Renyi-Ulam Games and Forbidden Substrings

Abstract

The Renyi-Ulam game is played between two players, the Seeker and the Obscurer. The Obscurer thinks of a number between 1 and $n$. The Seeker wishes to identify that number. On each turn, the Seeker asks the Obscurer whether her number belongs to a specific subset of the numbers from 1 to $n$. The Obscurer answers either yes or no, and her answer is either true or false. The series of truths and lies given by the Obscurer must conform to a restriction $R$ that the players have agreed on in advance. We give criteria on the restrictions $R$ that allow the Seeker to win. Then we apply our results to the study of restrictions characterized by forbidden substrings. In particular, we give a complete classification of all such restrictions characterized by two forbidden substrings, elaborating on Czyzowicz, Lakshmanan and Pelc's classification of all such restrictions characterized by one forbidden substring.