Two-variable logic
In mathematical logic and computer science, two-variable logic is the fragment of first-order logic where formulae can be written using only two different variables. This fragment is usually studied without function symbols.
Decidability
Some important problems about two-variable logic, such as satisfiability and satisfiability (logics)|finite satisfiability], are decidable. This result generalizes results about the decidability of fragments of two-variable logic, such as certain description logics; however, some fragments of two-variable logic enjoy a much lower computational complexity for their satisfiability problems.By contrast, for the three-variable fragment of first-order logic without function symbols, satisfiability is undecidable.
Counting quantifiers
The two-variable fragment of first-order logic with no function symbols is known to be decidable even with the addition of counting quantifiers, and thus of uniqueness quantification. This is a more powerful result, as counting quantifiers for high numerical values are not expressible in that logic.Counting quantifiers actually improve the expressiveness of finite-variable logics as they allow to say that there is a node with neighbors, namely. Without counting quantifiers variables are needed for the same formula.