We study the equational theory of Kleene algebra with complement w.r.t. languages. While the equational theory w.r.t. languages coincides with the language equivalence (under the standard language valuation) for terms of Kleene algebra, this coincidence is broken if we extend the terms with complement. In this paper, we present a finite relational semantics, which completely characterizes the equational theory above. As applications, we show that the equational theory is (1) undecidable and Π 0 1-complete for full terms; (2) decidable and PSPACE-complete if the complement only applies to variables or constants.