The algebra of Bonferroni bounds: discrete tubes and extensions

Bonferroni, or inclusion-exclusion, bounds and identities have a rich history. They concern the indicator function, and hence the probability content, of the union of sets. In previous work, the authors defined a discrete tube which yields upper and lower bounds which are at least as tight as the standard bounds obtained by truncating inclusion-exclusion identities at particular depths. Here, some connections to other fields are made, based particularly on the algebra of indicator functions. These leads to the consideration of the complexity of more general Boolean statements.