In mathematical logic and computer science, two-variable logic is the fragment of first-order logic where formula can be written using only two different variables. This fragment is usually studied without function symbols.
In logic, the term decidable refers to the decision problem, the question of the existence of an effective method for determining membership in a set of formulas, or, more precisely, an algorithm that can and will return a boolean true or false value that is correct (instead of looping indefinitely, crashing, returning "don't know" or returning a wrong answer).
My question is if it is decidable what is the method for determining membership in a set of formula and if it's undecidable tell me the reason.Also if there is any paper about this please inform me.
Thanks
The problem is dealt with in "On the Decision Problem for Two-Variable First-Order Logic," by Erich Grädel, Phokion G. Kolaitis and Moshe Y. Vardi. The Bulletin of Symbolic Logic, Vol. 3, No. 1 (Mar., 1997), pp. 53-69.
The answer is affirmative: Corollary 5.4 on Page 66 says: "The satisfiability problem for $FO^2$ with or without equality is NEXPTIME-complete," which implies that satisfiability for $FO^2$ is decidable in NEXPTIME.