Courcelle's theorem states that "Every graph property which is expressible in monadic second order logic is decidable in linear time for bounded tree-width graphs". Later it was extended to bounded clique-width graphs [CMR99].
The theorem also holds with linear time replaced by logarithmic space [EJT10] for bounded tree-width graphs. My question is: Does this theorem still hold for bounded clique-width graphs if we replace linear time with logarithmic space? Let me know of the recent progress in this direction.