We study the relationship between two kinds of higher-order extensions of model checking: HORS model checking, where models are extended to higher-order recursion schemes, and HFL model checking, where the logic is extended to higher-order modal fixpoint logic. These extensions have been independently studied until recently, and the former has been applied to higher-order program verification, while the latter has been applied to assume-guarantee reasoning and process equivalence checking. We show that there exist (arguably) natural reductions between the two problems. To prove the correctness of the translation from HORS to HFL model checking, we establish a type-based characterization of HFL model checking, which should be of independent interest. The results reveal a close relationship between the two problems, enabling cross-fertilization of the two research threads. Categories and Subject Descriptors F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic Keywords higher-order recursion schemes, higher-order modal fixpoint logic, model checking By the condition βj > 0 and Lemma 20, T IsZero k (β j #) is accepted from q0. Thus, we have the required result.

Guarded normal form requires occurrences of fixpoint variables in a µ-calculus-formula to occur under the scope of a modal operator. The literature contains guarded transformations that effectively bring a µ-calculusformula into guarded normal form. We show that the known guarded transformations can cause an exponential blowup in formula size, contrary to existing claims of polynomial behaviour. We also show that any polynomial guarded transformation for µ-calculus-formulas in the more relaxed vectorial form gives rise to a polynomial solution algorithm for parity games, the existence of which is an open problem. We also investigate transformations between the µ-calculus, vectorial form and hierarchical equation systems, which are an alternative syntax for alternating parity tree automata.

It is known that the model checking problem for the modal µ-calculus reduces to the problem of solving a parity game and vice-versa. The latter is realised by the Walukiewicz formulas which are satisfied by a node in a parity game iff player 0 wins the game from this node. Thus, they define her winning region, and any model checking algorithm for the modal µ-calculus, suitably specialised to the Walukiewicz formulas, yields an algorithm for solving parity games. In this paper we study the effect of employing the most straight-forward µ-calculus model checking algorithm: fixpoint iteration. This is also one of the few algorithms, if not the only one, that were not originally devised for parity game solving already. While an empirical study quickly shows that this does not yield an algorithm that works well in practice, it is interesting from a theoretical point for two reasons: first, it is exponential on virtually all families of games that were designed as lower bounds for very particular algorithms suggesting that fixpoint iteration is connected to all those. Second, fixpoint iteration does not compute positional winning strategies. Note that the Walukiewicz formulas only define winning regions; some additional work is needed in order to make this algorithm compute winning strategies. We show that these are particular exponential-space strategies which we call eventually-positional, and we show how positional ones can be extracted from them.

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