> You may recall that constructing an intersection automaton involves a cross product of NFAs
I'm not sure what you mean by "cross product" here, but there's nothing multiplicative involved. You just run both NFAs simultaneously. This gives you a number of states to track equal to the sum, not the product, of the two NFAs being intersected.
I'm not sure what you mean by "cross product" here, but there's nothing multiplicative involved. You just run both NFAs simultaneously. This gives you a number of states to track equal to the sum, not the product, of the two NFAs being intersected.