Extension and equivalence problems for clause minimal formulae

Inspired by the notion of minimal unsatisfiable formulae we first introduce and study the class of clause minimal formulae. A CNF formula F is said to be clause minimal if any proper subformula of F is not equivalent to F. We investigate the equivalence and extension problems for clause minimal formulae. The extension problem is the question whether for two formulae F and H there is some formula G such that F+G is equivalent to H. Generally, we show that these problems are intractable. Then we discuss the complexity of these problems restricted by various parameters and constraints. In the last section we ask several open questions in this area.