We analyse the complexity of propositional kernel resolution (del Val, 1999), a general method for obtaining logical consequences in restricted target languages. Different choices of target are relevant to important AI tasks, e.g. prime implicates, satisfiability, abduction and non-monotonic reasoning, and polynomial-size knowledge compilation. Based on a generalized concept of induced width, we identify new tractable classes for various targets, and show how to estimate in advance the complexity of every problem, under various atom orderings. This can be used to choose an ordering for kernel resolution. Two applications are discussed: estimating the number of prime implicates of any theory; and identifying tractable abduction problems.