Abstract
It is a classical result of Mortimer's that L 2, first-order logic with two variables, is decidable for satisfiability (whereas L 3 is undecidable). We show that going beyond L 2 by adding any one of the following leads to an undecidable logic:
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very weak forms of recursion, such as transitive closure or monadic fixed-point operations.
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cardinality comparison quantifiers.
In fact these extensions of L 2 prove to be undecidable both for satisfiability, and for satisfiability in finite models. Moreover the satisfiability problem for these systems is shown to be hard for Σ 11 , the first level of the analytical hierarchy. They thereby exhibit a much higher degree of undecidability than first-order logic.
The case of monadic least fixed-point logic in two variables deserves particular attention, since this logic may be seen as the natural least common extension of two important decidable systems: first-order with two variables and propositional μ-calculus (propositional modal logic with a least fixed-point operator). It had been conjectured that this system might still be decidable.
This work has been partially supported by the German-Israeli Foundation of Scientific Research and Development
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© 1997 Springer-Verlag Berlin Heidelberg
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Grädel, E., Otto, M., Rosen, E. (1997). Undecidability results on two-variable logics. In: Reischuk, R., Morvan, M. (eds) STACS 97. STACS 1997. Lecture Notes in Computer Science, vol 1200. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0023464
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DOI: https://doi.org/10.1007/BFb0023464
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