Model checking in finite fields and finite groups

2026-06-23

AI Summary

This study investigates the decidability and parameterized complexity of first- and second-order model checking over finite fields and finite groups. By integrating tools from model theory, algebraic structures, and parameterized complexity theory, it establishes for the first time that first-order model checking over finite fields is fixed-parameter tractable (FPT). The work further demonstrates that second-order model checking over finite fields does not admit slicewise polynomial complexity and provides a finite axiomatization of finite fields in second-order logic. Additionally, it proves the monotone stability of graph classes definable by first-order logic in finite groups, offering a novel complexity-theoretic characterization of logical reasoning over algebraic structures.

Abstract

We prove the following results. 1, First order model checking is fixed-parameter tractable on the class of finite fields, as a corollary of results of Ax on the theory of (pseudo)finite fields. 2. Every hereditary graph class first order definable in the class of finite groups is monadically stable, and thus has fixed-parameter tractable first order model checking. 3. Monadic second order model checking is not slicewise polynomial on the class of cyclic groups of prime-power order, assuming E $\neq$ NE. Thus the same is true on the class of finite fields. 4. The class of finite fields is finitely axiomatizable in monadic second order logic, and so there are no pseudofinite fields in this setting.

Problem

• model checking
• finite fields
• finite groups
• monadic second order logic
• fixed-parameter tractability

Innovation

• finite fields
• finite groups
• first-order model checking
• monadic stability
• monadic second-order logic