] Ethereum Research. Optimistic rollups. https://ethereum.org/en/developers/ docs/scaling/optimistic-rollups/, 2022. [2] Salvatore Ingala. Merkleize all the things. https://lists.linuxfoundation.org/ pipermail/bitcoin-dev/2022-November/021182.html, 2022. [3] Jeremy Rubin. CheckSigFromStack for 5 Byte Values. https://rubin.io/blog/2021/07/02/signing-5-bytes, 2021. 来源：金色财经lg...
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