Authors: Witold Charatonik ; Piotr Witkowski

We study the finite satisfiability problem for the two-variable fragment of first-order logic extended with counting quantifiers (C2) and interpreted over linearly ordered structures. We show that the problem is undecidable in the case of two linear orders (in the presence of two other binary symbols). In the case of one linear order it is NEXPTIME-complete, even in the presence of the successor relation. Surprisingly, the complexity of the problem explodes when we add one binary symbol more: C2 with one linear order and in the presence of other binary predicate symbols is equivalent, under elementary reductions, to the emptiness problem for multicounter automata.

• 03B20 - Subsystems of classical logic (including intuitionistic logic)
• 03B25 - Decidability of theories and sets of sentences
• 03C13 - Model theory of finite structures
• 03C80 - Logic with extra quantifiers and operators
• 68Q17 - Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
• 68Q19 - Descriptive complexity and finite models