Undecidability results on two-variable logics

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:

• very weak forms of recursion, such as transitive closure or monadic fixed-point operations.

• 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