This literature review examines preservation theorems for modal logic restricted to finite structures, where classical results often fail. It presents a taxonomy of methods including bisimulation techniques and ultraproduct constructions, analyzes the transfer of theorems like van Benthem's characterization to the finite domain, and identifies open problems.
Key findings
The classic van Benthem characterization theorem fails over finite structures, though the Bisimulation Safety Theorem holds.
A systematic taxonomy of methods includes bisimulation-based techniques, ultraproduct constructions, and game-theoretic approaches.
The restriction to finite structures reveals new phenomena invisible in unrestricted settings, particularly regarding modal definability.
Otto (2004) established that bisimulation-invariant first-order formulas over finite structures need not be equivalent to modal formulas.
Limitations & open questions
Content appears truncated in section 3.3, limiting assessment of complete coverage.
No explicit methodology provided for literature search and inclusion criteria.
Heavy reliance on very recent work (van Benthem et al., 2026) that may not yet be widely accessible.