Abstract
The model-checking games associated with fixed-point logics are parity games, and it is currently not known whether the strategy problem for parity games can be solved in polynomial time. We study Solitaire-LFP, a fragment of least fixed-point logic, whose evaluation games are nested soltaire games. This means that on each strongly connected component of the game, only one player can make nontrivial moves. Winning sets of nested solitaire games can be computed efficiently. The model-checking problem for Solitaire-LFP is Pspace-complete in general and Ptime-complete for formulae of bounded width. On finite structures (but not on infinite ones), Solitaire-LFP is equivalent to transitive closure logic. We also consider the solitaire fragment of guarded fixed-point logics. Due to the restricted quantification pattern of these logics, the associated games are small and therefore admit more efficient model-checking algorithms.
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Berwanger, D., Grädel, E. Fixed-Point Logics and Solitaire Games. Theory Comput Systems 37, 675–694 (2004). https://doi.org/10.1007/s00224-004-1147-5
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DOI: https://doi.org/10.1007/s00224-004-1147-5