Smart Proofs via Smart Contracts: Succinct and Informative Mathematical Derivations via Decentralized Markets

Modern mathematics is built on the idea that a proof should be be translatable into a formal proof, whose validity is an objective question, decidable by a computer. In practice, however, proofs are informal, succinct, and omit numerous uninteresting details: their goal is to share insight among a community of agents. An agent considers a proof valid if they trust that it could (in principle) be expanded into a machine-verifiable proof. A proof’s validity can thus become a subjective matter, possibly leading to a debate; if agents’ incentives are not aligned, it may be hard to reach a consensus. Hence, while the concept of valid proof is well-defined in principle, the process to establish a proof’s validity is itself a complex multi-agent problem. In this paper, we introduce the SPRIG protocol, which allows agents to propose and verify succinct and informative proofs in a decentralized fashion; the trust is established by agents being able to request more details at steps where they feel there could be problems; debates, if they arise, need to isolate specific details of proofs; if they persist, they must go down to machine-level details, where they can be settled automatically. A structure of fees, bounties, and stakes is set to incentivize the agents to act in good faith, i.e. to not publish problematic proofs and to not ask for trivial details. We propose a game-theoretic discussion of SPRIG interactions, illustrating how agents with different types of information interact, leading to a verification tree with an appropriate level of detail, and to the invalidation of problematic proofs, and we discuss resilience against various attacks. We then provide an in-depth treatment of a simplified model, characterize its equilibria and analytically compute the agents’ level of trust. The SPRIG protocol is designed so that it can run fully autonomously as a smart contract on a decentralized blockchain platform, without a need for a central trusted institution. This allows agents to participate anonymously in the verification debate, being incentivized to contribute with their information. The smart contract mediates all the interactions between the agents, and settles debates on the validity of proofs, and guarantees that bounties and stakes are paid as specified by the protocol. SPRIG also allows for a number of other applications, in particular the issuance of bounties for solving open problems, and the creation of derivatives markets, enabling agents to inject more information pertaining to mathematical proofs.