A Friendly Introduction to Ehrenfeucht-Fräıssé Games

Assuming some basic familiarity with ordinal arithmetic, we provide a friendly introduction to the theory of Ehrenfeucht-Fräıssé games. 1 What Is An Ehrenfeucht-Fräıssé Game? Ehrenfeucht-Fräıssé (EF) games were first developed in the 50’s and 60’s by Andrzej Ehrenfeucht and Roland Fräıssé. Although Fräıssé developed much of the background theory and some important applications in his doctoral dissertation [2], Ehrenfeucht [1] was the first to formulate these methods in terms of games. EF-games are interesting, easy to understand, and very useful for determining when two structures are logically equivalent. They frequently appear in top logic journals, providing interesting new results, as well as fresh new ways of looking at old theorems. So grab a pencil and paper, take your time, and learn a game! An EF-game calls for two players. These two players must meet different goals in order to win, so it’s important to keep them straight. We’ll call the first player Erin. She always goes first, and is sometimes called the ‘spoiler.’ Her goal in the game is to spoil the other player’s plan. We’ll call the second player Fred. He’s sometimes called the ‘copier.’ His goal is just to copy what Erin does. The only “pieces” required to play the game are two L-structures. In elementary intermediate logic, you learned that an L-structure consists of a domain (the “universe” of the structure), together with some other stuff (a relation, a function and a constant for each such item in the language L). In an EF game, the domain is where all the action is. The players begin the game by agreeing on how long they will play, say 3 turns. Each turn then consists ∗Email: [email protected]. Web: www.pitt.edu/∼bwr6.