AFAIK crime is reduced by: law and order, education, and welfare.
One of my latest issues will let me replace the police with military. Presumably this will have the effect of reducing law expenditure to zero, and massively increasing defense.
As much as I like the idea of a Gendarmerie, I am worried that the game will not recognize high defense expenditure in the crime calculation, and thus crime will sky rocket with no law and order.
So... can anyone confirm this?