# Simplification of a multi-index Boolean expression towards computation in fewer steps

Let $$x_{ij} \in \{0,1\}$$, $$1 \leq i \leq M$$ (typically, $$M = 2000$$), $$1 \leq j \leq N$$ (typically, $$N = 10$$), be Boolean variables. If possible at all, I would like to simplify the following expression $$\bigvee_{i_1,\dots,i_N} \left(\bigwedge_{k=1}^N x_{i_k k} \right),$$ where the join operation is taken over all indices $$(i_1,\dots,i_N) \in \{1,\dots,M \}^N$$, so that it can be computed faster instead of looping through all multi-indices.

Apologies if this is a well-known problem (with or without solution). This is very much outside of my field of expertise, so I am hoping someone around here could give some feedback on it. Feel free to edit the tags as you see more appropriate.

Thanks!

$$\bigwedge_{k=1}^{N} \bigvee_{i_k=1}^{M} x_{i_k k}.$$
This is a much simpler expression: $$NM$$ terms instead of $$M^N$$ terms, if you expand everything out.