It Is Easy to Be Wise After the Event: Communicating Finite-State Machines Capture First-Order Logic with "Happened Before"

TL;DR

This paper establishes a precise correspondence between communicating finite-state machines, a fragment of propositional dynamic logic, and first-order logic over message sequence charts, clarifying their expressive power and logical properties.

Contribution

It introduces a star-free propositional dynamic logic with loop and converse, proving its equivalence with first-order logic over MSCs and translating between CFMs and logical fragments.

Findings

Every first-order sentence over MSCs can be expressed in star-free PDL.

Star-free PDL sentences can be translated into equivalent CFMs.

First-order logic over MSCs has the three-variable property.

Abstract

Message sequence charts (MSCs) naturally arise as executions of communicating finite-state machines (CFMs), in which finite-state processes exchange messages through unbounded FIFO channels. We study the first-order logic of MSCs, featuring Lamport's happened-before relation. We introduce a star-free version of propositional dynamic logic (PDL) with loop and converse. Our main results state that (i) every first-order sentence can be transformed into an equivalent star-free PDL sentence (and conversely), and (ii) every star-free PDL sentence can be translated into an equivalent CFM. This answers an open question and settles the exact relation between CFMs and fragments of monadic second-order logic. As a byproduct, we show that first-order logic over MSCs has the three-variable property.