**1. Introduction**

For the specification of ongoing behaviours of reactive systems, the use of temporal logics has become one of the significant developments in formal reasoning [1–3]. However, interpreted over Kripke structures, traditional temporal logics can only quantify the computations of the closed systems in a universal/existential manner. In order to reason in multi-agent systems, we need to specify the ongoing strategic behaviours [4].

Since Alur and Henzinger [5] proposed alternating-time temporal logic (ATL/ATL∗) in 2002, strategy specification and verification has been an active research area in multiagent systems, artificial intelligence, and game theory. In recent years, there have been many extensions or variants of strategic logics proposed to reason about coalitional strategic abilities. For instance, in [6], Chatterjee et al. proposed strategy logic, which treats strategies as explicit first-order objects in turn-based games with only two agents; Mogavero et al. extended this logic with explicit strategy quantifications and agent bindings in multi-agent concurrent systems [7]; in order to reason about uniqueness of Nash Equilibria, Aminof et al. introduced a graded strategic logic [8]; in [9], Bozzelli et al. considered strategic reasoning with linear past in alternating-time temporal logic; and in [10], Belardinelli et al. studied strategic reasoning with knowledge. These logics are interpreted over concurrent game structures, in which agents act concurrently and instantaneously. Each agent acts independently and interacts with other agents. Formulas of these logics are used to specify an individual's or a group's strategic abilities.

Representation and Reasoning about Strategic Abilities with *ω*-Regular Properties. *Mathematics* **2021**, *9*, 3052. https://doi.org/10.3390/math9233052

**Citation:** Xiong, L.; Guo, S.

Academic Editor: R ˘azvan Diaconescu

Received: 22 September 2021 Accepted: 24 November 2021 Published: 27 November 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

In ATL/ATL∗, strategic abilities for coalition *A* (i.e., a set of agents) are expressed as *Aψ*, representing that coalition *A* has a group strategy to make sure that goal *ψ* holds, no matter which strategies are chosen by other agents outside of *A*, here *ψ* can be any temporal formula. A much more expressive strategic logic is Strategy Logic (SL) [6,7], which is a multi-agent extension of linear-time temporal logic (LTL) [11] with the concepts of agent bindings and strategy quantification. In SL, we can explicitly reason about the agent's strategy itself, allow different agents to share the same strategy, and also represent the existence of deterministic multi-player Nash equilibria.

However, on one hand, existing strategic logics are mainly based on the classical temporal logics. For instance, the underlying logics of ATL, ATL∗, alternating-time mucalculus (AMC) [5], and SL are temporal logic computational tree logic CTL [12], CTL∗ [3], *μ*-calculus [13], and LTL, respectively. However, they cannot express general *ω*-regular properties, such as "property *p* holds in any even steps in an infinite sequence, and holds in odd steps or not" [14].

On the other hand, the need of a declarative and convenient temporal logic, which can express any general *ω*-regular expression, is considered compelling from a practical viewpoint in industry [15]. In some papers, e.g., [16], the authors introduce regular expressions or automaton directly into LTL to express *ω*-regular properties. However, regular expressions or automaton are all too low level as a formalism for expressing temporal specifications. In 2011, Moshe Y. Vardi proposes a novel logic, named linear dynamic logic (LDL) [17], which merges LTL with regular expression in a very natural way and adopts exactly the syntax of propositional dynamic logic (PDL) [18]. LDL has three advantages:


In order to express any *ω*-regular properties in strategic logic, in [22], Liu et al. propose a logic JAADL to specify joint abilities of coalitions, which combines alternating-time temporal logic with LDL. However, in JAADL, the authors consider a very complex semantics and study the model checking complexity with imperfect recall for JAADL.

Similarly, to remedy the inability to express any general *ω*-regular temporal goal in strategic abilities in SL, we propose a novel strategic logic, called LDL-based Strategy Logic, abbreviated as LDL-SL. It can explicitly represent and reason about strategies and specify expressive strategic abilities for coalitions about more representative temporal goals, which can be general *ω*-regular properties. By combining LDL and SL, LDL-SL becomes a natural and intuitive strategic logic to specify more expressive properties. (In [23], the authors propose a strategy logic based on LDL interpreted over interpreted systems with bounded private actions.).

In this paper, we show that LDL-SL is much more expressive than SL and LDL and prove that the model checking complexity of LDL-SL is nonelementary-hard [24]. Moreover, we study fragments of LDL-SL and their model-checking complexities, and we define three types of strategic logics: ATL-like, one-goal, and star-free. The former two, which are fragments for LDL-SL, have the same expressivity as those based on LTL or CTL∗, and the model-checking problems are also the same. As for the last, firstly, we formally define the star-free LDL logic and prove it is equivalent with LTL. By this, we know that the corresponding star-free strategic logics are equivalent with those based on LTL/CTL∗. Furthermore, the model-checking problems of these new logics, based on LDL, are the same as those based on LTL/CTL∗. Furthermore, we show that the model-checking problem complexities of these logics are either 2EXPTIME-complete or nonelementary-hard.

Therefore, in any case, LDL can be viewed as a good and natural underlying temporal logic of strategic logics.

The paper is organized as follows. Section 2 introduces LDL, and its classical temporal logic fragments and then introduces the syntax and semantics of strategic logics. Section 3 indicates that LTL is equivalent with star-free fragment of LDL. In the next section, we propose the LDL-based strategy logic (LDL-SL) and give fragments of LDL-SL. Furthermore, we present the relations for expressivity among strategic logics. Moreover, the model checking problems for these new proposed strategic logics are considered. Finally, we present conclusions and future work.
