NP Reasoning in the Monotone $\mu$-Calculus

Satisfiability checking for monotone modal logic is known to be (only) NP-complete. We show that this remains true when the logic is extended with aconjunctive and alternation-free fixpoint operators as well as the universal modality; the resulting logic – the aconjunctive alternation-free monotone $\mu$-calculus with the universal modality – contains both concurrent propositional dynamic logic (CPDL) and the alternation-free fragment of game logic as fragments. We obtain our result from a characterization of satisfiability by means of B"uchi games with polynomially many Eloise nodes.