The complexity of propositional implication

The question whether a set of formulae Γ implies a formula φ is fundamental. The present paper studies the complexity of the above implication problem for propositional formulae that are built from a systematically restricted set of Boolean connectives. We give a complete complexity-theoretic classification for all sets of Boolean functions in the meaning of Post's lattice and show that the implication problem is efficiently solvable only if the connectives are definable using the constants {0,1} and only one of {∧,∨,⊕}. The problem remains coNP-complete in all other cases. We also consider the restriction of Γ to singletons which makes the problem strictly easier in some cases.