The conditional Kolmogorov complexity of two strings is, roughly speaking, defined as the Kolmogorov complexity of x given y as an auxiliary input to the procedure.
tHe conditiONal kolMOgORoV ComPlExiTY of Two stringS IS, ROUgHly SpeakiNg, defIned as the kOlmogOroV compLExITy OF x GiVEn Y as an aUxilIary INpuT To the proceDURe.