2-Xor Revisited: Satisfiability and Probabilities of Functions

The problem 2-Xor-Sat asks for the probability that a random expression, built as a conjunction of clauses , is satisfiable. We revisit this classical problem by giving an alternative, explicit expression of this probability. We then consider a refinement of it, namely the probability that a random expression computes a specific Boolean function. The answers to both problems involve a description of 2-Xor expressions as multigraphs and use classical methods of analytic combinatorics by expressing probabilities through coefficients of generating functions.