COOL 2 – A Generic Reasoner for Modal Fixpoint Logics (System Description)

. There is a wide range of modal logics whose semantics goes beyond relational structures, and instead involves, e.g., probabilities, multi-player games, weights, or neighbourhood structures. Coalgebraic logic serves as a unifying semantic and algorithmic framework for such logics. It provides uniform reasoning algorithms that are easily instantiated to particular, concretely given logics. The COOL 2 reasoner provides an implementation of such generic algorithms for coalgebraic modal ﬁxpoint logics. As concrete instances, we obtain in particular reasoners for the aconjunctive and alternation-free fragments of the graded µ -calculus and the alternating-time µ -calculus. We evaluate the tool on standard benchmark sets for ﬁxpoint-free graded modal logic and alternating-time temporal logic (ATL), as well as on a dedicated set of benchmarks for the graded µ -calculus.


Introduction
Modal and temporal logics are established tools in the specification and verification of systems.While many such logics are interpreted over relational transition systems, the semantics of quite a number of important logics goes beyond the relational setup, involving, for instance, probabilities [21,31], concurrent games as in alternating-time logics [1,38], monotone neighbourhoods structures as in game logic [36] and concurrent dynamic logic [39], or integer transition weights as in the multigraph semantics [5] of the graded µ-calculus [26].Coalgebraic logic [4] provides a uniform semantic and algorithmic framework for these logics, based on the paradigm of universal coalgebra [41].It provides reasoning algorithms ⋆ Supported by the ERC Consolidator grant D-SynMA (No. 772459) ⋆⋆

Supported by Deutsche Forschungsgemeinschaft (DFG) as part of the Research
Training Group 2475 (grant number 393541319/GRK2475/1-2019) and the project 'RAND' (grant number 377333057).⋆ ⋆ ⋆ Supported by Deutsche Forschungsgemeinschaft (DFG) under project MI 717/7-1 of optimal complexity at various levels of expressiveness, up to the coalgebraic µ-calculus [3,[22][23][24].These algorithms are parametric in the transition type of systems (weighted, probabilistic, game-based etc.) as well as in suitable choices of modalities specific to the given system type.Their instantiation to specific logics requires providing either a set of next-step modal tableau rules satisfying a suitable completeness criterion [46] or, more generally, a plug-in algorithm that determines satisfiability for an extremely simple one-step logic that describes the interaction between modalities, and consists of (conjunctions of) modal operators applied to variables only [30].
The COalgebraic Ontology Logic solver (COOL) provides reasoning support for coalgebraic logics based on these generic algorithms.The first version of the tool [15] provided reasoning support for fixpoint-free coalgebraic hybrid logic with global assumptions, using a global caching principle [13].In the present paper, we present COOL 2, which provides reasoning support for coalgebraic fixpoint logics, specifically for both the aconjunctive fragment and the alternationfree fragment of the coalgebraic µ-calculus.By instantiation, we obtain in particular the first implemented reasoners for the graded µ-calculus [27] (for which a set of coalgebraic modal tableau rules has been described in the literature [46]; however, this rule set has later turned out to be incomplete, cf.Remark 2.3) and the alternating-time µ-calculus [1].We describe the structure of the tool including implementational details, and present evaluation results, focusing on the graded µ-calculus and alternating-time temporal logic (ATL).Full details on the evaluation can be found in an extended version of this paper [18].

Related Work:
We have already mentioned work in coalgebraic logic on which COOL is based [3,13,[22][23][24]46].COOL is conceptually a successor of the Coalgebraic Logic Satisfiability Solver (CoLoSS) [2] but does not share any of its code.CoLoSS implements fixpoint-free logics, and is entirely unoptimised.The first version of COOL [15] has been evaluated on fixpoint-free next-step logics.
COOL does cover also various relational modal logics, for which there are numerous specialised reasoners, including highly optimised description logic reasoners such as FaCT++ [50], Pellet [48], RACER [19], and HermiT [12].As these systems do not support fixpoint logics, a comparison would be of limited value.In previous work, COOL has been evaluated on various relational fixpoint logics, and has been shown to perform favourably on Computation Tree Logic [24] (in comparison to reasoners featured in a previous systematic evaluation [14]), as well as on the aconjunctive fragment of the modal µ-calculus [23] (in comparison to MLSolver [11]).A reasoner for (next-step) graded modal logic has been evaluated against various description logic reasoners [49], using however the above-mentioned incomplete set of modal tableau rules.
For the same reasons, we refrain from evaluating COOL 2 against reasoners for coalition logic, i.e. the fixpoint-free fragment of the alternating-time µcalculus, such as CLProver [34].The only implemented reasoner for any fragment of the alternating-time µ-calculus that does include fixpoints still appears to be the tableau reasoner TATL for alternating-time temporal logic [6,7].TATL has been compared to COOL on random formulas in previous work [24].
where ♥ ∈ Λ has arity n and X ∈ Var.A formula χ is aconjunctive if for every conjunction ψ ∧ φ that is a subformula of χ, at most one of the formulas ψ and φ contains a free fixpoint variable X that is bound by a least fixpoint operator µX.While the logic does not contain negation as an explicit operator, full negation can be defined as usual; e.g.we have ¬♥ψ = ♥¬ψ and ¬µX.ψ = νX.¬ψ[¬X/X], using ¬¬X = X.
Both the theoretical satisfiability checking algorithm and its implementation in COOL 2 operate on the Fischer-Ladner closure [22,25,28] of the target formula.The alternation depth (e.g.[22,30,35]) of a formula is the maximum depth of dependent alternating nestings of least and greatest fixpoints within the formula.Formulas with alternation depth 1 are alternation-free.
Semantics: Formulas are interpreted over F -coalgebras, that is, structures where F : Set → Set is a functor determining the branching type of the systems at hand; thus ξ(x) ∈ F C encodes the transitions from x ∈ C, structured according to F .Modalities ♥ ∈ Λ of arity n are interpreted as predicate liftings, that is, families of maps ♥ U : (2 U ) n → 2 F U (for U ∈ Set) that assign predicates on F U to n-tuples of predicates on U , subject to a naturality condition [37,44].On a coalgebra (C, ξ), the semantics of formulas is defined inductively in the usual way for the propositional operators and fixpoints, and by A closed formula ψ is satisfiable if there is a coalgebra (C, ξ) and a state The standard modal µ-calculus [25] is obtained using the functor F = P(A) × P, where A is a fixed set of atoms, the similarity type Λ = {♦, , a, ¬a | a ∈ A}, and predicate liftings The expressive power of the modal µ-calculus is demonstrated by the formulas The former is a co-Büchi formula expressing the existence of a path on which p holds forever, from some point on; the latter formula expresses the Büchi property that there is a path on which the atom p is satisfied infinitely often.
(2) The graded µ-calculus [27] allows expressing quantitative properties with the help of modal operators n and [n], n ∈ N; formulas n ψ and [n]ψ then have the intuitive meaning that 'there are more than n successor states that satisfy ψ', and 'all but at most n successor states satisfy ψ', respectively.Its coalgebraic interpretation is based on multigraphs, which are coalgebras for the multiset functor [5].A graded variant of the above Büchi property is specified, e.g., by the formula νX.µY.(p ∧ n X) ∨ n Y , which expresses the existence of an infinite n + 1-ary tree such that the atom p is satisfied infinitely often on every path in the tree.
(3) The alternating-time µ-calculus (AMC) [42] extends coalition logic [38] with fixpoints and (modulo syntax) supports modalities D and [D], where D ⊆ N is a coalition formed by agents from the set N = {1, . . ., n} for some fixed n ∈ N; formulas D ψ and [D]ψ then state that 'coalition D has a joint strategy to enforce ψ' and that 'coalition D cannot prevent ψ', respectively.For instance, the formula νX.µY.νZ.(p ∧ D X) ∨ (q ∧ D Y ) ∨ (¬q ∧ D Z) expresses that coalition D has a joint multi-step strategy that guarantees that p is visited infinitely often whenever q is visited infinitely often.
Satisfiability Checking: We proceed to recall the satisfiability checking algorithm for the coalgebraic µ-calculus that forms the basis of the implementation within COOL 2. This algorithm adapts the automata-based approach to satisfiability checking for the standard µ-calculus, and generalises the treatment of modal steps by parametrizing over a solver for the one-step satisfiability problem of the logic, which concerns satisfiability of formulae with exactly one layer of next-step modalities [22].It thus avoids the necessity of tractable sets of tableaux rules for modal operators.Under mild assumptions on the complexity of the one-step satisfiability problem of the base logic at hand ('tractability'), the algorithm witnesses a, typically optimal, upper bound ExpTime for the complexity of the satisfiability problem; unlike a previous algorithm [4], the algorithm thus has optimal runtime also in cases where no tractable sets of modal tableaux rules are known, such as the graded (or, more generally, Presburger) µ-calculus (further cases of this kind include the probabilistic µ-calculus with polynomial inequalities [22] and the unrestricted form of the alternating-time µ-calculus with disjunctive explicit strategies [16]).
The algorithm constructs and solves a parity game that characterises satisfiability of the input formula χ.In this game one player attempts to construct a tableau structure for χ while the opposing player attempts to refute the existence of such a structure.Modal steps in this tableau construction are treated by using instances of the one-step satisfiability problem for the logic at hand, thereby generalising traditional modal tableau rules.The winning condition of the game is encoded by a non-deterministic parity automaton A χ , reading infinite words that encode sequences of step-wise formula evaluations (so-called formula traces) within a coalgebra; such words encode branches in the constructed tableau structure.Conjunctions give rise to nondeterminism in this automaton, and the parity condition of the automaton is used to accept exactly those words that encode sequences of formula evaluations in which some least fixpoint is unfolded infinitely often.To use the language accepted by A χ as the winning condition in a parity game, we transform A χ to an equivalent deterministic parity automaton B χ .This automaton then is paired with the tableau construction to yield a parity game in which the existential player aims to show the existence of a tableau structure in which all branches are rejected by B χ , and that is built in such a way that modalities always are jointly one-step satisfiable.To ensure the latter property, the modal moves in the game invoke instances of the one-step satisfiability problem of the base logic.For more details on one-step satisfiability and the overall algorithm, see [18,22].

Corollary 2.2 ([22]
). Suppose that the one-step satisfiability problem is tractable.Then the satisfiability problem of the corresponding instance of the coalgebraic µ-calculus is in ExpTime.
Remark 2.3.As mentioned above, previous algorithms for the coalgebraic µ-calculus (also implemented in COOL 2) rely on complete sets of modal tableau rules, specifically on one-step cutfree complete sets of so-called onestep rules [46]; such rules (in their incarnation as tableau rules) have a premiss with exactly one layer of modal operators and a purely propositional conclusion.A typical example is the usual tableau rule for the modal logic K: 'To satisfy a 1 ∧ • • • ∧ a n ∧ ¬ a 0 , satisfy a 1 ∧ • • • ∧ a n ∧ ¬a 0 '.It has been shown that the existence of a tractable one-step cutfree complete set of one-step rules implies tractability of one-step satisfiability [30], i.e. the approach via one-step satisfiability is more general.
As indicated in the introduction, a tractable one-step cutfree complete set of one-step rules for graded modal logic has been claimed in the literature [46,49] but has since turned out to be incomplete.(A similar rule for Presburger modal logic [29] has also been shown to be in fact incomplete [30].)

Implementation
The previous version COOL [15] only implements fixpoint-free (coalgebraic) logics, such as standard modal logic, probabilistic modal logic, or coalition logic.The main novelty of the new version COOL 2, described here, is the addition of fixpoint constructs to the previously implemented logics, supporting alternation-free and aconjunctive fragments of the resulting µ-calculi, and implementing on-the-fly solving to allow early termination support for treating modal steps both by tableaux rules (when a suitable rule set exists), and by one-step satisfiability checking (in the remaining cases) In more detail, COOL 2 is written in OCaml and implements the satisfiability checking algorithm described in Section 2, treating modal steps by solving instances of the one-step satisfiability problem 4 .For logics where a suitable set of modal tableau rules is implemented, those are used for the treatment of modal steps, rather than relying on one-step satisfiability (unless the user explicitly chooses otherwise); in these cases, COOL 2 essentially implements the algorithm described in [30].The current implementation supports the alternationfree and the aconjunctive fragments of the standard µ-calculus (both serial and non-serial), the monotone µ-calculus [20], the alternating-time µ-calculus (i.e.coalition logic with fixpoint operators), and the graded µ-calculus.Tractable tableaux rules are available for all cases except for the graded µ-calculus, for which COOL 2 uses the one-step satisfiability algorithm to decide satisfiability.
In particular, COOL 2 is the only existing reasoner for the graded µ-calculus (as well as the only reasoner covering the alternating-time µ-calculus beyond ATL).
The concrete logic used can be selected via a command-line parameter setting up the data structures in COOL 2 accordingly before parsing and checking the syntax of the given formula χ.COOL 2 then builds the determinised automaton B χ , yielding the parity game described above in a step-wise manner, repeatedly adding nodes in expansion steps that explore the game.In the case of simpler alternation-free formulas, the Miyano-Hayashi method [32] is used to construct B χ , resulting in asymptotically smaller games with a Büchi winning condition; for the more involved aconjunctive formulas, the implementation uses the permutation method for determinisation of limit-deterministic parity automata [9,23].Nodes in the constructed game are marked as either unexpanded, undecided, unsatisfiable, or satisfiable.
Optional solving steps may take place at any point during the construction of B χ , depending on runtime parameters of COOL 2; these steps compute the winning regions of the partial game that has been constructed so far and accordingly mark nodes as satisfiable or unsatisfiable, if possible.The reasoner terminates as soon as the initial node is marked satisfiable or unsatisfiable.If this does not allow for early termination, the game eventually becomes fully explored, at which point a final (obligatory) solving step for the complete game is guaranteed to mark the initial node, thereby ensuring termination.
We detail the implementation of the two main procedures within COOL 2.

Implementation of Expansion
Steps.The propositional expansion steps in the game construction for nodes v are performed using the propositional satisfiability solver MiniSat [8] to compute a word that encodes consistent propositional formula manipulations for v. Afterwards, the successor of v in B χ under this word is computed and added to the game.When the one-step satisfiability based algorithm of COOL 2 is used, modal expansion steps for nodes v create new game nodes for each subset κ of the modalities that are to be jointly satisfied at v; this is done by computing the successor of v in B χ that is reached by manipulating each formula from κ.
When the tableau-based algorithm of COOL 2 is used, the modal expansion step for a node v instead computes all applications of a modal rule matching v and inserts, for each such rule application, and each conjunctive clause κ in the conclusion of the rule application, the new game node that is reached from v in B χ by manipulating the modalities that constitute κ.Intuitively, using tableau rules reduces the search space by only adding nodes found in the conclusion of some matching rule application.
Any node that is added by some expansion step is initially marked as undecided.Crucially, all expansion steps perform on-the-fly determinisation, that is, given a game node v and a word that encodes a sequence of formula manipulations, the newly added game node is computed using only the information stored in v.

Implementation of Solving
Steps.A single solving step computes the winning regions in the parity game that has been constructed up to this point, and marks nodes accordingly.The game solving is done using either the parity game solver PGSolver [10] or a native implementation provided by COOL 2 that solves the game by fixpoint iteration.
If the one-step satisfiability-based algorithm is used, an assigned modal node v is satisfiable if its modalities are jointly one-step satisfiable in those successors of v that are satisfiable themselves.An enumerative representation of the game thus contains existential moves to all subsets Π of subsets of modalities of v that are sufficiently large for one-step satisfaction of the modalities of v, followed by universal moves to nodes induced by any κ ∈ Π; the full game thus is of doubly-exponential size.This can be avoided by inlining the modal steps, thereby evading the intermediate nodes Π.The winning region can then be computed in single-exponential time by using COOL 2's native fixpoint iteration over a function that computes the two-tiered modal steps in one go.
Decision procedures for the one-step satisfiability problems in the relational and the graded case are implemented in COOL 2 along the lines of the algorithms described in [22,Example 6] (in the graded case, nondeterministic guessing is replaced with a recursive search procedure).
If the algorithm based on modal tableau rules is used, the treatment of modal steps follows the tableaux-based algorithm that is given in [3].States v are satisfiable if for all rule applications that match v, the conclusion of the application contains a conjunctive clause κ such that the node induced by κ is satisfiable.COOL 2 also allows the user to specify the desired frequency of optional game solving steps, including the options once and adaptive.With the option once, no intermediate solving takes place so that the game is fully constructed and solved just once, at the very end of the execution.With the option adaptive, intermediate solving takes places, but the frequency of solving reduces as the size of the constructed graph increases; this option implements on-the-fly solving and allows for finishing early in cases where a small model or refutation exists.

Evaluation
We conduct experiments in order to evaluate the performance of the various algorithms implemented in COOL in comparison with each other, as well as in comparison with other tools (where applicable). 5Full definitions of all formula series used in the evaluation as well as additional experimental results can be found in the extended version of this paper [18].
Experiments: In a first experiment, we compare COOL 2 with the established reasoner FaCT++, which supports the description logic SROIQ(D) (subsuming fixpoint-free graded modal logic), using the following series of formulas from Snell et al. [49].
Intuitively, the satisfiable Cardinality(n) formulas express that there are at least 2n successors and that both q and ¬q are satisfied in at most n successors, each; similarly the unsatisfiable CardinalityU(n) formulas state that there are at least 2n successors, and that q and ¬q hold in at most n and n − 1 successors, respectively; the latter statements imply that there are at most 2n−1 successors, yielding a contradiction.Going beyond next-step formulas, we continue by devising various complex series of graded µ-calculus formulas that involve (nested) fixpoints and express non-trivial properties of graded trees, automata and games.
-We obtain a series of unsatisfiable formulas by requiring the existence of an n + 1-branching tree in which p holds everywhere while at the same time requiring that this tree contains some state with n + 2 successors that satisfy p: -Next we turn our attention to graded formulas involving parity conditions.We devise a series of valid formulas expressing that graded parity automata can be transformed to graded Büchi automata accepting a superlanguage of the original automaton: Here, Parity(n, k) encodes parity acceptance with k priorities and grade n while Buechi(n, k) expresses Büchi acceptance by a nondeterministic automaton that eventually guesses the maximal priority that occurs infinitely often; the negated formula ¬ParityToBuechi(n, k) is unsafisfiable.
-Rabin conditions are given by families of pairs i j , f j j≤k of sets i j , f j of states, and express the constraint that there is some j ≤ k such that states from i j (infinite) are visited infinitely often and states from f j (finite) are visited only finitely often.We can express Rabin conditions with k pairs (and onestep property ψ), Büchi properties and satisfaction of single Rabin-pairs by formulas Rabin(k, ψ), Buechi(f, ψ) and RabinPair(i, f, ψ), respectively.Then we obtain valid formulas stating that the existence of an n + 1-branching tree that satisfies the Rabin condition on each path implies that there is a path satisfying a simpler Büchi condition or a single Rabin-pair, respectively: -Coming to games, we specify the winning regions in graded Büchi and Rabin games by formulas BuechiGame(f, n) and RabinGame(k, n), respectively; in such graded games, players are required to have at least n winning moves at their nodes in order to win.The following valid formulas then express that winning strategies in graded Rabin games with k pairs guarantee that some node from i 1 ∪ . . .∪ i k is visited infinitely often: In a final experiment on alternating-time formulas, we compare COOL 2 with TATL [6] on the ATL example formulas given in [6] as well as on additional formula series.For instance, we turn the formula 1 Gp∧¬ 2 F 1 Gp (written here using ATL syntax) from [6] into a series Nest(n) with increasing number of nested operators; formulas then alternatingly are satisfiable and unsatisfiable:

Results:
We conducted all experiments on a virtual machine with four 2, 3GHz vCPUs processors and 8GB of RAM.We compare with a 64-bit binary of FaCT++ v1.6.5 and with TATL.We compute all results with a timeout of 60 seconds and average the results over multiple executions.For the execution and measurement we use hyperfine6 .Below, 'COOL' and 'COOL on-the-fly' refer to invoking COOL 2 with solving rate once and adaptive, respectively.Results for the Cardinality and CardinalityU series are shown in Figure 1 and Figure 2, respectively.From n = 10 and n = 8 onwards, COOL 2 outperforms FaCT++ considerably.An explanation for this could be that FaCT++ appears to treat multiplicities in a naïve way while COOL 2 employs the more efficient one-step satisfiability algorithm.Results for the unsatisfiable tree property are shown in Figure 3.As these formulas contain fixpoint operators, a comparison with FaCT++ is not possible.While COOL 2 is generally capable of handling quite large branching factors, this experiment showcases the drawbacks of on-the-fly solving in the case that a formula cannot be decided early so that repeated attempts of solving the game early lead to overhead computations.
Runtimes for COOL 2 (using on-the-fly solving) on the unsatisfiable series of parity formulas ¬ParityToBuechi(n, k) are shown in Figure 4.The results indicate that increasing the number of priorities k has a much stronger effect on the runtime than increasing multiplicities n in the modalities.This is in accordance with expectations as increasing k leads to much larger determinized automata and resulting satisfiabilty games, while increasing n only complicates the modal steps in the game while leaving the global game structure unchanged.
Results for the Rabin families of formulas are given in the table below, with † indicating a timeout of 60 seconds.COOL 2 is able to handle reasonably large formulas describing Rabin properties of automata and games, with the series for n = 1 expressing properties of standard automata (solved using tableau rules), and the series with n = 2 properties of graded automata with multiplicity 2 (solved using one-step satisfiability).
In accordance with previous experiments on random ATL formulas of larger sizes in [24], COOL 2 generally outperforms TATL by a large margin, starting from formulas containing at least five modalities or involving nesting of temporal operators; this trend is confirmed by Figure 5   In summary, COOL 2 shows promising performance in comparison to TATL and FaCT++, as well as for practical applicability.On graded formulas without fixpoints, COOL 2 scales much better than FaCT++ with regard to increasing multiplicities.In the presence of fixpoints, COOL 2 still scales well and can handle multiplicities that should be sufficient for practical use.The formula series ¬ParityToBuechi appears to show the limits of COOL 2 with the current implementation of graded one-step satisfiability checking.Nonetheless, our results indicate that COOL 2 is capable of automatically proving or refuting involved properties of (graded) ω-automata and games in reasonable time.

Conclusion
We have described and evaluated the current version COOL 2 of the CO algebraic O ntology Logic reasoner (COOL).Future development will include the implementation of additional instance logics, such as the probabilistic and graded µ-calculus with linear inequalities, as well as support for the full coalgebraic µ-calculus via on-the-fly determinisation of unrestricted Büchi automata, using the Safra-Piterman construction.

A.1 Details for Remark 2.3
The following set of modal tableau rules (one-step rules) has been claimed to be one-step cutfree complete [47,49]: The conclusion of the rule denotes a propositional formula (in conjunctive normal form) representing the Boolean function obtained by reading the numerical inequality as a constraint on Boolean variables p i , q i , with true interpreted as 1 and false as 0. In the relevant instance of the one-step logic, models consist of a set X, a valuation τ of the variables p i , q i as subsets of X, and a finite multiset µ over X.Given these data, the conclusion φ of (an instance of) the rule, a purely propositional formulas, is interpreted as a subset φ τ of X, using the Boolean algebra structure of the powerset.The premiss is evaluated w.r.t.satisfaction by µ, with the expected clauses for the propositional operators, and with n a, for a variable a, being satisfied if µ(τ (a)) > n, where µ(A) = x∈A µ(x) for A ⊆ X.The premiss is satisfiable over τ if φ τ = ∅, and the conclusion is satisfiable over τ if it is satisfied by some multiset µ over X in the sense just defined.
In the one-step logic, the rules are then applied to conjunctions of modal literals, i.e. formulas of the form either n a or ¬ n a, for a a variable, requiring an exact match of the rule premiss with a subset of the conjuncts (thus incorporating weakening into the rule application).Such formulas are called one-step clauses.The rules are read as tableau rules, i.e. to establish that the premiss is satisfiable, one needs to establish that the conclusions of all matching rule applications are satisfiable.
The rules are easily seen to be one-step sound, i.e. if for any rule instance matching a given one-step clause, the conclusion is propositionally unsatisfiable over a valuation τ , then the premiss is also unsatisfiable over τ .The rule set is one-step complete if, given a valuation τ in the powerset of X and a one-step clause χ, whenever all rule matches to χ have satisfiable conclusions, then χ is satisfiable.
We show that the latter property fails.To this end, consider the set X = {a, b, c, d}, propositional variables p A for all A ⊆ X, the valuation τ given by τ (p A ) = A, and the one-step clause χ consisting of the positive literals 2 p A for all A ⊆ X such that |A| = 2, and the negative literal ¬ 6 p X .First, note that χ is clearly unsatisfiable over τ : A multiset µ over X would have to satisfy µ(A) ≥ 3 for all A ⊆ X such that |A| = 2, so at least three of the four elements of X need to have multiplicity at least 2 under µ; moreover, if any element has multiplicity 0, then all others need to have multiplicity at least 3. Consequently, the total weight µ(X) is at least 7, so µ does not satisfy the negative literal ¬ 6 p X .On the other hand, none of the rule instances matching χ have unsatisfiable conclusions.The easiest way to see this is to note that the rules are, with fairly evident adaptations, still sound for a real-valued relaxation of the logic where µ may assume non-negative real values (of course, n a then means that µ(τ (a)) ≥ n + 1); under this semantics, however, χ is satisfiable over τ by taking µ(x) = 3 2 for all x ∈ X.

A.2 Additional Details for Section 2
We give additional details on syntactic notions for the coalgebraic µ-calculus, and on predicate liftings.Furthermore, we sketch the satisfiability checking algorithm, first introduced in [22]; here we give a presentation of the algorithm in terms of automata and games, tailored towards our implementation in COOL 2.
Fixpoint operators bind their variable, yielding notions of bound and free fixpoint variables; a formula then is closed if it does not contain any free variables.A formula is clean if every fixpoint variable is bound at most once in it.We restrict attention to closed and clean formulas (being aware of issues with formula size measures [28]).Variables X that are bound by µX then are µvariables, and variables bound by νX are ν-variables.A variable X is active in a formula ψ if X has a free occurrence in the formula that is obtained from ψ by exhaustively replacing free occurrences of fixpoint variables by their binding fixpoint formulas.E.g. for the formula χ = µX.(p ∨ ♦X) we have θ(X) = χ, and the closure of χ is a graph with nodes χ, p ∨ ♦χ, p, ♦χ and edges χ → p ∨ ♦χ, p ∨ ♦χ → p, p ∨ ♦χ → ♦χ, and ♦χ → χ.When we refer to the closure, we typically mean just the set of nodes of this closure graph.
We give more details on Example 2.1: A coalgebraic modelling of the graded µ-calculus [5] is obtained by using the functor F = G that maps a set C to the set GC = {θ : C → N | θ has finite support} of finite multisets over C, the similarity type The concurrent game functor F maps a set C to the set F C = {(S 1 , . . ., S n , f ) | ∅ = S i , f : i≤n S i → C}, where the S i are viewed as sets of available moves for agent i, and f an outcome function that evaluates joint moves of all agents.F -Coalgebras are concurrent game frames [38] Additional details regarding the satisfiability checking algorithm sketched in the main paper are as follows.
Depending on the syntactic structure of the input formula χ, it may be possible to employ simpler determinisation procedures for the construction of the automaton B χ , resulting in asymptotically smaller games.Currently known bounds for particular fragments of the coalgebraic µ-calculus are summarised in the following table, where n denotes the closure size of the target formula, k denotes its alternation-depth, and where NCBA and LDBA stand for nondeterministic co-Büchi automaton and limit-deterministic Büchi automaton, respectively.
We sketch the construction of the automata A χ and B χ to describe the coalgebraic satisfiability game.Recall that the accepted language L(A) of a parity automaton A = (Q, Σ, δ, v 0 , Ω) with priority function Ω : Q → N consists of those infinite words over Σ on which there is an accepting run of A, where a run is accepting if and only if the maximal priority (according to Ω) that is visited infinitely often is even.The logical connectives are captured by the alphabet of the automaton.Propositional connectives are treated by letters from a set Σ p , also encoding choices of disjuncts using letters of the shape (ψ 1 ∨ ψ 2 , b) where ψ 1 ∨ ψ 2 ∈ cl and b ∈ {1, 2}.For modalities, the automaton needs to work on the conjunctive satisfiability of a set of operators, so we put The alphabet of the automata is Σ = Σ p ∪ Σ s .Furthermore, we let choices = {w ∈ Σ * p | |w| ≤ n 2 } denote the set of propositional words of length at most n 2 .The nondeterministic parity automaton A χ = (cl, Σ, ∆, χ, Ω ′ ) reads sequences w ∈ Σ ω of formula manipulations and traces formulas through the closure.For the transition function we have, for example The priority function Ω ′ is defined by the alternation of least and greatest fixpoints in the formula, and A χ accepts words encoding infinite traversals through formulas where the outermost unfolded fixpoint is a least fixpoint.We now consider the deterministic parity automaton B χ = (D χ , Σ, δ, v 0 , Ω) that is obtained from A χ by co-determinisation (L(A χ ) = L(B χ )), noting that B χ is a parity automaton with at most 2nk priorities by the results from [40].Then B χ accepts infinite traversals through formulas for which there is no formulas trace (run of A χ ) on which the outermost fixpoint that is unfolded infinitely often is a least fixpoint.
The set D χ consists of macro-states, that is, data structures organising elements of cl in way that depends on the concrete determinisation construction that is used (see the table above), e.g., as Safra-trees [40] or permutations [9].The labelling function l : D χ → P(cl) assigns to each macro-state v ∈ D χ its label l(v), i.e. the set of formulas that occur in v.We denote by cores and states the sets of all states in D χ that have an unsaturated or a saturated label, respectively; here, a set of formulas is saturated if all its elements are either modalised formulas ♥ψ or ⊤, and unsaturated otherwise.We extend δ to words over Σ in the usual way and to subsets of Σ s by putting δ(v, A) = {δ(v, a) | a ∈ A} for v ∈ D χ , A ⊆ Σ s .
Definition A.1 (One-step satisfiability problem [33,43,45]).Let V be a finite set, Λ(V ) = {♥a | a ∈ V, ♥ ∈ Λ}, and let Θ ⊆ P(V ).We interpret a ∈ V and γ ⊆ Λ(V ) over Θ by We refer to the data (γ, Θ) as a one-step pair (over V ) and say that (γ, Θ) is satisfiable (over Example A.2. (1) For the relational modal µ-calculus (Example 2.1.1.),where Λ = {♦, }, the one-step satisfiability problem is to decide, for a given one-step pair (γ, Θ) over V , whether there is A ∈ γ Θ 1 , that is, a subset A ∈ PΘ such that for each ♦a ∈ γ, there is u ∈ A such that a ∈ u, and for each b ∈ γ and each u ∈ A, b ∈ u.Equivalently, one needs to check that for each ♦a ∈ γ there is u ∈ Θ such that a ∈ u and moreover b ∈ u for all b ∈ γ.
(2) For the graded µ-calculus (Example 2.1.2.), the one-step satisfiability problem is to decide, for a one-step pair (γ, Θ), whether there is a multiset β ∈ BΘ such that u∈Θ|a∈u β(u) > m for each m a ∈ γ and u∈Θ|a / ∈u β(u) ≤ m for each [m]a ∈ γ.This problem can be solved via a nondeterministic algorithm that goes through all u ∈ Θ, guessing multiplicities β(u) ∈ {0, . . ., m + 1} where m is the greatest index of any diamond modality m that occurs in γ.This multiplicity is used to update |V | counters that keep track of the total measure β([[a]] Θ 0 ) for a ∈ V and then forgotten.Once all multiplicities have been guessed, the algorithm verifies that β ∈ [[γ]] Θ 1 , using only the final counter values [27, Lemma 1].
Recall (e.g.[17]) that parity games are history-free determined infiniteduration two-player games given by data G = (V ∃ , V ∀ , E, v 0 , Ω), consisting of disjoint sets V ∃ and V ∀ of nodes belonging to the existential and universal player, respectively, a set E ⊆ V × V of edges (where V = V ∃ ∪ V ∀ ), an initial node v 0 , and a priority function Ω : E → N that assigns priorities Ω(e) to edges e ∈ E. A play is a finite or infinite sequence π = v 0 , v 1 , . . . of nodes such that (v i , v i+1 ) ∈ E whenever applicable.Finite plays are required to end in nodes without outgoing moves, and then are won by the player that does not own the last node in the play; infinite plays π ∈ V ω are won by the existential player if and only if the maximal priority visited in π is even (and by the universal player otherwise).The existential player wins the game G if she has a strategy to move at her nodes such that she wins every play that is compatible with this strategy; otherwise, the universal player wins G. Now we are ready to characterise satisfiability in the coalgebraic µ-calculus by parity games.Recall that B χ = (D χ , Σ, δ, v 0 , Ω) is a deterministic parity automaton with 2nk priorities.Definition A.3 (Satisfiability games).The satisfiability game G χ = (V ∃ , V ∀ , E, v 0 , Ω) for χ is a parity game with sets of nodes V ∃ = D χ and V ∀ = D χ × P(Σ s ).The other components of the game are defined by the following table, where Ω(v, τ ) denotes the maximal Ω-priority that is visited by the partial run of B χ that leads from v to δ(v, τ ) by reading τ letter by letter.
Thus the existential player attempts to show the existence of a specific subautomaton of B χ , intuitively by selecting, at each core node v of B χ , some nonmodal word τ that saturates v, leading to a state node.For modal steps at state nodes v, the existential player has to provide a set Ξ of letters (selecting a set of modal out-edges of v) such that the label of v is one-step satisfiable in the labels of the selected successors of v in B χ .The universal player then can challenge any letter κ ∈ Ξ and perform the corresponding modal step by moving to δ(v, κ).The existential player has to make her choices in such a way that for every compound word that results from the choices, the corresponding run of B χ is accepting (that is, does not contain infinite deferrals of least fixpoints), and never visits nodes with ⊥ in the label.Given a set G ⊆ D χ , we let win ∃ G and win ∀ G denote the winning regions for the existential and the universal player, respectively, in the partial game G χ | G that is obtained from G χ by removing all nodes that are not contained in G. Nodes in this partial game G χ | G may be undetermined so that we in general do not have win ∃ G ∪ win ∀ G = G.However, we do have win ∃ Dχ ∪ win ∀ Dχ = D χ .While the number of nodes in G χ | G is doubly exponential in n (Σ s is singly exponential, so P(Σ s ) is doubly exponential), we use a fixpoint computation over (subsets of) D χ [22] to compute the sets win ∃ G and win ∀ G in singly exponential time.
Definition A.4 (Game solving).Given a set G ⊆ D χ , we define the one-step solving function f where η i = µ for odd i, η i = ν for even i, where ν = µ and µ = ν, and where We note that nodes in the partial game G χ | G may be undetermined so that we in general do not have E G ∪ A G = G.However, we do have E Dχ ∪ A Dχ = D χ .Theorem A.5 ([22]).Existential player wins G χ if and only if χ is satisfiable, and Algorithm 1 computes the winning regions in G χ on-the-fly.
(The proof given in [22] elides the game formulation and instead shows directly that the fixpoint captures satisfiability; however, equivalence of the game and the fixpoint computation is straightforward.)

A.3 Additional Details for Section 4
The omitted formulas for the parity conditions in Section 4 are defined as follows.
Parity(n, k) = µX k .νX k−1 . . . ..νX2.µX1.These formulas express that 'there is an n + 1-branching tree starting here in which each path satisfies the parity condition encoded by the priorities p i '.This property implies that 'there is an n + 1-branching tree such that for every path there is an even priority p i that occurs infinitely often on the path and priorities larger than p i occur only finitely often on the path' :  The results of the comparison between COOL 2 and TATL on these formulas are shown in Figure 6.Furthermore, we turn formula 36, that is, the formula ¬ 1 p U ¬ 2 q U r into a formula series with increasing number of nested temporal operators, defined COOL on-the-fly TATL Fig. 6.Runtimes on selected ATL formulas [6] inductively by ψ(0) = ¬ 2 q U r ψ(i + 1) = ¬ (i mod 2) + 1 p i mod 2 U ψ(i) (i > 0) The according experiment results depicted in Figure 7 show that COOL 2 outperforms TATL on this formula series by a large margin, with runtime for COOL 2 remaining almost constant as n increases while TATL quickly runs into timeouts of 60 seconds.This presumably is due to the nesting of temporal operators.

Fig. 5 .
Fig. 5. Runtimes for the ATL series Nest(n) . Coalitions D induce sets S D = i∈D S i and S D = i∈N \D S i , and given s D ∈ S D and s D ∈ S D , the pair (s D , s D ) represents an element of i≤n S i .We use the modal similarity type Λ = { D , [D] | D ⊆ N } and the predicate liftings

Fig. 7 .
Fig. 7. Runtimes for ψ(n) which shows the stepped execution times for the series Nest that alternates between being satisfiable and unsatisfiable This function intuitively encodes one step in the partial game G χ | G that is obtained from G χ by removing all nodes that are not contained in G.The winning regions in G χ | G then can be characterised by the nested fixpoints E G and A G , defined by E