Probabilities of Boolean Functions given by Random Implicational Formulas

We study the asymptotic relation between the probability and the complexity of Boolean functions in the implicational fragment which are generated by large random Boolean expressions involving variables and implication, as the number of variables tends to infinity. In contrast to models studied in the literature so far, we consider two expressions to be equal if they differ only in the order of the premises. A precise asymptotic formula is derived for functions of low complexity. Furthermore, we show that this model does not exhibit the Shannon effect. Partially supported by the A.N.R. project BOOLE, 09BLAN0011, and by the P.H.C. Amadeus project Probabilities and tree representations for Boolean functions. Partially supported by the P.H.C. Amadeus project Probabilities and tree representations for Boolean functions, and by FWF (Austrian Science Foundation), National Research Area S9600, grant S9604 and ÖAD, grant F03/2010. the electronic journal of combinatorics 19(2) (2012), #P37 1