On the Boundedness Problem for Two-Variable First-Order Logic

A positive rst-order formula is bounded if the sequence of its stages converges to the least xed point of the formula within a xed nite number of steps independent of the input structure. The boundedness problem for a fragment L of rst-order logic is the following decision problem: given a positive formula in L, is it bounded? In this paper, we investigate the boundedness problem for two-variable rst-order logic FO 2. As a general rule, FO 2 is a well-behaved fragment of rst-order logic, since it possesses the nite-model property and has a decidable satissability problem. Nonetheless, our main result asserts that the boundedness problem for FO 2 is undecidable, even when restricted to negation-free and equality-free formulas '(X; x) in which x is the only free variable and X is a monadic relation symbol that occurs within the scopes of universal quanti-ers only. This undecidability result contrasts sharply with earlier results asserting the decidability of bound-edness for monadic Datalog programs, which amounts to the decidability of boundedness for negation-free and equality-free existential rst-order formulas (X; x) in which x is the only free variable and X is a monadic relation symbol. We demonstrate that our main result has certain applications to circumscription, the most well-developed formalism of nonmonotonic reasoning. Speciically, using the undecidability of boundedness for FO 2 , we show that it is an undecidable problem to tell whether the circumscription of a given FO 2-formula is equivalent to a rst-order formula. In contrast, the circumscrip-tion of every FO 1-formula is equivalent to a rst-order formula.