In the present pater, we consider the asynchronous parallelism on the SN P system, which is a representative of natural computing, and propose asynchronous SN P systems that perform logical and arithmetic operations. We first propose SN P systems that compute logical operations, which are NOT, OR, AND and EX-OR. The SN P systems work in O(1) sequential and parallel steps using O(1) neurons. We next propose a SN P system that computes addition of k binary numbers of m bits. The SN P system works in O(km) sequential steps or O(m) parallel steps using O(m) neurons. Finally, we propose a SN P system that computes multiplication of two binary numbers of m bits, and show that the SN P system works in O(m) sequential steps or O(m) parallel steps using O(m) neurons.