Weak Models of Distributed Computing, with Connections to Modal Logic

摘要

This work presents a classification of weak models of distributed computing. We focus on deterministic distributed algorithms, and we study models of computing that are weaker versions of the widely-studied port-numbering model. In the port-numbering model, a node of degree d receives messages through d input ports and it sends messages through d output ports, both numbered with 1, 2,.... d. In this work, VV_c is the class of all graph problems that can be solved in the standard port-numbering model. We study the following subclasses of VV_c: VV: Input port i and output port i are not necessarily connected to the same neighbour. MV: Input ports are not numbered; algorithms receive a multiset of messages. SV: Input ports are not numbered; algorithms receive a set of messages. VB: Output ports are not numbered; algorithms send the same message to all output ports. MB: Combination of MV and VB. SB: Combination of SV and VB. Now we have many trivial containment relations, such as SB in contained in MB in contained in VB in contained in VV in contained in VV_c, but it is not obvious if, e.g., either of VB in contained in SV or SV in contained in VB should hold. Nevertheless, it turns out that we can identify a linear order on these classes. We prove that SB in contained in MB = VB in contained in SV = MV = VV in contained in VV_c. The same holds for the constant-time versions of these classes. We also show that the constant-time variants of these classes can be characterised by a corresponding modal logic. Hence the linear order identified in this work has direct implications in the study of the expressibility of modal logic. Conversely, we can use tools from modal logic to study these classes.