The Entscheidungsproblem (decision problem) for modal logic

Question

The Entscheidungsproblem is identified with the decision problem for first-order logic that is, the problem of algorithmically determining whether a first-order statement is universally valid. http://en.wikipedia.org/wiki/Entscheidungsproblem

I would like to know how one could adapt a version of this problem to other logics, especially in the modal logic and if this is a positive or negative answer(in the case of first-order logic the answer is negative).

In general how could approach problems of decision in other logical(especially in the modal logic). Known works about this. Thanks

Answer 1

Well, you can see □,♢ as simply universal and existential quantifiers over universes, and modify all predicates (including equality) to include an additional argument for the universe. Then essentially anything that is true about the structure of propositional logic is also true about the structure of propositional modal logic.

Answer 2

See for example : Nino Cocchiarella & Max Freund, Modal Logic An Introduction to its Syntax and Semantics (2008), page 78 :

Decidability of $L_{at}$ and $S5$

The completeness theorems we proved for $L_{at}$ and $S5$ in §4.3 of chapter 4 and §5.1 of this chapter can be used to show that it is effectively decidable whether or not a formula is provable in either of these systems.

See in Patrick Blackburn & Johan van Benthem (editors), Handbook of Modal Logic (2007) all Chapter on MODAL DECISION PROBLEMS by Frank Wolter and Michael Zakharyaschev.

See in Robert Goldblatt, Mathematics of Modality (1993) Chapter 6.8 Finite Models and Decidability.

See in Dov Gabbay (editor), Handbook of Philosophical Logic (2nd ed) Vol_3 : Modal Logic, the Chapter on Advanced Modal Logic by M. Zakharyaschev, F. Wolter and A. Chagrov, sect.4.4 Undecidable properties of calculi, page 238-on.