The problem 2-Xor-Sat asks for the probability that a random expression, built as a conjunction of clauses $$x \oplus y$$xźy, 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.
['Élie de Panafieu', 'Danièle Gardy', 'Bernhard Gittenberger', 'Markus Kuba']
2-Xor Revisited: Satisfiability and Probabilities of Functions