In the original Ulam-Rényi game with m lies/errors, Player I chooses a secret number \({\bar{x}}\) in a finite search space S, and Player II must guess \({\bar{x}}\) by adaptively asking Player I a minimum number of binary questions. Up to m answers may be mendacious/erroneous or may be distorted before reaching Player II. In his monograph “Fault-Tolerant Search Algorithms. Reliable Computation with Unreliable Information”, F. Cicalese provides a comprehensive account of many models of the game and their applications in error-correcting coding with noiseless feedback. Since for \(m>0\) lies the game is not called off by contradictory answers, and repeated answers to the same question are more informative than single answers, the underlying logic of the game with m lies is not boolean. Indeed, the logic of Rényi-Ulam games is Łukasiewicz infinite-valued logic Ł \(_\infty \) . In this paper we consider Ulam-Rényi games with variable numbers of lies over infinite search spaces. We characterize the MV-algebras and the unital Specker \(\ell \) -groups given by the logic of these games.
