Distributed computation of a k P systems with active membranes for SAT using clause completion

In a work presented by Gazdag and Kolonits from 2013, it was shown that SAT can be solved in linear time in the number of variables using the clause completion method on a non-distributed P system with active membranes. A distributed P system with active membranes using the clause completion method solving SAT (denoted as k-$$\varDelta (n)$$) is presented in this work. k-$$\varDelta (n)$$ is a weak uniform solution to SAT that runs in linear time with respect only to the number of variables, n, of the Boolean formula $$\varphi$$. For the 2-component solution, 2-$$\varDelta (n)$$, we show that the communication cost is constant. But, increasing the number of components in k-$$\varDelta (n)$$, $$k \ge 3$$, would make the communication cost dependent on not just the number of components and the number of variables, but as well as the number of satisfying assignments to $$\varphi$$. We report that an exponential amount of resources (in terms of alphabet size and rules) are necessary to construct k-$$\varDelta (n)$$ solving SAT to obtain these reasonable communication costs.