We investigate the expressive power relative to three-valued and four-valued logics of various subsets of the set of connectives which are used in the bilattices-based logics. Our study of a language is done in two stages. In the first stage the ability of the language to characterize sets of tuples of truth-values is determined. In the second stage the results of the first are used to determine its power to represent operations. Special attention is given to the role of monotonicity, closure and freedom properties in classifying languages, as well as to maximality properties (for example: we prove that by adding any nonmonotonic connective to the set of four-valued monotonic connectives, we get a functionally complete set).