Write a "random" boolean expression, built from a conjunction of Xor clauses on a finite number $n$ of boolean variables: what is the probability that such an expression, chosen randomly, is satisfiable? i.e., that it (does not) computes the constant function False? Our approach to this classical problem is, starting from a uniform probability distribution on these boolean expressions, to define and compute asymptotically the probability distribution induced on the set of all boolean functions. Writing down a system of equations on the generating functions associated to the boolean functions, we use the structure of the dependencies between boolean functions to analyze this system and gain information on the structure of its set of singularities, thus opening the way to asymptotic analysis of the probabilities.

Work in common with Bernhard Gittenberger and Markus Kuba.