Renyi-Ulam Games and Forbidden Substrings

TL;DR

This paper analyzes the Renyi-Ulam game with restrictions on lies, providing criteria for winning strategies and classifying restrictions based on forbidden substrings, extending previous work from one to two substrings.

Contribution

It offers a complete classification of restrictions characterized by two forbidden substrings in the Renyi-Ulam game, building on prior results for one substring.

Findings

Criteria for Seeker to win under certain restrictions

Complete classification of restrictions with two forbidden substrings

Extension of previous classification from one to two 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.