**Conclusions and Discussions, Acknowledgment and References**

CGAAL: A Distributed On-The-Fly ATL Model Checker: The Definitions You Should Know About by@heuristicsearch

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by Aiding in the focused exploration of potential solutions.April 26th, 2024

This paper is available on arxiv under CC 4.0 license.

**Authors:**

(1) Falke B. Ø. Carlsen, Department of Computer Science, Aalborg University, Denmark & [email protected];

(2) Lars Bo P. Frydenskov, Department of Computer Science, Aalborg University, Denmark & [email protected];

(3) Nicolaj Ø. Jensen, Department of Computer Science, Aalborg University, Denmark & [email protected];

(4) Jener Rasmussen, [email protected];

(5) Mathias M. Sørensen, [email protected];

(6) Asger G. Weirsøe, [email protected];

(7) Mathias C. Jensen, Department of Computer Science, Aalborg University, Denmark & [email protected];

(8) Kim G. Larsen, Department of Computer Science, Aalborg University, Denmark & [email protected].

We recall the definitions of concurrent games and alternating-time temporal logic.

A computation starting in the state *q* is called a *q*-computation.

**Alternating-time Temporal Logic** The alternating-time temporal logic (ATL) [1] is defined with respect to a finite set Π of propositions and a finite set Σ = {1,...,k} of players. An ATL formula is given by the abstract syntax:

where p ∈ Π is a proposition and A ⊆ Σ is a set of players. The operators ⟪⋅⟫ and J⋅K are path quantifiers. We will refer to them as "enforce" and "despite", respectively. The ◯ ("next") and U ("until") are temporal operators. Additional temporal operators like ♢ ("eventually") and □ ("invariant") are derived as usual.

Given a state *q* of a game structure S, we write *q* ⊧ φ to indicate that the state *q* satisfies the property described by φ. The satisfactory relation ⊧ is defined inductively:

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