Consistent histories: enforcing linearity and excluding double-spending amongst collaborating controllers

We address the consistent histories problem: how to ensure that linear digital resources (such as tokens of fiat currency, of cryptocurrency, records of title, exclusive locks, and soon) remain linear across execution traces that span multiple independent collaborating controllers (state machines), without a single trusted global authority.

This problem underlies a wide range of practical challenges, including atomic swaps,cross-chain bridges, interchain messaging, and tokenised settlement. A solution to this problem would address an important outstanding theoretical issue in the mathematics of banking and cryptocurrency.

We propose a mathematical framework based on a novel notion of sequential located semilattices with Cut. In this model, each controller (communicating state machine) maintains a bounded semilattice of times (= some abstraction of controller state), such that incomparable elements join to a distinguished top element representing inconsistency.

Controllers attest to each other’s times via an attestation operation that is controller-preserving, expansive, and contradiction-preserving. A carefully designed family of flags on times enables us to introduce a Cut operation, which prunes history, thus reducing storage cost.

Our main result states that any inconsistent execution trace contains an active inconsistent index (an inconsistency that has not been cut away). We further provide an even stronger absolute consistency criterion, which has the advantage of being independent of trust assumptions, at the expense of being a little harder to prove.

In particular, Cut is safe: the only way to remove byzantine behaviour from history is to introduce new byzantine behaviour into the history, so that if a pruned history is safe, then the original history must have been safe.

Our framework is constructive in the sense that we exhibit explicit models, including one which is close to a realistic implementation, such that times are tuples carrying cryptographic digests of attested controller states.

The mathematics in this paper has been formalised in Lean 4 and the formalisation is available online.