The canonical problem explored by computer scientists in designing these protocols, the Byzantine Generals Problem, is itself an exercise about liars in government.
This article focuses on technologically ensuring the veracity and execution of those steps of legal procedure which are capable of such enhancement (formalizeable or objective aspects which I call "dry", in contrast to the many inherently subjective and non-syntactical "wet" aspects of the law): securing chains of evidence, securing chains of command, securely recording and publicizing the ownership and transfer of property, and so on.
A set of ideas I have for procedural law, or "government" broadly defined, is that many of its dry steps might be based on Byzantine fault tolerance protocols along with cryptographic protocols that form tamper-evident structures such as unforgeable chains of evidence. I describe some of these and related protocols further here, here, and here, but I will describe the basic idea of Byzantine fault tolerance here.
The basic idea of a Byzantine fault tolerant protocol is that it is a highly distributed peer-to-peer protocol robust from a certain fraction or less of its participants lying about information originally observed or created by one or a small subset of them. The fraction varies based on various assumptions of the model, but common figures are 1/3 and 1/2 for information originating from one node assuming that node is truthful. If the fraction required for successful collusive lying is not achieved (and such an attack requires either informed negotiations occurring before this protocol step or negotiating the collusion in a single step, the latter possible to avoid by assuming fraud if messaging is abnormally delayed), the liars are detected and can be excluded from future participation in the network. In a less formal sense, Byzantine fault tolerant protocols are simply distributed, peer-to-peer networks with dense communications (in the least efficient but most secure versions, every node sends every bit of information to ever other node) in order to protect against minorities of colluding liars, and to detect and exclude any liars who have not reached the threshold of collusion and thus can be excluded from the network.
Byzantine fault tolerance protocols are not as strong as cryptographic protocols. They can also suffer from the sock puppet problem (also called in some literature the "Sybil" problem), in which one or a few liars control a much larger and sufficient fraction of network nodes, if the participants are not strongly identified as unique individuals. Thus where it is possible, we should augment these dense peer-to-peer protocols with or use instead stronger cryptographic schemes such as hashing and a variety of cryptographic signatures. If the Byzantine protocol is overcome by collusive liars in a way that cannot be detected before sufficient collusion occurs or prevented by cryptography, some outside manual "meta-protocol" is required to figure out who is lying and repair the network or create a new network containing the truthful state. For some kinds of communications, digital signatures and a chain of evidence based on cryptographic hash chains are a much stronger security against forgery. Byzantine protocols, with their imperfect detection and exclusion of liars, are to be relied on only where the lie is of a nature not amenable to prevention by cryptographic chains of evidence.
Sensors and effectors can be readily hooked up to these high-integrity networks. Cryptography can, for example, provide us an unforgeable chain of evidence from a security camera to our computer displays and an unforgeable return chain of command from our mice to a gun or a jail cell lock. Cryptography can also secure smart contracts with the local officials: a judge declares you bailable, said authorization being transmitted to your jail door. Your girlfriend fills out a web form which pays the bail bondsman with a credit card. The bondsman's computer debits her account and then puts up the digital bond, and the jail door opens. You're out and I don't get to date your kind-hearted girlfriend.
One popular piece of secure government that many people have worked on is secure voting.
Several years ago I sketched an important sub-protocol of liar-resistant government, namely secure property titles (or, more generally, secure public registries). Such titles could, of course, include titles to political as well as real, personal, and intellectual property, and physical security devices such as sensors and weapons could be controlled based on them. In addition to to the cryptographic integrity of the records themselves, the public title registry can follow any rules of transfer in at least a Byzantine failure resistant way. Normal title transfers, signed over by the former owner, would be cryptographically strong.
Besides the obvious real property titles, domain names, and so, on, these registries could securely record and transfer the shares of a corporation. Bit gold, my sketch of an electronic currency that minimally relies on trust in any one person or organization, achieves this minimal vulnerability by using secure property titles. Satoshi Nakamoto has implemented BitCoin which very similarly uses a dense Byzantine fault tolerant peer-to-peer network and and cryptographic hash chains to ensure the integrity of a currency.
Making a number of important legal and political functions liar-resistant is on the horizon, and bits and pieces of this task are already being implemented.