We propose an interpretable rule-based classification system based on ideas from Boolean compressed sensing. We represent the problem of learning individual conjunctive clauses or individual disjunctive clauses as a Boolean group testing problem, and apply a novel linear programming relaxation to find solutions. We derive results for exact rule recovery which parallel the conditions for exact recovery of sparse signals in the compressed sensing literature: although the general rule recovery problem is NP-hard, under some conditions on the Boolean 'sensing' matrix, the rule can be recovered exactly. This is an exciting development in rule learning where most prior work focused on heuristic solutions. Furthermore we construct rule sets from these learned clauses using set covering and boosting. We show competitive classification accuracy using the proposed approach. Copyright 2013 by the author(s).