---
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.'
author:
- Benedikt Bollig
- Marie Fortin
- Paul Gastin
bibliography:
- 'lit.bib'
title: 'It Is Easy to Be Wise After the Event: Communicating Finite-State Machines Capture First-Order Logic with “Happened Before”'
---

Introduction
============

First-order (FO) logic can be considered, in many ways, a reference specification language. It plays a key role in automated theorem proving and formal verification. In particular, FO logic over finite or infinite words is central in the verification of reactive systems. When a word is understood as a total 