Product logic and probabilistic Ulam games

Connections between games and many-valued logic have been shown first by Mundici [M] for the case of the Rényi-Ulam game and à Lukasiewicz logic. Given a finite set Ω of cardinality N (called the search space) and a natural number n, the Rényi-Ulam game G(N,n) is the following: a player, called Responder, chooses an element of Ω called the secret. The other player, called Questioner, has to guess the secret on the ground of binary questions of the form: Is the secret in X? with X ⊆ Ω. Responder has to answer all of them with a maximum of n lies. (As regards to winning strategies, etc., we can suppose without loss of generality that Ω = {1, 2..., N}. This is why we use the notation G(N,n) instead of G(Ω, n)). In [M], Mundici codes the information contained in a sequence σ of questions-answers (called record in [CM]) by means of the function fσ from Ω into [0, 1] defined as follows: say that a pair (Q, A), of questions-answers, where Q is Is the secret in X? with X ⊆ Ω, and A ∈ {Y ES, NO} falsifies x if either x ∈ X and A = NO or x / ∈ X and A = Y ES. Let for every x ∈ Ω, hx be the number of questions-answers in σ which falsify x (repetitions are taken into account). Then fσ is defined, for every x ∈ Ω, by fσ(x) = n+1−h σ x n+1 . As observed by Mundici, the truth-value function fτ corresponding to the juxtaposition τ of two records σ and ρ is the pointwise à Lukasiewicz conjunction fσ ̄ fρ of fσ and fρ. Moreover truth-value functions can be partially