Worlds of Events: Deduction with Partial Knowledge about Causality

TL;DR

This paper introduces a proof-theoretic model for reasoning about causality with partial knowledge in distributed systems, proving key properties and establishing bisimilarity concepts for consistent causal deduction across devices.

Contribution

It presents the first formal causality model for distributed partial knowledge, with proofs of computability, consistency, and bisimilarity-based deduction rules.

Findings

Partial knowledge leads to a weaker causality model than classical.

Devices deduce causality similarly if fed with the same information and starting bisimilar.

Proofs of bisimilarity results are simplified by induction on sequence length.

Abstract

Interactions between internet users are mediated by their devices and the common support infrastructure in data centres. Keeping track of causality amongst actions that take place in this distributed system is key to provide a seamless interaction where effects follow causes. Tracking causality in large scale interactions is difficult due to the cost of keeping large quantities of metadata; even more challenging when dealing with resource-limited devices. In this paper, we focus on keeping partial knowledge on causality and address deduction from that knowledge. We provide the first proof-theoretic causality modelling for distributed partial knowledge. We prove computability and consistency results. We also prove that the partial knowledge gives rise to a weaker model than classical causality. We provide rules for offline deduction about causality and refute some related folklore. We define two notions of forward and backward bisimilarity between devices, using which we prove two important results. Namely, no matter the order of addition/removal, two devices deduce similarly about causality so long as: (1) the same causal information is fed to both. (2) they start bisimilar and erase the same causal information. Thanks to our establishment of forward and backward bisimilarity, respectively, proofs of the latter two results work by simple induction on length.