According to a theorem of Courcelle monadic second-order logic and guarded second-order logic (where one can also quantify over sets of edges) have the same expressive power over the class of all countable $k$-sparse hypergraphs. In the first part of the present paper we extend this result to hypergraphs of arbitrary cardinality. In the second part, we present a generalisation dealing with methods to encode sets of vertices by single vertices.

• 03B25 - Decidability of theories and sets of sentences
• 03D05 - Automata and formal grammars in connection with logical questions
• 68Q45 - Formal languages and automata