Learning of monotone functions with single error correction

Learning of monotone functions is a well-known problem. Results obtained by V. K. Korobkov and G. Hansel imply that the complexity φM (n) of learning of monotone Boolean functions equals C n ⌊ n / 2 ⌋ $\begin{array}{} \displaystyle C_n^{\lfloor n/2\rfloor} \end{array}$ + C n ⌊ n / 2 ⌋ + 1 $\begin{array}{} \displaystyle C_n^{\lfloor n/2\rfloor+1} \end{array}$ (φM (n) denotes the least number of queries on the value of an unknown monotone function on a given input sufficient to identify an arbitrary n-ary monotone function). In our paper we consider learning of monotone functions in the case when the teacher is allowed to return an incorrect response to at most one query on the value of an unknown function so that it is still possible to correctly identify the function. We show that learning complexity in case of the possibility of a single error is equal to the complexity in the situation when all responses are correct.

On the Lyapunov Exponent of Monotone Boolean Networks †

Boolean networks are discrete dynamical systems comprised of coupled Boolean functions. An important parameter that characterizes such systems is the Lyapunov exponent, which measures the state stability of the system to small perturbations. We consider networks comprised of monotone Boolean functions and derive asymptotic formulas for the Lyapunov exponent of almost all monotone Boolean networks. The formulas are different depending on whether the number of variables of the constituent Boolean functions, or equivalently, the connectivity of the Boolean network, is even or odd.

Bounds for the average-case complexity of monotone Boolean functions

We consider average-case complexity of computing monotone Boolean functions by straight-line programs with a conditional stop over the basis of all Boolean functions of at most two variables. For the set of all