SCAN is complete for all Sahlqvist formulae. Goranko, V., Hustadt, U., Schmidt, R., & Vakarelov, D. In Berghammer, R., Möller, B., & Struth, G., editors, Revised Selected Papers of the 7th International Seminar on Relational Methods in Computer Science and the 2nd International Workshop on Kleene Algebra [Bad Malente, Germany, 12-17 May 2003], volume 3051, of LNCS, pages 149-162, 2004. Springer.
SCAN is complete for all Sahlqvist formulae [pdf]Paper  abstract   bibtex   
SCAN is an algorithm for reducing existential second-order logic formulae to equivalent simpler formulae, often first-order logic formulae. It is provably impossible for such a reduction to first-order logic to be successful for every second-order logic formula which has an equivalent first-order formula. In this paper we show that SCAN successfully computes the first-order equivalents of all Sahlqvist formulae in the classical (multi-)modal language.

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