Model checking of strategic ability under imperfect information is known to be hard. The complexity results range from NP-completeness to undecidability, depending on the precise setup of the problem. No less importantly, fixpoint equivalences do not generally hold for imperfect information strategies, which seriously hampers incremental synthesis of winning strategies.In this paper, we propose translations of ATLir formulae that provide lower and upper bounds for their truth values, and are cheaper to verify than the original specifications. That is, if the expression is verified as true then the corresponding formula of ATLir should also hold in the given model. We begin by showing where the straightforward approach does not work. Then, we propose how it can be modified to obtain guaranteed lower bounds. To this end, we alter the next-step operator in such a way that traversing one's indistinguishability relation is seen as atomic activity. Most interestingly, the lower approximation is provided by a fixpoint expression that uses a nonstandard variant of the next-step ability operator. We show the correctness of the translations, establish their computational complexity, and validate the approach by experiments with a scalable scenario of Bridge play.

In this paper we introduce Strategy Logic with simple goals (SL[SG]), a fragment of Strategy Logic that strictly extends the well-known Alternating-time Temporal Logic ATL by introducing arbitrary quantification over the agents' strategies.  Our motivation comes from game-theoretic applications, such as expressing Stackelberg equilibria in games, coercion in voting protocols, as well as module checking for simple goals. Most importantly, we prove that the model checking problem for SL[SG] is PTIME-complete, the same as ATL. Thus, the extra expressive power comes at no computational cost as far as verification is concerned.

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